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This book is based on a thesis by Kundan Misra which satisfied the requirements for the degree of Master of Science (Research), University of New South Wales (UNSW). The project was conducted under the supervision of Professor James Franklin. The research started in 2008, the thesis was submitted for examination in August 2011, changes requested by examiners were completed in September 2012, and in the same month the university decided to award the degree. The thesis available at this page has minor post-submission amendments and improvements. The submitted version is at the UNSW library here.
© Copyright 2012 Licenced under the Creative Commons Attribution Licence For licence details, see www.creativecommons.org.au |
The author may be contacted at kundan at austi dot org or kundanmisra at yahoo dot com
Abstract
Modernity began in Leibniz’s lifetime, arguably, and due to the efforts of a group of philosopher-scientists of which Leibniz was one of the most significant active contributors. Leibniz invented machines and developed the calculus. He was a force for peace, and industrial and cultural development through his work as a diplomat and correspondence with leaders across Europe, and in Russia and China. With Leibniz, science became a means for improving human living conditions. For Leibniz, science must begin with the “God’s eye view” and begin with an understanding of how the Creator would have designed the universe. Accordingly, Leibniz advocated the a priori method of scientific discovery, including the use of intellectual constructions or artifices. He defended the usefulness and success of these methods against detractors. While cognizant of Baconian empiricism, Leibniz found that an unbalanced emphasis on experiment left the investigator short of conclusions on efficient causes. Leibniz worked outside, but complemented, the current of formal reasoning and empiricism which was developing in scientific circles during his lifetime. He supported the development of methods for calculation and demanded precise reasoning, while arguing that it was folly to omit the Neoplatonic orientation from science. Indeed, without Neoplatonism there would be no modernity. Leibniz’s Neoplatonic course complemented his work with machines. Leibniz crystallised the Neoplatonic orientation as a pragmatic humanist agenda, and merged it with national imperatives for developing science. Leibniz’s policy orientation is aligned with the Hermetic conception of Man as magus, who ultimately can control even the stars. The industrial-scientific age which followed Leibniz is a testament to the success of his life’s work.
Chapter 2: Neoplatonism and the awakening of science
Chapter 3: Ushering
in modernity
Chapter 5: The
Neoplatonist and Empiricist schools
Chapter 6: Discovery
and deduction
Chapter 7: Science
by thought, and reality and substance.
Chapter 8: The
ongoing role of thought and the ongoing creation of the Best
Chapter 9: A priorism in science
Chapter 10:
Conclusion – Leibniz’s humanist and Neoplatonic agenda
Appendix 1: To the greater glory of God
Appendix 2: A proof
from basics
Chapter 2:
Neoplatonism and the awakening of science
The first Neoplatonists were “new Platonists”
Hermeticum and magic in Europe
The other kind of Neoplatonism, and operations
Chapter 3: Ushering
in modernity
Leibniz’s relationship to modernity
Laws that function without divine intervention
Modernity and discoveries achieved by reason
Broad-based influence of reason on culture
Conflicting ideas on the nature of Reason
Cusa’s ideas on mind gain traction in science
Vis viva or effect-producing force
Machines are indispensable to the concept of modernity
Political influences on and of Leibniz
Seeking to understand Creation as a whole
Geometry as a tool to understand Creation as a whole
Understanding Creation as a whole, and improving upon it
Philosopher scientists and French nation-building
An era of pro-nuclear
advocacy
Participating with God in Creation
Minds can successfully search for truth
Sense perception and Empiricism
Existence and attainability of truth: Rationalism and Neoplatonism, vs
Empiricism
Religious or moral truth vs scientific truth
Neoplatonism, Empiricism and Atheism
Chapter 5: The Neoplatonist
and Empiricist schools
Metaphysical calculations and a method of Analysis for concepts in physics
Invaluable medieval speculations
Leibniz saw the need for more powerful methods
Reason gives value to observations
Kepler, Neoplatonism and precision
The Empiricist school and the influence of Paolo Sarpi
Leibniz comments on the experimental program of the (British) Royal Society
Newton’s prescribed method of discovery
Evolution of Newton’s method of discovery
Newton against hypothesis and towards mathematicization
Kepler and Leibniz in Newton’s method
Consequences of mandating empirico-logic
Empirico-logic in other domains
Reasoning other than empirico-logical
Chapter 6: Discovery
and deduction
Discovery is often metaphysics
Archimedes had the same problem
All things and no things – at the same time
Does the infinite exist? It doesn’t really matter.
Do Platonic ideas exist? It doesn’t really matter.
Science may use indemonstrables
Global architecture determines local mechanics
Chapter 7: Science
by thought, and reality and substance.
Dream vs reality, and substance
vs phenomena
Ideas and “the Best” are outside all that exists
The apparent governance over actual things by ideas
The relationship between ideas and actual things
Inexorable necessity created by the requirement of “the Best”
Simple monads and compound monads
Passive vs active, or body vs soul
Immaterial realm is in harmony with material realm
Human souls differ from souls of beasts
Monads are the same as one another
Chapter 8: The ongoing
role of thought and the ongoing creation of the Best
Choices God makes, continuously
First kind of choice: God mediates between monads
Second kind of choice: God chooses the best possible from what is possible
Unanswered question re first kind of choice
Consideration re first kind of choice: God’s choices vs intelligent monads’
choices
Question re second kind of choice
Answer to the question re second kind of choice: what God does from
moment-to-moment
Creating the universe anew in each moment
Is it “Hello Humphrey Appleby” or, Does God control everything while
appearing not to?
Is everything planned in advance?
Is a priorism better than
empirics?
For humans, ideas and empirics are interdependent
Scientific comprehensibility of God’s plan
Intention and the Scholastic Leibniz project
God’s thoughts are made real by his will
Thoughts that soul monads have
Humankind as a subject of physical science
“Ideas” (in the strict monadological sense) about humankind
Foundations of a programme for discovery
Chapter 9: A priorism in science
Example 1: The infinitesimal and the calculus
Example 2: The Fifth Postulate and Non-Euclidean geometry
Example 3: Elliptical orbits in a heliocentric solar system
Alternative views of the Creator’s mind
Chapter 10:
Conclusion – Leibniz’s humanist and Neoplatonic agenda
Broadening the field of debate for Platonism
Considering the big questions first
A new metaphysics as a blueprint for the good in human society
Flexibility in scientific method, and the ultimate journey
Appendix 1: To the greater glory of God
Appendix 2: A proof
from basics
This publication
is based on my MSc (Research) thesis awarded by the University of New South
Wales through the School of Mathematics and Statistics in 2012, supervised by
Professor Jim Franklin. There are minor changes from the original thesis,
mostly in the last page and section of the Conclusion.
The project
began with the question of how Leibniz’s metaphysics influenced his
mathematical work. We found that Leibniz was just one of a number of thinkers
who did science by beginning with plausible and quasi-theological first
principles regarding creation as a whole. Scientific questions then act as pieces
in a jigsaw puzzle in a quest for understanding creation as a whole and the
role of humanity in it. Mathematics is one tool available to humans not merely
to carry out calculations but to seek truth. Leibniz was one of a line of
thinkers who proceeded in this way. Significant forebears of Leibniz include
Nicolaus of Cusa and Johannes Kepler.
This
orientation did not arise spontaneously but has ancient roots. In particular,
there are many parallels between Leibniz’s philosophy and the Hermetic corpus.
Through the analysis of Frances Yates, it is understood that the putative
writings of Hermes Trismegistus were central to the Neoplatonic revival and
associated Humanist Renaissance. It seems that Leibniz developed and
systematised proto-science of figures such as John Dee, referred to by
Cornelius Agrippa as “real artificial magic”. The origin and potential of such
so-called “magic” which included proto-science and early engineering was
appreciated by Giordano Bruno and the circles he influenced. Further understanding
the debt that Leibnizian thought owes to Bruno and Egyptian “magic” is a
promising direction for further work.
During the
research, the dichotomy between thought and sense perception arose many times.
Of course, this has been discussed since Hermes and was addressed in detail in Plato’s
The Republic and by nearly every
philosopher in Classical Athens and since. Generally speaking, the Platonists
are oriented towards a priori
discovery – using the mind to construct hypotheses – whose outcomes can be
tested in controlled experiment. In the construction of hypotheses,
mathematical preconceptions of the universe play a central role because they
imply structure which becomes the foundation of predictive hypotheses which can
be tested. General sense perception is distrusted as not just unreliable but
misleading. As an inventor and pragmatist himself, Leibniz saw the necessity
for experiment and described how he would catalogue all desired experiments
when trying to understand a particular issue in physics. The distinction is
that Leibniz did not expect answers to come from experiment. Rather,
experiments were merely an aid to the a
priori thought process.
We found
exciting open questions which if addressed would present the opportunity for
leaps in science. One example is a serious conception of “the Best” in the
context of Leibniz’s best of all possible universes doctrine. Another example is
further detail on the pre-established harmony, wherein God intervenes from
moment-to-moment to ensure the best possible correspondence between the
thoughts of incorporeal substance (“mind”) and phenomena exhibited by corporeal
substance (“body”). Another is the question of Man as a physical scientific
force in the universe. Another is that which has received much attention but
little progress since Leibniz himself, which is the formulation of a Universal
Characteristic, or symbolic calculus whereby all kinds of problem can be solved
in a systematic way.
Copenhaver:
Copenhaver, B.P. Hermetica Cambridge
University Press 1992
Klemm: Klemm,
F. (Singer, D. W. trans.) A History of
Western Technology Allen and Unwin, London 1959
Loemker: Loemker,
L. E. Gottfried Wilhelm Leibniz:
Philosophical Papers and Letters 2nd ed. D. Reidel, Dordrecht Holland 1969
Monadology: Latta, R. (trans.), Leibniz, G.W. Monadology Oxford University Press 1898
edition
Theodicy: Huggard, E.M. (trans.), Leibniz, G.W. Theodicy §Project Gutenberg edition 2005
accessed at www.gutenberg.com
Wiener:
Wiener, P.P. (ed.) G. W. Leibniz
Selections New York: C. Scribner, 1951
Wootton, D. Paolo Sarpi: Between Renaissance and
Enlightenment Cambridge University Press, 2002
Yates 1964: Yates, F. Giordano
Bruno and the Hermetic Tradition Routledge and Kegan Paul Ltd London 1964
Yates 2009: Yates, F. Giordano
Bruno and the Hermetic Tradition Routledge Classics, digital reprint 2009
In
fact, … final causes may be introduced with great fruitfulness even into the
special problems of physics, not merely to increase our admiration for the most
beautiful works of the supreme Author, but also to help make predictions by
means of them which would not be as apparent, except perhaps hypothetically,
through the use of efficient causes. Philosophers have in the past perhaps not
sufficiently observed this advantage of final causes.
It
must be maintained in general that all existent facts can be explained in two
ways – through a kingdom of power or efficient causes and through a kingdom of
wisdom or final causes; that God regulates bodies as machines in an
architectural manner according to laws of magnitude or of mathematics but does
so for the benefit of souls and that he rules over souls, on the other hand,
which are capable of wisdom, as over citizens and members of the same society
with himself, in the manner of a prince or indeed of a father, ruling to his
own glory according to the laws of goodness or of morality. Thus, these two
kingdoms everywhere permeate each other, yet their laws are never confused and
never disturbed, so that the maximum in the kingdom of power, and the best in
the kingdom of wisdom, take place together.[1]
G. W. Leibniz, Specimen
Dynamicum 1695
We introduce some terminology
and distinctions that are peculiar to the philosophy of science, and which we
use throughout the thesis. In the following list of terms, we group associated
terms rather than following a strict alphabetical order.
In the thesis, we will
capitalise terms relating to new concepts introduced by Leibniz and others
which have not yet developed into methods in general use. Examples are Analysis
as Leibniz uses the term, versus the more general “analysis” in day-to-day usage.
Another example is Metaphysics as Leibniz understands it, versus mathematics
which Leibniz generally used in the same sense as everyone else. Another
example is “the Best” where best refers to the actuality of structures in the
universe as designed by God as “the best possible universe”. In any case, we
hope that the reader does not set too much store by capitalisation.
A posteriori One is said to formulate a
hypothesis or to gain understanding a posteriori if one does so after and
pursuant to or as a result of actual experience or observation of physical
events or empirical evidence.
A priori One is said to
understand something a priori or prior to experience by working it out in one’s
mind.
Architectonic Leibniz’s adjective for the planned architecture and creation of the
universe and the laws by which it operates.
Averroes A Muslim thinker credited or discredited with extending Aristotle’s
ideas into positivism. St Thomas Aquinas who was active approximately a century
later was an ardent critic, and wrote that Averroes and extended Aristotle’s
ideas far beyond what Aristotle intended.
Averroism The doctrine that formal deduction is the only valid
way to reason. It is named after its leading proponent Averroes, a Muslim
thinker, who adopted Aristotle’s idea and extended them. Thomas Aquinas
disagreed with Averroes’ reading of Aristotle claiming that Averroes had
commandeered Aristotle to support his own agenda or outlook on the art of
thought whether philosophical, scientific, artistic, etc. Averroist reasoning
or logic became increasingly influential. By the 17th Century the
University of Padua, formerly humanist, became a centre for teaching and
promoting Averroes’ method. Averroism was an attempt to interpret Aristotelian
teachings without concern for consistency with Christian theology. The Latin
Averroists at the University of Paris “seemed, at least to their critics, to
abandon the Scholastic attempt to reconcile the newly translated texts of
Aristotle with the dictates of Christian orthodoxy.”[2]
Baconian The doctrine that experimental effort to discover,
record and categorise the testimony of the senses is the way that science
should be done. Early figures in English scientific circles prior to the Royal
Society, especially Francis Bacon, promoted this outlook.
Empiricism The doctrine that what is knowable, and even what is worthy of
consideration, includes only what is perceptible. Galileo was perhaps the fork
in the road, for he had great success in science, especially mechanics, and
apparently through the extensive use of experiment. Galileo was both Kepler’s friend
and, financially, was supported by Paolo Sarpi who was supposedly the first to
look through Galileo’s telescope. Sarpi was an influential gentleman,
politician and statesman and wrote a rudimentary version of what later appeared
as the rules of Newtonian science. Arguably, it was in Galileo and Sarpi that
the empiricism of Newtonian science first took form. Today empiricism is
associated with Francis Bacon with whom Sarpi corresponded, and with Isaac
Newton and John Locke who were born after the deaths of Bacon and Sarpi. Empiricism
was pronounced in English scientific circles and is often associated with
Bacon, Isaac Newton and John Locke. The Royal Society of London for Improving Natural
Knowledge (“Royal Society”) was founded in 1660 out of English scientific
circles and, accordingly, empiricism is dominant in the orientation of Royal
Society. There are also strains in 17th Century English science
which are almost anti-empiricist. Henry More and Robert Boyle are notable
examples who will be discussed in this thesis.
Empiricist A scientist or philosopher thinker
who believes in a more a posteriori
approach to science.
Conception, concept or
notion (Leibniz’s usage) The content of a thought abstracted
from the thought, or the meaning of a thought in terms mathematical, symbolic,
logical or otherwise regarded with coherent and rational structure as far as
possible. Nearly all human thoughts are inexact, indistinct, confused or
defective in some way. Some thoughts are clearer, more coherent and less
confused than others.
Idea (Leibniz’s usage) A
concept that is clear, distinct, self-consistent and consistent with every
other idea. Ideas do not exist
independently of minds, but only exist in minds, whether those of souls or of
God. For Leibniz, ideas are not subjective, are not social constructs and are
not constructed but only discovered. For Leibniz, ideas are concepts which are
coherent. The ultimate test of coherence is God’s design of the best possible
universe which is the actual universe that we are living in. Thus, an idea has
explanatory capability.
Ideal A perfect or pure version of a thing or concept,
including mathematical concepts and geometrical constructions.
Perception (Leibniz’s usage) A thought or the way in which something is understood by a
mind. This is contrary to the standard use of the word to refer to perceiving
through the bodily senses.
Thought (noun, Leibniz’s
usage) Used interchangeably with “perception”. It
is a thought is a mental image or, more generally, a mental impression.
Rational thought (verb) The wilful process of creating mental impressions, and of
bringing structure to concepts. It might also be called “intellection” or
“creative mentation”. This is the process through which hypotheses are made,
which are often tantamount to discoveries though to earn the label “discovery”
confirmation via experiment or mathematical deduction or both is usually
needed. For Leibniz, holding all concept and information related to a problem
or circumstance in one’s mind in a single thought was a prerequisite to the
understanding which precedes a new subsuming higher hypothesis. For Kepler, it
was seeking after the manifestation of harmony in the context being analysed
that provided the superior or most fruitful impetus towards the higher
hypothesis. Each is an example of an a
priori process.
Metaphysics The study of the relations, constraints and principles that underlie
the physical universe and that underlie physics. Metaphysics may also refer to
philosophical rules that govern one’s assumptions in and approach to physical
science. The Stanford Encyclopedia of
Philosophy says that the term derives from Aristotle’s Metaphysics.[3] Whether or
not this is true, Aristotle was not the first to deal with metaphysics nor did
subsequent thinkers, such as Leibniz, agree with Aristotle’s Metaphysics on all points or even on
fundamental points. The Stanford
Encyclopedia of Philosophy says that metaphysics is about things that do
not change, in contradistinction to physics which is about the physical world
which does change. However, it is not at all certain that metaphysics is not
dynamic or not evolving. In order to make progress in metaphysics, it is
generally necessary to step outside of any confinement by assumptions derived
from the physical universe. This may lead to considerations concerning the
origin of the physical universe which quickly leads to questions of theology and
matters of how the Creator or God works and, indeed, of what God is. Such
questions were addressed fearlessly by thinkers including Nicolas of Cusa,
Johannes Kepler, Leibniz and many others.
Nominalism The understanding that concepts and ideas are merely thoughts and
do not actually exist as the things that they are expressing. Rather, concept
and ideas exist only in the mind not as objects in the physical universe which
are subject to the laws of physics or of metaphysics.
Ontology Part of metaphysics that is more
interested in what exists and does not exist, with an emphasis on entities or
“things” expressed by Platonic ideas.
Neoplatonism The outlook that Platonic ideas are valid, truth is
discoverable and what is perceptible via the senses is mere opinion, while
truth is discoverable through Reason. It has the connotation of emphasis on
that which the senses cannot perceive. As a result of this, in part, it has
been confused with mysticism and irrationality. This is correct only if
stepwise deduction is considered the only valid form of rational thought, and
many - including Leibniz - would disagree. That is, Neoplatonism validates
modes of thought other than stepwise deduction under the umbrella term Reason.[4]
Neoplatonist A Neoplatonist is an adherent of variants of Plato’s ideas who
appeared several centuries after Plato, or a follower of one of these adherents
of Plato. Neoplatonism is often mixed with Hermetism, early Christian thought
and even Caballa. Leading Neoplatonists include Plotinus, Cardinal Nicolaus of
Cusa and even Cosimo de Medici.[5] Some
Neoplatonists, such as Plotinus, studied the Corpus Hermeticum which was attributed to Hermes Trismegistus.
Newtonian The doctrine that empiricism and stepwise deduction
from sensory observations is the way that science should be done. This doctrine
downplays the role of forming hypotheses in science. The rules of Newtonian
science appear to be an extended version of rules of Averroist logic.[6]
Platonic idea To Kielkopf who undertook a thorough review of Wittgenstein’s
foundations of mathematics, an “absolute platonist [sic]” holds that “the
domain of mathematical objects themselves exist independently of minds.”[7]
The same can be said of Platonic ideas more generally, even ones that are not
mathematical.
Platonist A Platonist is a follower of Plato’s ideas in their pure
form as expressed in Plato’s own writings such as The Republic. This includes the ability to discover truth through
reason and dialogue, that universal truths exist and the doctrine of
reminiscence.
Pythagorean A follower of Pythagoras, the well-known philosopher. The
significance of being a Pythagorean would be greatly deepened if it were true
that Pythagoras had been a disciple of Hermes Trismegistus as some scholars
argue.
Rationalism The doctrine that knowledge of
the physical universe is primarily attained through intellectual reasoning.
Reality (Leibniz’s usage) All that
is. This comprises God and the universe. The universe comprises the physical
universe and all that is incorporeal. The physical universe comprises “body
monads” while the incorporeal comprises “soul monads”. For Leibniz, “body” and objects as people perceive
them are not real. Rather, “coherence is the sign of truth, its cause is the
will of God, and its formal reason is the fact that God perceives that
something is the best, i.e. most harmonious, or, that something is pleasing to
God. And so divine will is itself the existence of things, so to speak.”[8]
For something to exist, it requires that God willed it to exist. As far as
Leibniz is concerned, the test for humans of truth is coherence and the best
test of coherence is the ability to predict future phenomena.[9]
Monad (Leibniz’s concept) A part of
reality. An indissoluble piece of substance. Monads are all that exists, aside
from God. Passive monads have perceptions or thoughts but are sluggish in
perceiving and thinking. Active monads have perceptions, thoughts and memory.
There is a continuum from the passive to the active, with physical matter that
is lifeless such as rocks being passive and called body monads, amoebae being
more active, animal souls being still more active, and human souls being even
more active. A collection of body monads controlled by a mind, such as the body
monads comprising the physique of a human being, have a corresponding soul
monad. The soul monad is the same thing as the mind. The soul monad is linked
to the collection of body monads through a connection called the
pre-established harmony. This is a pre-prepared correlation between the soul
monad and the body monads. This makes it appear that the soul controls the body,
when it fact the actions of the two have been established in advance to ensure
that they move in lockstep. This correlation of actions between the incorporeal
and corporeal domains is part of the design of the universe as the best of all
possible worlds.
Substance (Leibniz’s usage in the context of monads) A collective or
aggregative term for what is real. It includes physical and incorporeal
substance. Physical substance includes passive matter and all kinds of
radiation. Incorporeal substance comprises thinking (or “active”) souls, and
these govern physical substance.
“The best” (Leibniz’s usage) The
most superior and good way of designing or doing anything. It is in accord with
ideas. It is the standard used by God to make decisions. It is that which is
capable of existing. The Best makes sense over series events, not in a
“snapshot”. Thus, hardship appears to be other than the Best but in a wider
context provides to be desirable and good in the impact on the clarity and
distinctness of the perceptions of minds, and the closeness of mental
conceptions to ideas. In a priori sense this must be the case. In the
context of passive matter such as the motion of planets or of an entire solar
system through a galaxy, and speaking simplistically, the Best manifests itself
as a concept of harmony.
Reminiscence (Leibniz’s
usage) The doctrine that all knowledge has
always been with each person, and “acquiring” it is a matter of remembering. This
was presented in Plato’s Meno by a
boy with no education solving a geometrical problem purely by the aid of
prompts. Leibniz thought that this made complete sense. He wrote, “I am in no
wise in favour of the Tabula rasa of Aristotle; and there is something sound in
what Plato called reminiscence. There is even something more, for we have not
only a reminiscence of all our past thoughts but also a presentiment of all our
future thoughts.”[10] We will
refer to a proponent of the doctrine of Reminiscence as a “Recallist”.
Renaissance humanism The human-centred
outlook of the historical period and movement called the Renaissance. It was
especially prominent in Italy but found all over the Continent and the British
Isles. This outlook touched science, theology, art, literature and many –
perhaps all – domains of human endeavour. It is associated with a renewed
interest in Roman and Greek classics, as well as with an interest in the
putative writings of Hermes Trismegistus. The Hermetic humanist was typically
also more inclined towards classical Hellenic writings. Not all men of letters
were interested in all three categories. Those interested in Roman classics
were considered “grammarian pedants” by those who found truth in Hermetic
writings. The Roman (Latin) humanist hearkened back to the golden age of Cicero
and regarded the Middle Ages as barbarous. By contrast, there were Hermetic
humanists such as Ficino who revered Platonists from the Middle Ages such as
Thomas Aquinas.[11]
Teleology The final or ultimate
cause of a phenomenon or event, or the study which seeks to understand the
final or ultimate cause of a phenomenon or event.
Some concepts are best understood in terms of their
relationship to other concepts, and some concepts even exist primarily as a
counter to other concepts. We introduce some pairs or dichotomies, and some
triples or trichotomies.
Dichotomy 1: Objectivism vs Subjectivism
An objectivist understands that there are truths that transcend
time, space and social norms. Indeed, the definition of truth is something that holds
everywhere, at all times and for all people. That is, truth is absolute truth.
An objectivist may also be called an absolutist.
To a subjectivist, truth
varies from person to person and, to an even greater degree, from culture to
culture. In particular, morals vary from person to person and culture to
culture. This variation is valid, and does not detract from the legitimacy of a
truth for the person/s for whom it holds. In short, truth is relative. Thus, a subjectivist
may also be called a relativist.
Dichotomy 2: Theist vs Atheist
Does a person believe in God
or not? A theist believes in a single God and an atheist does not believe that
God exists. Nearly all of the historical thinkers in this dissertation are
theists. What is more important than whether or not a thinker believes in God
is their conception of God. For example, do they consider the mind of God to be
unknowable or is God a closed book who, are far as human are concerned, acts
arbitrarily. An atheist typically has to believe that the universe has always
existed, because once the creation of the universe is posited, then there has
to be a Creator. An atheist can concur with a theist who believes in a rational
God on many matters of science, if the atheist believes in a rational universe
that has always existed.
Dichotomy 3: Gnostic vs Ignorantist
How close can we come to
God? Gnostic means that we can have direct experience of God. Ignorantist means
that we can only ever indirectly understand God; while we can approach God and
get closer and closer, we will always be infinitely distant from God. We coin
the term ignorantist based on the title of Nicolaus of Cusa’s De Docta
Ignorantia (“On learned ignorance”). Petrarca before Cusa wrote on ignorance[12] and it is
known that Cusa had a copy of Petrarca’s work.[13]
Trichotomy 1: hylarchic
vs mechanist vs animist
How does God
relate to or connect with the physical universe? The hylarchic principle is
that God is in the universe, giving the physical universe a real spiritual
quality. God is intervening in the functioning of the universe from
moment-to-moment, perhaps in a structured or ordered way. To a mechanist, God
is not in the universe but made the universe so that it can function without
God’s ongoing intervention.
An animist believes that the
physical universe contains divine entities; indeed, that everything in the
universe is a divine entity in itself. For example, even a rock is not only
alive but is a spiritual entity with divine qualities. Pantheism is similar to
animism. A pantheist considers the world as a whole to be a single, divine
entity. There are no pantheists in this dissertation.
Trichotomy 2:
Neoplatonist vs Rationalist vs Empiricist
A Neoplatonist or “new
Platonist” considers that there is a world of ideals which can be discovered
and understood only by the mind. These ideals or perfect versions of things are
templates for what we find in the physical world. The physical world comprises
combinations of ideals so complex that it appears to not mirror the world of
ideals at all. A Neoplatonist is necessarily an absolutist.
Rationalism says that truth
can be discovered by way of thought without any sensory input or physical
observation. Clearly, a Neoplatonist is a rationalist, but a rationalist is not
necessarily a Neoplatonist. An empiricist believes that what is known comes to
a person via the physical organs of their five senses.
An empiricist cannot be a
Neoplatonist because a Neoplatonist believes that what the senses perceive is
no different from mere opinion and is therefore of no value in uncovering
truth. A Neoplatonist could be at least a quasi-empiricist because the sensory
input might help prompt or stimulate thoughts that help lead us to truth;
however, a Neoplatonist cannot be entirely empiricist because thought – not
sense perception – must necessarily play the crucial mediating role before
truth may be uncovered.
The relationship between
rationalism and empiricism is the same as the relationship between Neoplatonism
and empiricism.
Relationship of concepts to one another
Relations arising from Trichotomy 2 which
fall out from the definitions are:
Neoplatonist ==> Rationalist
Empiricist ==> ¬ Neoplatonist
Empiricist ==> ¬ Rationalist
Neoplatonist ==> ¬ Empiricist
Rationalist ==> ¬ Empiricist
Leibniz’s “empiricism” is simply that sensory
observation prompts thoughts and checks reasoning, including the soundness of
reasoning. Empiricism says that the senses give us knowledge and that thoughts
are only needed to organise the knowledge that the senses give us. We see that
this is at odds with Leibniz’s position as explained in his New Essays on Human Understanding.
The doctrine of Reminiscence is
connected to one’s conception of the soul, especially the continuity, antuiqity
and origin of the soul and its knowledge. Therefore, Reminiscence holds a
tenuous connection to the other concepts:
Rationalist =/=> Recallist
Recallist =/=> Rationalist, yet Leibniz is both
Rationalist and Recallist.
Platonist ==> Recallist, from Meno and this is Leibniz’s position
Neoplatonist =/=> Recallist, though
some were, like Leibniz.
Definitions do not do
justice to the richness of the ideas expressed by the terms which are best read
in a context. However, the intention of this background is to ensure that the
reader has seen the terms once and has some familiarity with them when they are
first encountered in this dissertation.
The thesis begins with the
historical milieu in which Leibniz worked, and the earliest projects he was
engaged in after meeting Christian Huygens. Next we situate the goals of
scientific discovery which were being debated in Leibniz’s lifetime in terms
that crossed over into the theological. We then turn to the philosophical
battles over how science should be conducted into which Leibniz pitched. With
that background, we will be ready to consider scientific method including how
philosophical positions influenced the way in which scientific problems were
approached. The questions addressed then broaden, quite naturally, into the
broadest questions such as the role of thought in science and which kinds of
notions should be encouraged in scientific endeavour and which not.
This argument in this thesis
agrees with how Jürgen Lawrenz’s 2007 PhD thesis at the nearby University of
Sydney Leibniz: Double-Aspect Ontology
and the Labyrinth of the Continuum situates Leibniz among these three
’isms:
1.
Realism: objects and events exist in
the universe, and their properties and relations obtain, irrespective of our
beliefs about them and independently of our ability to discern them;
2.
Idealism: the world is comprehended
wholly in the form of mental representations;
3.
Phenomenalism: our knowledge of the world
is mediated entirely by sense impressions.
Lawrenz says that Leibniz is
realist, idealist and not phenomenalist, and nothing in this thesis contradicts
those outcomes. That Leibniz is realist is almost immediate. That he is
idealist follows from the exclusive place given to mind in understanding and
even – via the pre-established harmony – in experience. That Leibniz cannot be
phenomenalist also follows from his conception of the role of mind.
Neoplatonism and the ideas
that it encompassed is so important in understanding the context in which
Leibniz worked that it deserves its own chapter. So in Chapter 2, “Neoplatonism
and the awakening of science”, Neoplatonism is introduced along with the many
ideas that are encompassed by its contribution. This includes proto-science,
“magic” and an interest in the Hermetica or the writings of Hermes Trismegistus.
Equally if not more important, but secondary in this thesis, is the revival of
interest in Aristotle, and the interpretation and reinterpretation of Aristotle
by people subject to such processes as a millennium of Christianity and several
centuries of Islam. What was significant about the renewed interest in
Hermetica and a serious approach to magic was a structured approach to what
Yates calls “operations” or human intervention in the physical universe.[14] This leads
to the next chapter.
In Chapter 3 “Ushering in
modernity”, Leibniz’s role in nascent nation-building efforts of Louis XIV are
examined. It is argued that the expression of rationalism in machine-building
and inanimate power sources define the beginning of modernity. If this argument
is correct, then it follows that Leibniz was a central figure in the
commencement of modernity. In modernity, the engagement in ever more efficient
and powerful kinds of “operations” is systemised and given a central place in
society, policy and all aspects human endeavour. In early modernity, the nature
of Man as magus was first institutionalised like never previously in history
with the construction of machines and projects to “operate” on an unprecedented
scale for the benefit and ongoing realisation of Man’s purpose and role in the
universe.
In Chapter 4 “Idea of
truth”, the pursuit of truth is introduced with the reason versus empiricism
dichotomy which is indistinguishable from idealism versus phenomenalism. The
chapter introduces scientific personalities of decisive importance to the
subsequent history of science: Johannes Kepler, Galileo Galilei, Paolo Sarpi
and Isaac Newton. It is explained that for Leibniz the search for scientific
truth is inseparable from the investigator’s ability to understand the nature
of the Creator of the universe. This is a Neoplatonic orientation. Contrary to
this, a junction in science is found in Paolo Sarpi in whom a stream separate
from the Neoplatonists that emphasises Empiricism, and which coincides with
Atheism, is introduced into Western science.
In Chapter 5 “The
Neoplatonist and Empiricist schools”, the concept of calculation natural
evolution of the Neoplatonic school is introduced. This includes calculation
with metaphysical concepts. Lawrenz writes of Leibniz that one can, “Open any
page of his writings to see physics happily consorting with metaphysics – it is
the pattern of his thinking.” In this chapter, it is explained that Leibniz had
no choice in this, for it is an inexorable consequence of his integrated conception
of how the capabilities of the human mind understand and commandeer reality.
Leibniz regarded mathematical calculation as but one aspect of reasoned thought.
Leibniz promoted the project to construct a Universal Characteristic that would
supply an answer to any question, not only in physical science but also about
moral differences, through unambiguous calculation.
In Chapter 6 “Discovery and
deduction”, Leibniz’s recommended process for discovery is described. The
Leibnizian prescription is to hold all the considerations relating to a problem
in one’s mind in a single thought. It takes considerable time and thoroughness
to be able to achieve this. This is posited as a top-down method in contrast to
stepwise reasoning also known as discursive reasoning or, in some contexts, as
dialectic. Much is owed to Nicolaus of Cusa’s De Docta Ignorantia or “On Learned Ignorance” which argues that
stepwise reasoning does not lead to new knowledge. Rather, the approach that
leads to breakthroughs is to seek the higher hypothesis which subsumes a
problem and its related sub-problems – such as all previously understood
phenomena and a new class of anomalous observations – in a new whole with a
simpler but more widely-encompassing explanation. To hold all considerations on
some topic in one’s mind in a single thought is the way Leibniz thinks that God
works. This is considered in the next chapter when God’s choosing of “the Best”
of all possible universes is dissected.
In Chapter 7 “Science by
thought, reality and substance”, Leibniz’s theory of reality in toto – physical and incorporeal – is
put on the table. Its comprehensiveness is astounding and would be more
astounding had multiple attempts not been made in the past. Liberty is taken in
many footnotes to point out similarities between Leibniz’s theory and Hermes
Trismegistus. Leibniz’s doctrine that this physical universe is the best of all
possible worlds is part of his scientific conception of the universe and of
humanity’s role in it. It is shown that mathematics is necessarily incomplete
unless a rigorous conception of “the Best” can be formulated. The chapter is
left hanging with important open questions which suggest that the a priori method is the only way to
proceed when faced with really fundamental questions.
In Chapter 8, it is
explained how the Leibnizian framework draws mind and humanity into the domain
of physical science. The thoughts of soul monads have a role in the physical
universe. The chasm between the perceptions of minds and the actions of body
comes into view. Leibniz spanned this divide with pre-established harmony,
which requires God to intervene every instant to mediate between mind and body
in “the Best possible way”. The fact that God mediates at all seems to be a
glaring self-contradiction by Leibniz since he always held that the Best
possible universe cannot require God’s ongoing intervention. The contradiction
is resolved by the fact that the universe is re-created in every instant
according to the thoughts of soul monads, via the agency of the pre-established
harmony which God shepherds along in the best possible way.
Until science is expanded to
subsume the pre-established harmony, science is – to a significant degree – condemned
to being an exercise in shadow boxing, or pursuing truths lacking in context
and lacking in metaphysical basis. Arguably, we cannot even understand the
mechanism of the physical universe without incorporating the pre-established
harmony into physics. The fact that the universe unfolds in “the Best possible
way” implies that God has built some kind of engine of intention into the
physics of the universe. Of course, one obvious intentional part of the
universe is humankind itself. Thus, the status and role of humankind as a
physical force in the universe – as a law of physics, so to speak – warrants
attention in future research.
In Chapter 9 “A priorism in science”, three successful
scientific projects that relied primarily on a priori thought are described. The examples chosen are the
infinitesimal, non-Euclidean geometry and elliptical orbits in a heliocentric
solar system. In all of these, but especially in the first and third, the
thinkers concerned pursued a priori methods
deliberately, and were self-conscious advocates for their own approach. In
developing non-Euclidean geometry, the thinkers were departing from the Fifth
Postulate in a conscious effort to create a geometry that is self-consistent
and functional. Beyond that, Riemann in particular understood that any new
geometry must be viable in the physical universe. For Leibniz, the physical
universe is the test of truth because, as explained above, the physical
universe expresses “the Best”.
In the concluding chapter,
the humanist project in which Leibniz was engaged is discussed. It is explained
how his metaphysical and philosophical work, and his science and mathematics,
furthered the aims of this project. This project refines and furthers a conception
of Man as magus which was “modern” for the 17th and 18th
Centuries. This survives today in the policies of scientific-industrial nation
state. Such policies were seen in early form under the leadership of Louis XIV
and Tsar Peter the Great. It is not a coincidence that Leibniz played an active
role in the nation-building efforts of Louis XIV’s France and Tsar Peter the
Great’s Russia, as described in this thesis.
It will be argued in the course of this thesis that
Leibniz represented a continuation of the influence of Neoplatonism as well as
being a conscious referent to the original writings of other movements and
schools of thought, such as those of Plato, Aristotle and their followers. When
referring to actual figures such as Plato and Aristotle, the meaning is clear.
What, though, is Neoplatonism? The term encompasses a body of intellectual
disciplines, mindsets and networks of co-thinkers. I think of Neoplatonism as a
movement that helped create much larger shift, like lava flowing between two
tectonic plates. To use phrases that might mean little at present, Neoplatonism
includes Hermetism, Cabala, medieval magic, memory methods, the use of symbols
and construction of languages, alchemy, proto-science, proto-engineering and a
bit of Platonism.
After the fall of Rome in 476 CE,[15]
intellectuals looked to Athens, Egypt and other ancient cultures for wisdom and
to increase their knowledge in almost any and all domains. Those earlier
cultures were regarded as more enlightened, virtuous and generally elevated
than the present. As Yates put it, the farther back in history a seeker
explored, the purer and better the minds found. The closer to the present, the
more benighted and corrupt. Even the standout philosopher of Rome, Cicero,
leaned heavily on the philosophers of Athens. De Natura Deorum reflects Plato’s The Republic in many ways. After Rome, the tendency and indeed the
imperative to look backwards in time increased.
The takeover of Athens by the Romans did not eliminate
adherents of Plato. Indeed, Plato’s Academy continued to teach students. The
Greek thinkers Plotinus (c.204/5 – 270 C.E.), Porphyry (c.234–305 C.E.) and Iamblichus
(c.242-327 C.E. or 250-330 C.E.[16])
are regarded as the founders of Neoplatonism. These can be regarded as the
first kind of “Neoplatonist”. The term has a literal meaning in the sense of a
series of teachers known for their interpretations of Plato. Indeed many if not
all actually taught in Plato’s academy. This line of teachers included Speusippus
of Athens was the son of Plato's sister Potone; he became head of the Academy when
Plato died in c.348/347 BCE and remained its head for eight years.[17]
In the early 4th century C.E., Iamblichus moved the major Neoplatonist school
from Rome to Syria. Thereafter, Neoplatonism flourished mainly in the Eastern
Roman Empire, with centres in Athens, Alexandria, and Pergamum (now in Turkey).
In the 5th century C.E. one of its greatest teachers was Proems, at Athens. This
was also where the well-known Neoplatonist Proclus (c.412–485 C.E.) taught. During
the same century the writings of St. Augustine firmly established its influence
in Christian theology. St Augustine embraced some parts of Neoplatonist
teaching while filtering out parts considered to be contrary to his Christian
theology.
The Neoplatonists were not unmolested by the powers
that were. While Constantine was the first Christian emperor of Rome, his
nephew Julian who succeeded him was a Neoplatonist, a pupil of Aedesius, who
had in turn been taught by Iamblichus.[18]
Emperor Julian was sympathetic to Neoplatonism but his successor Emperor
Theodosius was not. Under Theodosius, the leading Neoplatonist teacher Hypatia
was dragged from the academy and murdered in 414/415 C.E. (there are differing
accounts of the year) by a mob of Christian monks under the control of Bishop Cyril
of Alexandria. Ostensibly, Hypatia’s crime might have been to attempt to
publicly hold the light of reason to the “metaphysical allegories from which
Christianity had borrowed its dogmas”.[19]
It is likely that the threat posed by Neoplatonism was much deeper, and that
the real threat was not to Christianity but to the Roman Empire and to the
system of Empire in general. However, these ideas are speculative and further
discussion is beyond the scope of this thesis. Finally, under Emperor
Justinian, the last seven significant Neoplatonic teachers Hermias, Priscianus,
Diogenes, Eulalius, Damaskias, Simplicius and Isidorus were unable to continue
the Neoplatonic school in Athens.
The early Neoplatonists just mentioned were aware of
the Egyptian writings especially those of Hermes Trismegistus and there were
Neoplatonists who absorbed Christian mysticism. The first Christian
Neoplatonist to write in Latin was Victorinus, who was converted to
Christianity in 360 C.E.[20]
In the 400s C.E., Pseudo-Dionysius, expounded a Christian mysticism based on
Neoplatonism that influenced later Christian mystics. Four centuries later, a famed
medieval Neoplatonist named John Scotus Erigena; he was the Irish theologian of
the 9th century C.E. The Cambridge Neoplatonists are a group from the late 17th
century C.E. who sought to counteract the scientific materialism then
prevalent.[21]
Following the first Neoplatonists, the Arab
philosophers were active for at least a century or so were also active in
reviving ancient philosophy both Greek and Egyptian. This was an intermezzo
between the various wars and wrecking operations that beset the West between
the fall of the Roman Empire and the Renaissance.
Avicenna (Ibn Sina 980-1038) wrote on logic and
philosophy, while also making original discoveries in medicine. Avicenna is
often cast as Platonist mediated by the Neoplatonists,[22],[23],[24]
but he also adopted ideas from the Stoics such as Zeno and Chryssipus, and from
Aristotle and his disciples.[25],[26]
Avicenna also tried to reconcile the orthodox Islamic doctrine of the
“theologians” of his culture with the philosophers. Al Ghazali (1058-1111)
ridiculed attempts of writers such as Avicenna who tried to reconcile Islamic
doctrine with philosophy. However, he spared neither the Islamic doctrinal
theologians nor the philosophers. In talking about the philosophers, he was
targeting contemporary proponents of Aristotelian doctrine as they interpreted
it. Platonic ideas seemed to be outside his purview.[27]
It was not only the “third force” of Plato that Al Ghazali let alone. Kilcullen
at Macquarie University says that the Muslims were “neo-Platonists” presumably
that they nearly all were neo-Platonists, though it is unclear where this means
students of Plato or whether it means Neoplatonists.[28]
There was also the “big influence of Hermetic and gnostic literature and ideas
on the Arabic world and particularly on the Arabs of Harran.”[29]
Further, “Talismanic magic was practised by those Arabs, and the influence came
through the Sabeans who were immersed in Hermetism, in both its philosophical
and its magical aspects.”[30]
The Picatrix was probably written in
the 12th Century, and while it lists magical images and procedures,
it does so in a philosophical setting similar to if not borrowed from the Corpus Hermeticum and the Asclepius.[31]
The Picatrix circulated during the
Italian Renaissance and references to it are found in the writings of Pico the
younger, among others. A copy was found in the library of Pico Della Mirandola (1463-1494).
It was included in a list of works “codex” on magic copied in 1488 in Kraków,
probably by a university student referred to as Egidius. The codex includes
Hermetic works and the Picatrix. From the commentary given in the codex, one
commentator concludes that “in all probability he [Egidius] knew the content of
the Picatrix”.[32] Rabelais
(c.1494-1553) came to its defence to it in one of his famous satires, Pantagruel. Rabelais poked fun at those
who shun the Picatrix by referring to
“Father Devil Picatrix, doctor of the faculty diabolical”. (Pantagruel III, 23)
While the Picatrix was written in the 1100s, we would
expect it not to have arisen spontaneously but to more likely have followed a
century and probably more of thought, teaching.
During the 1100s Averroes
(Ibn Rushd, 1126-1198) which was guided or at least given initial direction
from Abu Bakr (Ibn Tufayl) chief physician to the monarch Abu Ya‘qub Yusuf
(c.1168-1169). Averroes was from an influential family possibly with a certain
religio-political orientation. For example, the distinctions of his grandfather
Abul-Walid (1058-1126) was that he was chief justice, wrote a definitive legal
text, and leaned on the monarch Ali Ibn Yusuf to be tougher on the Christians
of Andalus.[33]
Averroes ostensibly wrote in response to Al Ghazali taking exception to his
ridicule of Aristotelian philosophers saying that those philosophers and Al
Ghazali had not understood Aristotle. Averroes used persuasion not attack. His
main target seemed to be those engaged in the practice of law. Lawyers, judges
and clerics alike were wedded to Islamic doctrinal theology in the practice of
law to the extent that even the monarch Abu Ya‘qub Yusuf never publicly
announced his support for philosophy. Nonetheless, possibly under the
persuasion or at least influence of Ibn Tufayl himself, the monarch Abu Ya‘qub
Yusuf allowed Ibn Tufayl to commission Averroes to write commentaries on
Aristotle that were accessible to the reading public.[34]
From the 9th
Century up to Averroes there had been a cloud of suspicion alternating between
theology and philosophy. Ibn Masarra (883-931 C.E.) is regarded as the first
Andalusian philosopher. The zeitgeist was against them, and so he and his
disciples only survived by living as hermits. Hourani writes that Ibn Masarra
introduced a pseudo-Empedoclean pantheism.[35] Empedocles
(c.494/5-435 BCE) is one of the few pre-Socratic philosophers whose original
writings have been found.[36] In the
late 11th Century and into the 12th Century, philosophy
found increasing favour to the detriment of theology. In fact, the monarchs
encouraged the study of Malikite law and banned theology.[37] Malikite
law is a school of law founded by Malik ibn Anas (c.710-795 CE) which
recognizes a range of sources of law.[38] Ibn Bajja better
known as Avempace (c.1095-1138 CE) was the first Andalusian philosopher to make
direct use of the works of Plato and Aristotle. Philosophers were still subject
to suspicion. For example, ibn Wahib a contemporary of Avempace, ceased to
speak about philosophy openly out of fear for their lives.[39] It appears
that ibn Sina also known as Avicenna attempted to play a conciliatory role but
to Al Ghazali both the theologians and philosophers were wrong as well as being
irreconcilable. Al Ghazali is best known for his Tahafut al falasifa (“The Incoherence of the Philosophers”). While
this sounds like a dogmatic and politically-charged title, Al Ghazali was not a
dogmatic contrarian. He wrote many other works which describe, explain and
grapple with both theology and philosophy.[40] When
Averroes replied to Al Ghazali 80 years later, it was only to defend Aristotle
not for the most part any other philosopher and not (Islamic) theology.
It has been mentioned that
Picatrix reached Europe and had some Hermetic effect on intellectuals there
centuries later. So too did Avicenna, Al Ghazali and Averroes. The reception of
Averroes was not entirely favourable in the increasingly Christianized Europe
due in part to Averroes’ advocacy of ideas such as that the universe has always
existed and was not created, and the impermanence of the individual human soul.[41] In 1270,
Saint Thomas Aquinas took the trouble of writing “On there being only one
intellect” in Paris while occupying a Dominican chair of theology as Regent
Master. Aquinas argued that intellect is a faculty of the soul that animates
the human body, and that there is not only a single separate intellect that
suffices for and furnishes all humans, but that each individual human soul has
its own intellect.[42] Moreover,
Aquinas argued, Aristotle would have agreed whereas Averroes had misinterpreted
Aristotle. Aquinas presented Aristotle as consistent with Christian metaphysics
on essential points.
Averroes is sometimes
presented as having washed over Europe and fully infiltrated institutions such
as the University of Padua which became a “stronghold” of Averroism. Martin
explains that the truth is not so simple. (Martin, C. 2007) Intellectuals were
as averse to an Averroes dogma as they were to any other. Rather, Averroes was
often quoted in order to indirectly assert a secular orientation without
inviting arrest, torture and death at the stake by the Inquisition.[43] Indeed, it
is strange that no text by Averroes was placed on the Vatican’s Index Librorum Prohibitorum (“List of
Prohibited Books”) especially given Aquinas’ thorough case against the coldly
argued anti-Christian position of Averroes’ Middle
Commentaries on Aristotle’s De Anima (“On the soul”).
Returning to the influence
of pre-Athenian thought, we have already mentioned the Corpus Hermeticum. This was and is ascribed to Hermes Trismegistus
though the circuitous route between Hermes Trismegistus and the documentation
of Hermes’ teachings in the Corpus
Hermeticum is subject to debate. It is reported by some that Hermes was
Egyptian,[44]
and he is sometimes associated with the Egyptian god Thoth with the head of an
Ibis. It has been argued that there were three different Hermes Trismegistuses
the third of whom was the teacher of Pythagoras, while others dispute whether
Hermes was ever a real person.
It has already been
mentioned that the Arabs of Harran were steeped in Hermetic lore at least from
the 12th Century when the Picatrix was written.[45] The
Picatrix was in part a magical text with symbols that can be used as talismans.[46] In the 13th
Century, Abulafia wrote a tract of Cabalistic symbols and combinations of
letters as an aid to understanding, meditation and memory.[47] In 1305,
the Picatrix was translated into Latin by a scholar named Sloane.[48] Yates
quotes at length from Book IV Chapter 3 of the Picatrix:
There are
among the Chaldeans very perfect masters in this art they affirm that Hermes
was the first who constructed images by means of which he knew how to regulate
the Nile against the motion of the moon. This man also built a temple to the
Sun, and he knew how to hide himself from all so that no-one could see him,
even though he was inside it. It was he, too, who in the east of Egypt
constructed a City twelves miles long… Around the circumference of the City he
placed engraved images and ordered them in such a manner that by their virtue
the inhabitants were made virtuous and withdrawn from all wickedness and harm.
The name of the City was Adocentyn [in the Arabic original, al-Ašmūunain].
Giordano Bruno 279 years
later wrote about the Greek origins of European civilisation affirming that its
origins were not Judaic but Chaldean. In Bruno’s work The Expulsion of the Triumphant Beast published in 1548, he wrote,
“Do not infer that the sufficiency of Chaldean magic comes from Cabala; indeed,
there has never been anyone who could pretend that the Egyptians could have
taken any principles from the Judaic corpus.”[49]
Giordano Bruno was a
philosopher, activist, teacher, Dominican friar and expert on the art of
memory. He was born in Nola Italy in 1548 and burned at the stake in 1600. He
began a period of almost 20 years as a fugitive from the Inquisition at the age
of 28 when he fled the Dominican monastery where he was studying following a
tip-off that he would be apprehended by the Inquisition for possessing
forbidden books. He then lived as an itinerant teacher and writer. Patronised
by Henri III of France, he went to London as an organiser and kind of emissary with
a letter of introduction from Henri III to the French Ambassador. This
introduction afforded Bruno diplomatic protection. While in England, Bruno was
active in Hermetic and Neoplatonic circles, with known highlights including a
lecture at Oxford and strong relations with Fulk Greville a teacher of
Shakespeare. It is believed that Bruno taught his art of memory to Greville who
passed it on to Shakespeare. Elizabethan actors were required to have powerful
memories due to the short times needed to learn lines and the fact that a
single actor would typically play many roles in a play. While acting circles
were necessarily full of skilled mnemonists, Shakespeare was respected even in
acting circles for his memory. It is not too much of a stretch to attribute
this to Bruno’s memory methods.[50]
Bruno is credited with
bringing a modern outlook on the universe, as full of stars which were suns
like our own. Each sun, according to Bruno, had many planets and many of those
were peopled with intelligent beings who probably worshipped their own gods. Like
that of most of the people mentioned in this thesis, the story of Bruno is an
epic in itself. We introduce Bruno because he is mentioned in this thesis in
connection with the revival of the teaching of Hermes, the role of a rational
kind of magic, and the use of symbols as an aid to thought.
Consistent with Bruno’s view
is one circulating in the circles of “New Age spiritualism” that the wisdom of
the Cabala is merely what the Jews took with them of Egyptian wisdom when they
left Egypt.[51]
Magic, talismans, symbols,
languages and the art of memory in the 16th Century were on some occasions
treated separately and at other times together. There are many studies that
show why it is artificial to separate these domains. Credit goes to Yates for
showing how these threads were drawn together by Bruno in Art of Memory.[52] However,
there were many figures other than Bruno, such as Marsilio Ficino, Pico della
Mirandola, perhaps nearly every European thinker from the Neoplatonists up to
Leibniz who would not have been able to study one area without examining the
others too. This does not mean that every contributor to language such as Dante
or to the art of memory such as Bruno was an aspiring magician.
Michael White writes that though
Bruno “was fully cognizant of the power of magic ritual and the occult
tradition”, nonetheless Bruno “was convinced by very little of the occult
canon.” In fact, “he knew much of it was superstition, wild fantasy, and
wishful thinking.” However, it did indirectly play a role. “To Bruno, as to
many great thinkers after him, the occult was primarily a useful tool, a key
that would open doors into arenas of thought and hidden depths of the human
psyche. Along the occult path he found tracks, roughly hewn, that led to
revelation and inspiration. Alchemy held no interest for Bruno; he was never
motivated by experiment and was not drawn by the search for the philosophers’
stone, the dream of limitless wealth. Neither did he practice ritualistic magic
or necromancy; indeed, he often mocked practicing astrologers and many of the
irrational precepts of witchcraft.”[53]
In the first half of the 15th
Century, Cosimo de Medici was financing expeditions for ancient texts and the
translation of those texts obtained. While Athenian writings such as those of
Plato or Aristotle or their disciples were valued, the earlier philosophers of
Egypt, the Chaldeans and others were even more highly valued. As mentioned
above, the perception was that more ancient meant purer and better. More than a
hundred years after Medici, Bruno continued to hold that the best of European
philosophy emanated from Egypt, and not merely because “older is better” was a
catchy epithet. He understood that Ancient Egypt was the source of Greek philosophy, implying that even the Greeks were a
dilution of the best of Egypt. In his The
Expulsion of the Triumphant Beast published in 1584, Bruno wrote, “ We
Greeks recognize Egypt, the great monarchs of literature and nobility, as the
parents of our epics, metaphors and doctrines”.[54],[55]
Returning to Cosimo de Medici, the monk Leonardo da
Pistoia sent manuscripts of the Corpus Hermeticum, which were putatively
written by Hermes Trismegistus himself to Medici. Medici sent them on to Ficino
in c.1462 asking Ficino to pause on the translation of Plato and to translate
the Hermetic manuscripts as quickly as possible so that Medici could read them
for he was terminally ill.[56] Ficino’s
translations of the works were published in 1471. The works were the Asclepius or De voluntate divina “On Divine Will”, the Pimander or De sapientia et
potestate Dei “On God’s Wisdom and Power”, and the Asclepii Definitiones.[57] Ficino
(1433-1499) holds a significant place in the history of theology, religion,
magic and philosophy.
In 1614, Isaac Casaubon
proved that the Hermetic manuscripts were no older than the second or third
century C.E.[58]
However, by that time, they had already had 140 years to transform the thinking
of intellectual and influential circles throughout Europe.
To use Yates’ words, Hermes
Trismegistus, through these proxy writings, authorised by his antiquity the
revival of forms of magic.[59] That
revival included the surrounding philosophy including doctrine of the universe
and the active place of humankind in it.
Ficino the translator of the
manuscripts for Cosimo de Medici has an enthusiasm for “magna naturalis” or
natural magic or the magic of nature which, ultimately, can only mean the
science or, at least, proto-science albeit undoubtedly with a mystical overlay.
Pico della Mirandola began his career under Ficino’s influence and inherited
Ficino’s enthusiasm for magia naturalis.[60] Pico
perhaps partly due to the Judaic origins of the Old Testament account of
Creation in the Torah effected a marrying of Hermetic and Cabalistic magic.
This, writes Yates, “was to have momentous results, and the subsequent
Hermetic-Cabalistic tradition, ultimately stemming from him [Pico], was of most
far-reaching importance.”[61]
Many have heard of a
controversial 900 theses published in Rome centuries ago. It was Pico who wrote
them and in 1486 took them to Rome proclaiming that he was ready to prove in
public that they were all reconcilable with one another. Twenty-six of the
theses were on Cabalist and/or natural magic. One thesis says the old-style kinds
of magic should be forbidden, but that magia
naturalis is a good, allowable magic. In another thesis, Pico says good
magic is in part the practical science of nature.[62]
In his survey of “learned
magic” according to manuscripts from Central European medieval libraries, Láng
writes “It might sound surprising, but ‘necromancy’ in [one of its meanings] was
long a successful applicant for denoting a widely accepted part of science. The
great summary of Arabic magic, the Picatrix,
defines it in a rather wide and naturalistic sense as the science dealing with
all the things that are hidden from the senses or from the intellect, the
functioning of which most people do not understand.”[63]
While Ficino obscured the
Hermetic origins of the magia naturalis that he promoted, Pico writes in his
oration On the Dignity of Man, “It is
really the magic of the Asclepius
that I am talking about, and I glory in Man the Magus as described by Hermes
Trismegistus.”[64]
What can Man the Magus do? He says, “The Magus the earth to heaven, that is to
say the force of inferior things to the gifts and properties of supernal
things.” Indeed, Pico begins this oration with the words of Hermes Trismegistus
to Asclepius, “The Magus, oh Asclepi, is a miraculous man.”
It is at the point of Pico’s
work that we have reached the Neoplatonic age. It is in this sense that
Neoplatonism is used in this thesis. It encompasses a confluence of ideas and
philosophies. Some might call the Pico era the Neoplatonic humanist Renaissance
par excellence at whose heart of
which we find Hermes Trismegistus. The fact that we find Hermes at the heart of
that era raises warning bells about whether the Renaissance was in fact
Christian or even humanist. Thanks go to Yates for pointing this out.[65]
Ultimately, it was about the power of humans acting individually and, later, in
concert. Rather than Christian, perhaps it was Hermetist with a Christian
sheen. It was as pantheist as it was humanist, with humans playing a special
role in the matrix of the natural universe since humans could understand,
imitate and thence manipulate and bend the power of nature to the collective
will of humans.
The entire current of
“magic” led into the domain of “operations” as Yates calls it or the
post-Renaissance idea of Man consciously intervening in the physical universe
to change it or amend it for Man’s benefit or for Man’s beneficial ends. An
intermediate kind of magic which Yates refers to as “astral magic” is
unmistakeably tending towards operations. It seeks “escape from astrological
determinism by gaining power over the stars, guiding their influences the
direction the operator desires.”[66] Walker
explains Ficino’s theory that is Stoic in origin of a spiritus mundi (“cosmic spirit”) which provides a channel of
influence between planets and stars, and the world in which humans live.
Methods of influencing the stars and planets were detailed by Ficino.[67]
In 1510, Henry Cornelius
Agrippa wrote his De occulta philosophia
which he published in 1533. In fact, this work was a survey of renaissance
magic including Natural Magic which was the subject of Book I.
Agrippa did the same as
Ficino did in the sense of explaining methods for influencing the stars.
However, while Ficino tries to force his magic in to a Christian framework,
Agrippa does not.[68] Rather,
arguing the converse, Agrippa says explains that some Christian practices
including prayer itself were forms of magic, some of which were legitimate and
effective. Of these, some were effective in their own right while others were
effective in influencing people and, though people, events, but only because
people were emotionally affected by the rituals and practices.
While the perception of the
cosmic order of the Agrippan Magus is almost identical to the medieval perception,
Man has now changed and so also has Man’s role in the cosmic order. Man is “no
longer the pious spectator of God’s wonders in the creation, and the worshipper
of God himself above the creation.”[69] Man is now
an “operator” who “seeks to draw power from the divine and natural order.”[70] In an
image by Robert Fludd of Man’s Art is shown as a monkey which imitates Nature.[71] The
apparent loss in dignity is more than exceeded by the gain in power by
“becoming the clever ape of nature, who has found the way nature works and by
imitating it, has obtained her powers.”[72] Man has
“learned how to use the chain linking earth to heaven.”[73]
Contemporaneously with
Agrippa, John Dee had his career as perhaps the consummate Renaissance Magus.
Dee was interested in the applied sciences, and built devices such as a flying
crab for a college stage play.[74] He was
certainly a mathematician as far as was possible in the 16th
Century, having written the preface to the first English translation of Euclid,
by Billingsley.[75]
The book included pop-up figures of 3D geometry.[76]
Nearly a century after
Agrippa, Tommaso Campanella wrote Magia e
Grazia which is mostly on religious magic. Perhaps this is one of the
reasons why Campanella was imprisoned in Rome by the Inquisition for
twenty-five years suffering both torture and solitary confinement.[77] In Magia e Grazia, Campanella classifies
different kinds of magic one of which he calls “real artificial magic”. He
enumerates examples such as when “Architas [sic] made a flying dove of wood” or
“when Daedalus made statues which moved through the action of weights or of
mercury” or “to make a head which speaks with a human voice”.[78] Dampening
the enthusiasm of those engaged in artificial intelligence in today’s computer
science, Campanella also writes, “But such forces and materials can never be
such as to capture a human soul.”
We have attempted to show
how Neoplatonism and the magic talked about – if softly – during the
Neoplatonic age led to the rise of mechanical devices which are not far from
the art of invention, engineering and the age of mechanisation. This is one
point where Leibniz enters our story, as he shall do in the next chapter.
Yates contrasts the
operational role of Man with the Greek position. Yates argues that the Greeks
were not particularly interested in operations:[79]
The Greeks
with their first class mathematical and scientific brains made many discoveries
in mechanics and other applied sciences but they never took whole-heartedly,
with all their powers, the momentous step which western man took at the
beginning of the modern period of crossing the bridge between the theoretical and
the practical, of going all out to apply knowledge to produce operations. Why
was this? It was basically a matter of the will. Fundamentally, the Greeks did
not want to operate. They regarded operations
as base and mechanical, a degeneration from the only occupation worthy of the
dignity of man, pure rational and philosophical speculation. The Middle Ages
carried on this attitude in the form that theology is the crown of philosophy
and the true end of man is contemplation; any wish to operate can only be inspired
by the devil. Quite apart from the question of whether Renaissance magic could,
or could not, lead on to genuinely scientific procedures, the real function of
the Renaissance Magus in relation to the modern period (or so I see it) is that
he changed the will. It was now dignified and important for man to operate; it
was also religious and not contrary to the will of God that man, the great
miracle, should exert his powers. It was this basic psychological reorientation
towards a direction of the will which was neither Greek nor mediaeval in
spirit, which made all the difference. [emphasis in original]
Even Zeno the putative
founder of the Stoics saw the aim of attaining full understanding of the
universe as a self-governing system was to “rise to complete wisdom and attain
perfect ethical conduct.”[80] At least
per Cicero’s interpretation of Zeno’s doctrine – according to Hunt – there was
an element of stasis in the Stoic doctrine and conception of the universe which
might have tended away from operations. Surprisingly, it may be in the domain
of what is called magic that we see the earliest signs of the orientation
towards operations in any culture.
On the other hand, an
argument for the operational Greeks can be put. Greeks such as Archytas of
Tarentum (400-350 BCE) and Archimedes of Syracuse (c.287-212 BCE) were
impressive as inventors, and these two are that we know about which readily
come to mind. Could it have been that the Greeks were so beset by war from the
Persians and Romans, and/or distracted by foolish policy such as that which led
to the Peloponnesian Wars, to achieve their potential in operations?
A modern parallel can be
drawn. Consider the once highly “operational” USA with the Apollo landing,
plans for the colonisation of Mars and dam-building projects on a grand scale.
Contrast that history with the veering of American policy away from operations
from the Vietnam War onwards.
It is has been written that
Archimedes preferred contemplation in abstract geometry to inventing machines
and was only compelled to invent machines by his monarch, King Hero, to defend
Syracuse against the Roman army led by Marcellus (214-212 BCE) during the
Second Punic War. Nonetheless, for every potential operator like Archimedes who
had to be imposed upon to engage in operations, there might have been ten who
were able and who preferred to operate rather than to meditate. Arguably, had
the Greek civilisation not been prevented from flourishing by institutions such
as the Persian and Roman Empires, then they would have realised and expressed
an intention to “operate”.
This leads to a discussion
of nations or “the state” as operator, Magus or magician commandeering the capabilities
of entire populations for operations in grand projects.[81] Further
discussion on this topic is beyond the scope of this thesis, though the topic
is worthy of further effort.
The related current also
intertwined with magic but also running in to the ocean of “Man as operator” is
number or mathematics. Many of those who took “magic” seriously as a means
whereby Man may “operate” also took mathematics seriously. Indeed, inventors
from Archytas of Athens to Archimedes of Syracuse to John Dee of England to Huygens
and Leibniz were equally well-known for their contribution to mathematics.
Johannes Trithemius, Abbott
of Sponheim, was a friend and teacher of Agrippa. Trithemius’ book the
“Steganographia” was printed in 1606. It is a tract about cryptography and
Cabalist angel magic. It presents methods for summoning networks of angels to
enable the transmission of messages using telepathy. Part of the process of
summoning these angels involved many pages of calculations involving the
numbers that represented the angels.[82]
It is consistent with the
central role of number in the discipline of magic that Ficino includes
Pythagoras amongst the prisci magi
given the mathematical achievements of the Pythagoreans. Pico’s 900 theses include
fourteen that, according to Pico, followed from the mathematics of Pythagoras.
Yates concludes that “Renaissance magic was turning towards number as a
possible key to operations”[83] meaning,
in our language, that without mathematics there would be no science or
engineering.
Leibniz is famous for his
contribution in creating the infinitesimal calculus. The relevance to this
chapter is that Leibniz’s calculus was part of his larger vision for a
Characteristica Universalis which would be a language that serves not only for
numerical calculation but for reasoning in general.
Ramón Lull (1232/5 – 1316)
sought a language with which truths could be communicated to infidels who did
not know Latin and with illiterates who knew no written language. Lull
considered such questions as how many anagrams could be formed from a finite alphabet.
For example, there are n!
permutations of n letters. Lull
called it the ars combinatoria or
“art of combinations” which 400 years later became one of Leibniz’s first
sources.[84]
In 1660, Leibniz wrote his Dissertatio de
arte combinatoria which used concepts from Lull.[85] Eco
explains that aside from Leibniz’s vision for a universal characteristic, his
ars combinatoria had a great deal in common with the many project for universal
language undertaken over centuries up to Leibniz’s career. However, Leibniz’s
search for a characteristic emphatically was not a search for a universal language
any more than the infinitesimal calculus can be used for day-to-day
communication. Peckhaus asks the question whether Leibniz envisioned a
universal language for reasoning or a universal characteristic.[86]
Leibniz thought about what
would be the best way of providing a list of primitives and ultimately an
alphabet of thoughts. Leibniz described an encyclopaedia as an inventory of
human knowledge which might provide the basic material for the art of
combination. Leibniz wrote that the greatest aid for the mind could be to
discover a small set of thoughts from which an infinity of other thoughts might
issue in order, just as the symbols for all numbers are obtained from the
symbols 0-9.[87]
Leibniz’s work in this area
was a continuation of a long line of philosophers and philologists who had
dreams akin to those of Leibniz of finding a tool for human thought in the form
of language. We do not wish to paraphrase the comprehensive work of Eco but
merely to provide some signposts.
Symbols that putatively had
a magical purpose were also used in the art of memory with only the pragmatic
purpose of assisting in memory. Certain symbols stood out for their magical
purpose and also because they were striking as well as forbidden and,
therefore, memorable. Bruno is of course the standout figure among these.
Ficino is not a good example because, unlike Bruno, he actually believed in the
magical power of the images.[88] Such
images were regarded by the clergy as demonic whether officially or otherwise.
Augustine had already banned the Egyptian gods whether presented as gods or
recast in modern costume.[89]
The art of memory was not
only about memory but about thought. Similarly, Leibniz’s encyclopaedia project
was about recording knowledge, but it could only be useful if undertaken in
tandem with a scientia generalis or
“general science” from which the whole encyclopaedia could be derived.[90] Leibniz
equated the scientia generalis with
the scientia felicitatis or the
search for wisdom.[91] A tool for
this science was the Leibniz proposed a battery of related and indispensable
disciplines such as created about creating it with the aid of a tool for
reasoning which he called the “universal characteristic” about which more below.
Leibniz’s programme had some
similarity to George Dalgarno’s bifurcated project for a universal language,
culminating in his Ars signorum of
1661. Eco says that Leibniz was “perhaps the only scholar who considered Dalgarno
respectfully”.[92]
For Dalgarno, a universal language had first to accommodate the plane of
content, that is, a classification of all knowledge. Second, it required an
expression level or a grammar to denote the content.[93] John
Wilkins ran a similar project on behalf of the Royal Society which started in
1668. Eco’s assessment is that while Wilkins “accomplished what Dalgarno only
promised to do”, Wilkins’ achievement was limited to an image of the universe
“designed by the Oxonian culture of his time.”[94]
Bruno’s memory methods
converged with his pitch into the quest for the perfect language which was at
one time thought to be the language of Adam. The perfect language was believed
to allow metaphysical and scientific secrets to be unlocked. Bruno thought that
the hieroglyphic language of the Egyptians was superior to the alphabetic
languages. This is contrasted with Leibniz who thought that alphabetic language
were superior to pictographic languages like Mandarin because one need not be
highly educated to know and alphabetical language but need only learn the 26 or
so letters to get started. No doubt, Bruno’s knowledge of the usefulness of
symbols as markers that are pregnant with meaning and his active use of them to
great effect in his memory techniques was to Bruno a provable demonstration of
the power of symbols. Moreover, Yates says that Bruno was steeped in
understanding and belief in the magical power of certain symbols. To the extent
that this is correct, the magical power of symbols will have added a new raft
of power to symbols for Bruno. Leibniz’s argument makes sense as far as getting
started in a language is concerned, but moving past the rudiments one does need
a certain amount of education to understand the meaning associated with words
beyond a minimal vocabulary. In Bruno’s defence, a well-formed symbol can
convey meaning even to an illiterate person. Further, if symbols are formed in
a sensible and intuitive way, then someone trained in the rudiments of a
symbolic language should be able to make some sense of “advanced” symbols even
without advanced training.
Plotinus and Iamblichus seem
to imply that they too, like Bruno, prefer the hieroglyphic language of the
Egyptians.[95]
The search for the best language to aid the process of revelation or as an aid
to uncovering metaphysical truths “the perfect language” was on. Those who
studied the art of memory along with occult symbology with or without magical
content were also involved the search
for the perfect language. The study of the accessibility of symbolic versus
alphabetic languages is another topic that is beyond the scope of this thesis
but worthy of further study.[96]
As explained above, the
“magical” disciplines led into the domain of “operations” as Yates calls it or
the post-Renaissance ideas of Man consciously intervening in the physical
universe through science and engineering. This current in magic was really magic
minus mysticism and minus occultism. The search for the universal
characteristic was part of this. That is, rather than symbolism with occult
meaning and purpose, the idea was a structure of symbols – indeed, a language –
that would not only be an aid to reasoning or to rational thought, but that
would in effect as a gateway to reason. For a moment, consider Reason as an
abstract body of “correct answers” or “Leibnizian ideas” provided. The right
answers would be calculated by the universal characteristic, or – more
correctly – would be forced out by the relationship of the structure and design
of the language to Reason. The imperative “let us calculate” using Leibniz’s
envisioned characteristic was more a charge to sit down and use the
characteristic to obtain the correct answer.[97] The reason
was built into the system, whereas “calculating” using the system was
relatively mechanical; one would merely “crank the handle” to get the answer
out. This would not be a stretch for Leibniz since he developed and designed a
mechanical calculating machine that was eventually built. Leibniz certainly did
not think that such automatic calculation methods had to be restricted to
questions involving numbers.
Franklin gives an overview
more specific to diagrammatic reasoning.[98] It may be
that the universal characteristic would be a diagrammatic language. Frege’s
work towards Leibniz’s vision is for a diagrammatic language.[99] Category
theory also uses diagrams, but is only intended to apply to specifically to
algebra.
To conclude, Neoplatonism
largely encompasses what is discussed above regarding symbols, magic and
Egyptian thought. The last item largely means Hermetism, Platonism itself, and
more. Proto-science grew out of Neoplatonism, from which science grew shaking
off its mystical magical roots while certain aspects of its roots remained such
as the interest in language and the desire to create the best possible language
or characteristic that aids reasoning and acts as a gateway to propositions
concerning theology, metaphysics and science that are correct. This is about
acquiring the ability to discover a principle before the effect of that
principle has been discovered or observed. This is in effect the idea of an a priori approach to science. Further to
that approach, tools of thought aid purposeful creative thought and structured
reasoning. Many will disagree with an attempt to encapsulate Neoplatonic revolution
in a single sentence. However, it is arguable that the many currents which meet
in Neoplatonism effected recognition of the goal, purpose and capability of
humanity to influence, affect, build in, improve and – indeed – control the
physical universe. That is, to “operate”.
Leibniz appears at the tail-end of the Renaissance,
born shortly before the end of the Thirty Years War which was concluded with
the 1648 Peace of Westphalia. The Thirty Years War was not an historically
isolated eruption. Bloodshed fuelled by misguided religious passion marred much
of the 16th Century. That the senseless slaughter above all else required
a revolutionary change in thinking had already been noticed by one born nearly
one hundred years before Leibniz, named Giordano Bruno whom we met in the
previous chapter. As Michael White wrote, “The Wars of Religion provided a
harsh backdrop to Bruno’s entire adult life and added further turmoil to the
usual privations and struggles of sixteenth-century common folk. Wherever Bruno
travelled within Europe, doctrinal intolerance and endemic slaughter in the
name of God reassured him that only a spiritual and intellectual revolution
could ever disassociate religion from murder, horror, and endless pain.”[100]
Moving into the 17th Century, the Thirty
Years War and the peace which ensued affected intellectual circles, which
included Leibniz’s teachers. In
particular, it was apparent that an intellectual order was needed which would ensure
tolerance and an enduring peace. This certainly affected the philosophical and political outlook of Jakob
Thomasius (1622-1684) the jurist, historian and philosopher whom Leibniz says
is “the most celebrated German Peripatetic”.[101]
Indeed, Thomasius was known as an important conciliator.[102]
Thomasius is particularly relevant because Leibniz was taught and mentored by
Thomasius from when Leibniz entered the University of Leipzig at the age of 14.
The need to conciliate recurs so frequently in Leibniz’s writings that Christia
Mercer coined the phrase “conciliatory eclecticism” to describe one of the main
drivers of Leibniz’s philosophy.[103]
In a letter to Thomasius in April 1669, Leibniz explained that the Aristotelian
and the mechanical philosophies could be reconciled, and suggested a conception
of substance to this end.
Averroism was growing in strength. Leibniz was a
reader of classical Greek texts in his youth, and also read medieval writers
and Renaissance Humanists.
European-centred civilisation was extending its
knowledge of the world beyond Europe. Columbus had landed in the New World 150
years before and the Republic of Letters had access to educated persons far
beyond Europe. Leibniz was a diplomat by profession. He had international
influence, and corresponded with leaders and men of letters across Europe and
Eurasia, and in China.[104] That Leibniz corresponded with Peter the Great is not
surprising considering that a month after Leibniz met Peter the Great, Leibniz in
1712 was appointed by Peter to the Russian Justizrat.[105]
Leibniz was in such frequent correspondence with Jesuits in China that he wrote
in a letter to Princess Sophia Charlotte to whom he was close in 1697, “I will
thus have a sign placed at my door, with these words: bureau of address for
China, because everyone knows that one has only address me in order to learn
some news.”[106]
Science had recently made strides through Kepler and
Galileo. Galileo was affected by the Reason-oriented mould of Kepler and the empirico-deductive
mould of Averroism. With the success of Galileo and the influence of his patron
Paolo Sarpi, who was Atheist and an Averroist, the empirico-deductive approach
to science was being promoted in opposition to Kepler’s Reason-oriented approach
which was also represented by Leibniz’s second (after Jakob Thomasius) mentor Christian
Huygens. Invention of machinery was burgeoning in Leibniz’s day, and Leibniz
took an active interest in the design and building of machines.
We see that while Leibniz never announced his scientific
programme, he drip fed it over his career and developed his method over his
career.[107] While
his mathematics and physics were developed over time, the seeds of certain
aspects of Leibniz’s metaphysics such as the doctrine of the best of all
possible worlds and the principle of sufficient reason were present in his
earliest writings. Arguably, those earliest ideas were animated by Leibniz’s
passion to advance humanity by pursuing knowledge. Since Leibniz was born into
“the century of genius”,[108]
the seventeenth century, he found many debates and programmes to contribute to
as well as starting a few of his own. If one is looking for a declaration of Leibniz’s
scientific agenda, it is set forth most cohesively in summary form in his letters.
While the empirico-deductive approach was rising on
the Continent and in Britain, Leibniz was a torch-bearer for the Keplerian
approach to science. Continental Platonism was imbued with science, and
scientific discovery was its central concern. Their definition of the Discovery
process is part of their definition of the relationship between God, Man and
the physical universe, which is not reconcilable with empirico-deductivism.
Popularly, modernity encompasses a lessening of
importance of forms of tradition, and increasing importance of new abstractions
such as nation states and corporations.[109]
The popularly promoted idea of modernity encompasses capitalism,
industrialisation, the nation state, scientific experiment, secularization and rationalization.[110]
The Thirty Years War which ended two years after Leibniz was born is regarded
as being at the beginning of modernity.[111]
This might be because it ushered in the Westphalian system of nation states
with the 1648 Treaty of Westphalia.
Modernity is also regarded as encompassing
rationality, an absence of excessive religiosity, the use of logical deduction
and reliance on precision in taking observations especially with sensitive
instrumentation. Leibniz
promoted all of these things. He also promoted industry, science, the nation
state and rationality. At the same time, he recognised the Neoplatonic roots of
science. However, he was no alchemist or magician. Leibniz was motivated by the
power of rationality when applied to pragmatic scientific and industrial
operations. He saw even greater power in the application of lucid rationality
to metaphysics and philosophy, because it is in these intellectual acts that
the largest questions that trouble humans can be dealt with thereby allowing
great leaps in science, thence in industry and thence in the human condition.
The rise of modernity saw a
decline in the sway of religious fundamentalism and this went hand-in-hand with
the rise in the use of and high regard for rational thought. This does not mean
that people believed in God any less. Rather, what changed was the conception
of God. The rising conception of God was of a rational God whose every act is
for good reason. Leibniz tirelessly promoted the idea that the universe is the
best possible universe, because it was made by a Creator who designed it so that
it would unfold in the best possible way.
Leibniz believed that God acted through objective laws to create the
universe, and these objective laws can be understood whereby the universe is accessible
to human understanding and modification. In criticism of the philosophers
Scaliger, Sennert and Sperling, Leibniz wrote to his former teacher Thomasius,
“…they conclude that God produces creatures rather from his own active power
than from the objective and, so to speak, passive power of nothing. In their
opinion, therefore, God produces things out of Himself and is thus the primary
matter of things. But you will judge more correctly on this subject.”[112]
In his “Discourse on metaphysics” of 1686, Leibniz
writes, “when we say that things are not good by any rule of excellence but
solely by the will of God, we unknowingly destroy, I think, all the love of God
and all his glory. For why praise him for what he has done if he would be
equally praiseworthy in doing exactly the opposite?” Rather, “the eternal
truths of metaphysics and geometry, and consequently also the rules of
goodness, justice, and perfection” are the consequences of God’s understanding
rather than of his will, and God’s understanding does not depend upon his will.[113]
Essentially, geometry and harmony are beautiful in themselves not because God
has chosen them and, moreover, God chose them as they are because of his
perfect understanding.
Leibniz’s opposition to Henry More’s hylarchic animism
includes, among other arguments, that the idea is superfluous,[114]
is not distinctly conceived and leaves many things to be explained unlike the
commonly believed notion of individual souls.[115]
Leibniz says that space is not a real absolute being and is certainly not God
himself. Moreover, since space has parts, it does not belong to God. Leibniz
then says that space is merely relative as time is, and that it is merely an
order of coexistences as time is an order of successions.[116]
Newton expresses a kind of hylarchic animism similar
to Henry More’s. By contrast, Leibniz has said that God is not in the universe
and the universe exists independently of God. Newton, on the other hand, writes
that God is omnipresent, not only virtually but substantially. Newton says that
God constitutes space and time. Leibniz says that space and time do not even
exist, but are creations of the human mind.[117]
Of God, Newton writes, “He endures for ever, and is everywhere present; and by
existing always and everywhere, he constitutes duration and space. … He is
omnipresent, not virtually only, but also substantially, for virtue cannot
subsist without substance. In him are all things contained and moved; yet
neither affects the other: God suffers nothing from the motion of bodies;
bodies find no resistance from the omnipresence of God.”[118]
Of course, Newton is self-contradictory because he also posits a mechanical
clockwork universe,[119]
and Leibniz disagrees with this side of Newton too:[120]
Sir Isaac Newton, and his followers, have
also a very odd opinion concerning the work of God. According to their
doctrine, God Almighty needs to wind up
his watch from time to time: otherwise it would cease to move. He had not, it
seems, sufficient foresight to make it a perpetual motion. Nay, the machine of
God’s making is so imperfect, according to these gentlemen, that he is obliged
to clean it now and then by an
extraordinary concourse, and even to mend
it, as a clockmaker mends his work…
The idea that God acts through objective laws is
fundamental to modernity,
and a defining feature of it. The objective action of God’s mind through
universal principles that are discoverable by human minds makes the universe
the legitimate and natural locus of proactive intervention, change and
improvement by humans. This marks the onset of modernity. Of course machines
(modes of mechanised work and transport)
and increasingly powerful energy sources are media for such human proactive
action.
Modernity often refers to lifestyle or amenities of
modern life being made available to as wide a section of the population as
possible. It can also mean a departure from superstition and religious
fundamentalism, and an embrace of rationality. This implies the ability to
answer questions using a thought process rather than dogma, which is a rather
Platonic-dialogue oriented technique. It is a rational process whose pursuit is
open to any human mind, and that process can lead if not to a final answer then
at least to greater clarity and an incomplete answer. Leibniz took this further
with his quest for a Universal Characteristic which would allow questions to be
answered using a calculating procedure. This, however, was distinct from the
“art of discovery”.[121]
The Universal Characteristic would make a process of reasoning available to all
people who had a basic education and were armed with pen and paper.[122]
It would be a calculus which depends on the analysis of ideas, and which is
more important than the calculi of arithmetic and geometry. He says, “its
formation seems to me one of the most important things that can be undertaken.”[123],[124]
To a 21st Century
mind, the idea that the earth revolves around the Sun rather than vice-versa
seems a “modern” one. However, what is perhaps the feature that more powerfully
gives it the distinction of “modern-ness” is the fact that a human mind
discovered it through reasoning and by considering the paradoxes in the
previously prevailing belief system. Praise for this goes to Kepler. However,
one of the most salient features of Kepler’s reasoning is hypothesis. Of
course, Kepler was an able mathematician. Yet, fundamental to Kepler’s process
of reasoning was a presumption of harmony, the formulation of a hypothesis, and
then the undertaking to test the hypothesis and check its complete conformance with observations.
Kepler and Galileo’s outlook typified the “new
mechanical philosophy”[125]
(“NMP”) which was part of the
development of Rationalism which is to be discussed below in Chapter 3. Brown
reports that Erhard Weigel who taught some mathematics to Leibniz at the
University of Jena shared with Leibniz a desire to reconcile Christianity,
Aristotle and the NMP. While Weigel regarded the NMP of Kepler, Galileo and
such like as an extension of Aristotle,[126]
we will later argue that Rationalism is better regarded as an evolution out of
Neoplatonism rather than as an extension of Aristotle.
Kepler and Galileo’s outlook typified the “new
mechanical philosophy”[127]
which developed hand-in-hand with Rationalism, which will be discussed further
in Chapter 3. Brown reports that Erhard Weigel who taught Leibniz mathematics
at the University of Jena shared with Leibniz a desire to reconcile
Christianity, Aristotle and the new mechanical philosophy. Weigel might have
seen the new mechanical philosophy of Kepler, Galileo and such like as an
extension of Aristotle. However, we will later argue that Rationalism was for
the most part an evolution of ideas from Neoplatonism.
What is critical is the pre-thought that went into his hypothesis
formulation; clearly, Kepler did not pick a hypothesis “out of the air”. Kepler
had metaphysical presumptions in how the universe must be designed, and in what
the Creator’s intention must have been in designing the universe arising from
his overall understanding of God, geometry, Man and Man’s place in the
universe. An interconnection between God and geometry was present in Kepler’s
thoughts, and to that he owed a debt to the Pythagoreans and Plato. What did
Kepler add to the traditions of Pythagoras
and Plato? Why could Pythagoras and
Plato not have completed Kepler’s work? Perhaps they “only” lacked the observational
data that Kepler had access to, and the numerical methods work of Tycho Brahe
and Kepler himself. Of course, the data and methods involve enormous suites of
resources. However, arguably, the Pythagoreans and Plato worked with a similar
metaphysical orientation and set of assumptions about the nature of the
universe to Kepler.
Leibniz recommended Kepler’s
a priori method, “The most perfect method involves the discovery of the
interior constitution of bodies a priori
from a contemplation of God, the author of things. But this method is a
difficult one and not to be undertaken by anyone whatever.” Further, “Some
hypotheses can satisfy so many phenomena, and so easily, that they can be taken
for certain. Among other hypotheses, those are to be chosen which are the
simpler; these are to be presented, in the interim, in place of the true
causes.”[128]
We cannot say that the
ancient Greeks were not used to formulating hypotheses. Archimedes might have
done so mentally without committing it to writing. Archytas might have done so in
working out how to double the cube.[129]
Most of all, Plato in his dialogues makes an art and game of proposing hypotheses
for testing.[130]
It is clear in Leibniz’s work on the mine pump, simply
through his multiple attempts, that he must have followed a process of
hypothesis testing. Similarly with Papin in his letters to Leibniz, and his
several steam engine design proposals to various persons of significance.
Leibniz was familiar with hypothesis formulation in Astronomy, perhaps directly
from reading Kepler.[131]
A cohesive reason-based metaphysics is associated with
the foundations of modernity – counterposed against, say, a literalist reading
of religious scriptures. Such a metaphysics provided preconceptions about the
structure of the universe. The current discussion surrounding cosmic radiation
and the lack of empty space is forcing re-examination of the foundations of
physics.[132] Yet Leibniz
himself had much to say on the non-emptiness of space,[133]
and much of it was derived from the principle of sufficient reason which
emanated from a quasi-theological position on how God necessarily must think.
An absence of superstition
was seen in many cultures prior to extended European civilisation. Thus,
modernity cannot be defined by a lack of superstition or even by a lack of
reason. Even the modern era’s broad-based facility of reason among the populace
is not definitive because there have been entire civilisations prior to the
present in which a large proportion of the population used methods which today
could be described as scientific or at least with ingenuity, with regard to
astrogation, navigation, ship-building or boat-building, fishing, domestication
of animals, etc. Thus, modernity is simply this civilisation’s manifestation of
reason/rationality. There are parts of the world in which superstition can be
found even today. However, there were, to varying degrees, parts of the world –
national cultures, say – certainly through the 19th and 20th Centuries
in which reason was dominant. Of course, on the other hand, irrationality even
exists in the Western world in the 21st Century even in professional
and social circles with a high-level of formal education.
Leibniz had something to do
with the establishment of a dynamic which saw the broad-based influence of
reason on culture and the infusing of post-17th Century culture with reason. Leibniz’s letters
to national leaders, such as the Czar of Russia, indicate a concern to uplift
the minds of the population as a whole. Leibniz’s personal drive in this
direction is illustrated by Kutateladze when
he describes the debt owed by Russian science to Leibniz:[134]
Science
in Russia had started with the foundation of the Academy of Sciences and Arts
which then evolved into the Russian Academy of Sciences of these days. The turn
of the sixteenth and seventeenth centuries is a signpost of the history of the
mankind, the onset of the organized science. The time of the birth of scientific
societies and academies accompanied the revolution in the natural sciences
which rested upon the discovery of differential and integral calculus. The new
language of mathematics brought about an opportunity to make impeccably precise
predictions of future events.
To
the patriotism of Peter the Great and the cosmopolitanism of Leibniz we owe the
foundation of the Saint Petersburg Academy of Sciences as the center of Russian
science. Peter and Leibniz stood at the cradle of Russian science in much the
same way as Catherine I and Euler are the persons from whom we count the
history of the national mathematical school in Russia. We must also acclaim the
outstanding role of Leibniz who prepared for Peter a detailed plan of
organizing academies in Russia. Leibniz viewed Russia as a bridge for
connecting Europe with China whose Confucianism would inoculate some necessary
ethical principles for bringing moral health to Europe. Peter wanted to see
Leibniz as an active organizer of the Saint Petersburg Academy, he persuaded
Leibniz in person and appointed Leibniz a Justizrat with a lavish
salary.
This is an early conception
of the General Welfare of the population of a nation which was also found in
the American Declaration of Independence and Constitution. Leibniz’s New Essays Concerning Human Understanding
in response to John Locke’s Essays on
Human Understanding indicates this contention of “life, liberty and the
pursuit of happiness” versus “life, liberty and property”.[135],[136] (In the
next chapter, we shall meet Paolo Sarpi whose influence Leibniz opposed.
Robertson writes that “there can be no doubt” that Sarpi “anticipated Locke in
the sphere of metaphysics” and that “there is reason for thinking” that Sarpi
“supplied Locke with the germs of many of the ideas which we find expanded” in
Locke’s writings.[137]) In this
way, Leibniz’s thought was instrumental in the creation of national cultures
which fostered the pursuit of knowledge, invention and an expanding scientific
understanding by the participation in these activities of as many members of
the population as possible. Leibniz’s idea was not only that society and
nations should be organised to promote these things but that these pursuits
were the main and perhaps only reason for the existence of society and nations,
and are the end around which all political activity and legal principles for
organising people should be directed.
Let us explore the idea that Reason and its
application are at the heart of modernity. Before doing so, a digression on
what is meant by “reason” is needed. In particular, reason is counterposed to
deduction, and more will be said on the distinction in Chapter 6 “Discovery and
deduction”.
Reason as Plato exhibited it in The Republic is
not logical deduction for the character Socrates raises new points from “left
field” that could not have been deduced and thereby commence entirely new lines
of enquiry and, often, confounds his interlocutors.
We first introduce Cardinal
Nicolaus of Cusa (1401-1464) because he revolutionized the understanding of human reason,
the pursuit of knowledge by humans, and helped define the domain of what is
knowable or in which humans seek knowledge.
Cusa was educated in a
region of Germany where the Brotherhood of the Common Life was active though possibly
not in a school directly run by the Brotherhood. An overview of the Brotherhood
and its circles will help place the reader in the milieu in which Cusa worked.
The Brotherhood was a
religious order founded by Gerard Groote (1340-1384). In 1374, Groote, at the age
of 34 having hitherto lived in relative luxury, experienced religious
conversion following a sickness, and entered the Carthusian monastery as a
guest to participate in their severe regimen of prayer, fasting and manual
labour. Ultimately he departed from the Carthusians but thenceforth lived an
austere life of prayer, study and preaching. Groote was from a wealthy merchant
family. After his parents died as victims of the Black Plague, he left one of
the estates he inherited to the Carthusians. He ceded part of the house in
which he lived to poor women so establishing a community known as the Sisters
of the Common Life. The sisters supported themselves through agricultural and
artisan pursuits, and gained great expertise in agriculture and sewing. They
built a flourishing dairy business and earned an impressive income from sewing
and knitting. Discipline and obedience to the two matrons was expected. The
Sisters became a powerful centre of Church reform.[138]
Groote also established a
monastery for the Brotherhood of the Common Life, which provided superior
education to capable children regardless of family background. The Brotherhood was
dedicated to educating capable children in the Greek classic and in the
Christian tradition of St Augustine. The Brotherhood ultimately established
over a hundred schools across Europe and influenced many more. The order sought
to provide education regardless of background and sought to transform the
potential of the population of Europe in the wake of the Black Plague.
Erasmus of Rotterdam also had an education heavily
influenced by the Brotherhood. Erasmus along with others created the Devotio Moderna movement which catalysed
a revolution in the Church. This “New Devotion” or “Modern Piety” ultimately
swept up such personalities as Luther, Calvin and Loyola.[139] While Nicolaus of Cusa was
only peripherally connected with the Devotio
Moderna,[140] he played an equally
important role in history.
In 1418 the Brotherhood was
charged with heresy. Jean Gerson the former chancellor of the University of
Paris defended the Brotherhood, making appeal to the Christian concept of imago viva Dei and the duty of Christians
to act in the imitation of Christ. Gerson’s further significance is underlined
by the fact that he later wrote the educational curriculum for the young Louis,
the future King Louis XI of France, with emphasis on the study of St Augustine’s
City of God. Thomas á Kempis
(1380-1471) was a follower of Groote, and expressed the ideals of the Brethren
in The Imitation of Christ
(c.1418-1427). Nicolaus of Cusa’s theology has been called that of Thomas á
Kempis but in philosophical language.[141]
The Brotherhood and its
supporters were significant in the movement that supported the independence of
France. It might not be an accident that Jeanne d’Arc who was so central in
liberating France from England grew up in Domremy adjacent to German towns
where the Brotherhood was active.
Nicolaus of Cusa, usually
referred to as “Cusa”, was a very significant reformer and activist from his
position within the church hierarchy. Cusa wrote the Concordata
Catholica in 1433 during the Council of Basel which ultimately failed. In the
Concordata, the concepts of human
rights and national sovereignty can be found in seminal form. More than 200
years later, the concept of national sovereignty became the basis of the 1648
Treaty of Westphalia. What is now known as the “Westphalian principle of
national sovereignty” is the foundation for international relations in the 21st
Century.
Cusa was active in initiating the Council of
Florence which brought together the Roman and Eastern Orthodox Churches, and
concluded in 1439. It was due to Cusa that the doctrine of the Filioque (i.e. “and from the Son”) was
agreed at the Council of Florence. This is the doctrine that the Holy Spirit
proceeds from the Father and from the Son, not only from the Father. This
ensures a greater accessibility of the Holy Spirit to humanity.
While God an infinite being created the universe,
humans have access to ideas concerning the infinite. Humans can get closer to
God, and the human mind can always get a better grasp of Creation. This is the
same as the consequence of the doctrine of the Filioque, that humans are able
to come closer to God. Indeed, the Holy Spirit proceeds from the Son to reach
us. The spiritual relevance might be more obvious. However, in the intellectual
and scientific sense, the human mind can progressively come to better
understand the universe, and this is nothing but the process of scientific
discovery. Cusa explained the competence of human thought for scientific
understanding even of God’s creation as a whole though his work De Docta Ignorantia which he defended in
correspondence with Johannes Wenck against attacks by Wenck.
Most significant for this thesis, Cusa conceived the
role and position of mind in relation to Creation or the physical universe, so
establishing an epistemological foundation for science. The premise of De Docta Ignorantia is the relationship
of the finite with the infinite. The human mind is finite compared with the
infinity of God. The physical universe, which is God’s creation, is also
infinite. This might not mean that the physical universe is infinite in size,
but in possibilities. Thus, the human mind can never fully understand the
universe nor can it fully grasp God. However, this does not mean that there is
no point in trying. On the contrary, there is every point in trying. Since a
circle is like an infinite-sided polygon, a polygon with a finite number of
sides can never match it but it can get progressively closer.[142] This is an analogy only,
but one that is as useful as it is simple to fight back against those who argue
that humanity is condemned or destined to be passive or ineffective in a
universe which is governed by the diktat of a God who can never be understand.
It is a powerful argument against regarding God and the forces of nature as
playing with human in an arbitrary or whimsical way.
In explaining the human ability to formulate a
“higher hypothesis”, the analogy of the circle is used again. By remaining
within the framework of existing thinking, we behave like a geometer who
restricts himself to a regular polygon. How can he achieve the smooth circle? The
circle is not an infinite-sided polygon but a qualitatively different object. A
higher power, so to speak. Humans have the ability to transcend the existing
“system of regular polygons” and move up to the circle thus solving the
conundrum of the day, be it political, artistic, social, cultural or – what is
most relevant to this thesis – scientific.
How does one find or move to that higher
hypothesis? The ability to do so is fundamental to progress of all kinds. Thus,
renaissances are typically renaissances of human thought, and progress explodes
not in one but in all domains: political, artistic, social, cultural,
scientific. Indeed, the separation between these domains is artificial, for all
are refined and advanced by the capabilities of the human mind.
Cusa might have been the first to say that
inspiration is a valid source of scientific insight and, ultimately, of new
knowledge. Indeed, Cusa argued that it is the only source of new knowledge, and
he counterposed it to discursive logic. Translations of Cusa, such as Jasper
Hopkins’ translation, use “discursive reasoning” but the context makes clear
that “discursive logic” or even “deductive logic” was his meaning.[143]
Kepler consciously used
Cusa’s prescription in formulating his hypotheses for the working of the solar
system yet we do not see inspiration in Kepler so much as an a priorist belief
in universal harmony combined with impressive rigor or discipline in his
calculations.[144]
The testing of
the hypotheses was conducted largely mathematically. Leibniz’s method in
discovering the calculus was very different, but he was equally supportive of
the use of hypothesis. Meli defends Cassirer’s thesis of 1902, that “the
legitimacy of hypotheses in natural philosophy and mathematics was defended by
Leibniz exactly as Kepler had done in astronomy. In their philosophical systems
phenomena assume a new dignity and the true hypothesis becomes the instrument
for binding them to the laws of knowledge.”[145]
Cusa was the first to say
that the universe is infinite.[146]
Bruno who was born 84 years after Cusa died also proclaimed the infinity of the
universe. Fifty years after Bruno’s death, Henry More was gripped by the idea
of the infinity of the universe, not necessarily adopted from Cusa or Bruno
directly, and in particular the infinity of space. As mentioned elsewhere in
this thesis, the infinity of space inexorably led Henry More to the idea that
God is in space. Newton was in partial agreement with these conceptions of More.
Kepler consciously used Cusa’s
prescription in formulating his hypotheses for the working of the solar system
yet we do not see inspiration in Kepler so much as an a priorist belief
in universal harmony combined with impressive rigor in his calculations. The
testing of the hypotheses was conducted largely by calculation. Leibniz’s
method in discovering the calculus was very different.
The gaining of understanding and following it up with
the potent exercise of mind on the physical universe brings us into the domain
of modernity. This certainly was not something that had never been thought
before. For example, Plotinus (3rd Century C.E.) wrote that since Intelligence
is the base of all, understanding is a prerequisite to the exercise of noös
(spirit) on söma (body).[147]
The gaining of understanding and following it up with the potent exercise of
noös on söma brings us into the domain of modernity. Central to modernity was
insight into and consequent control over the physical universe. Kepler’s
purpose was to better understand Creation. For a human to even consider that it
was for them to understand the universe could have been regarded as blasphemous
arrogance that put him/her on the same footing as God. Kepler’s theological
justification, perhaps to keep himself out of trouble with church authorities,
was that by better understanding God’s creation we can better glorify God.[148] Similarly, Leibniz wrote that “the greatest
usefulness of theoretical natural science, which deals with the causes and
purposes of things, is for the perfection of the mind and the worship of God.”[149]
What Kepler did not say was that scientific development contributes to the
public good, perhaps because Kepler’s work in understanding the heavens did not
have a clear pragmatic application in his time. Leibniz could see that science
and engineering contribute to the public good, and he united Kepler’s position
with the public good, saying, “To contribute to the public good and to the
glory of God is the same thing.”[150]
Understanding principles of
nature precedes surpassing or controlling nature, and the motivation for
gaining that understanding is to surpass and control nature. As mentioned in
the previous chapter, it allows us to ape nature and thence master it. That it
is natural for humankind to do so was also a consequence of the
Egyptian/Hermetic conception of Man as magus who can control even the stars. That
it is humankind’s right to do so was the message of the Renaissance bound up
with Christian theology, resulting from the conception that humanity is made in
the image of the Creator.[151],[152] Leibniz
explained, “It seems to me that the aim of all humankind should chiefly be nothing
other than the knowledge and development of the wonders of God and that it is
for this reason that God has given humankind dominion over this globe.”[153] Humanity
consciously using its superior status over all of nature is the defining
characteristic of modernity and
crystallises what the Renaissance worked towards.
Speculation
and contemplation, including the exercise of reason, do not on their own define
modernity. Whittaker says “the happiness involved in [the speculative life] ...
is regarded as something that necessarily goes with mere thinking and
understanding” in reference to “a self-conscious theory of [the speculative
life as set forth] ... as at the opening of Aristotle’s Metaphysics”. However, there is no mere thinking or understanding,
for action does not stop there.[154] Understanding
is a prerequisite to:[155]
(a)
Organise men
justly and accordingly arranging government in the best way possible which is
no small thing because it can mean the difference between violent tyranny, for
example, and enlightened republic, and
(b) Build machines, ships and
cities, undertake agriculture, and modify the physical universe in significant
ways and even on a vast scale.
There is a “difference between nature and art, that is
to say, between the divine art and ours”. Machines built by God or nature are
machines in their smallest parts ad
infinitum. By contrast, the constituents of, say, a wheel give no
indication of the use for which the wheel is intended and indeed are not
machines – and certainly are not alive – in any real sense.[156]
Modernity has seen increasing – and in some cases exponentially increasing –
power of “human art”. Leibniz assisted this process with his concept of vis viva.
Normally, vis
viva belongs in a discussion on the history of physics. It is known in the
context of the debate between Leibniz and the Cartesians on the kinetic energy formula
mv2 versus the Cartesian
momentum mv (i.e. mass times speed,
or m|v|)
which was corrected to mv (mass times velocity) in 1668 by
the non-Cartesians John Wallis, Christopher Wren and Christian Huygens.[157]
Vis viva which literally means
“living force” is today regarded as kinetic energy quantified by mv2. It will be seen below that
vis viva meant more Leibniz than this
formula. The concept certainly was the subject of debate, and Leibniz fired the
opening salvo by attacking the Cartesian concept of mv in his 1686 essay
entitled “Brief Demonstration of a Notable Error of Descartes”.[158]
The discussion fits in this chapter, “Ushering in modernity”, because for
Leibniz the effect that a projectile or explosion could produce was what
mattered and that, in turn, was more a result of velocity than of mass though
mass plays a role.
Leibniz is not only concerned velocity but also with
acceleration and the effect produced by sudden deceleration. Leibniz writes,
“by effect I mean here not any effect
whatever but that for which force is expended or consumed and which may
therefore be called violent.”[159]
(emphasis in original) Leibniz then explains what kind of effect he does not
mean and which machines are, by comparison, “harmless”. Leibniz says, “the
ancients had a knowledge of dead force only, and it is this which is commonly
called mechanics, which deals with the lever, the pulley, the inclined plane
(…), the equilibrium of liquids, and similar matters concerned only with the
primary conatus of bodies in itself, before they take on an impetus through
action. Although the laws of dead force can be carried over, in a certain way,
to living force, yet great caution is necessary, for it is at this point that
those who confused force in general with the quantity resulting from the
product of mass by velocity were misled because they saw that dead force is
proportional to these factors.”[160]
According to Meli’s interpretation, “for Leibniz living force could be
represented as the integral of dead force times an infinitestimal distance”.[161]
Leibniz agrees that mv is a valid concept with
many uses and is only explaining that mv2 is something
qualitatively different. To compare the two, Leibniz writes, “The force which a
heavy body exercises in moving along a perfectly horizontal plane is not of
this kind [living force], because however far such an effect is prolonged, it
always retains the same force, and though we use the same principle in
calculating this effect also, which we call harmless,
we now exclude it from consideration.”[162]
(emphasis in original) Leibniz then goes on to discuss how the “force”, or
kinetic energy as we would call it today, of a tiny projectile is absorbed by
and affects even the largest of bodies.
The debate between conservation of momentum or mv
and conservation of kinetic energy or mv2 was an important
aspect of the vis viva discussion.
However, arguably, to Leibniz’s mind the more powerful “take away” was the
power of explosive force versus the relative “harmlessness” of “dead force”
exemplified by the uniform motion of a heavy body along a perfectly horizontal
plane. The question then becomes – how do we generate and make us of explosive
force? This will lead us into the below discussion of steam power.
First, however, understanding the principles of “effect”
is prerequisite to machine-building or mechanical engineering, since a machine harnesses
that effect in a deliberate way. In turn, modernity embraced the task of consciously
harnessing the laws of physics through machines or otherwise to do useful work
in the service of human ends. This is in the same way that the calculus is the
use of symbols as tools of thought in the service of human ends.
Leibniz’s concern with the ability to
do useful work is shown where he argues that the difference between mv and mv2 was:,[163]
not worthless to consider, nor are they
quibblings over words, for they are of the greatest importance in comparing
machines and motions. For example, if power is obtained from water or animals
or from some other cause, by which a weight of 100 pounds is kept in constant
motion so that within a fourth of a minute it can be made to complete a circle
of 30 feet diameter, but someone else maintains that a weight of 200 pounds can
in the same time complete half the circle with less expenditure of power, his
calculation seems to yield a gain; but you ought to know that you are being
deceived and getting only half the power.
While this argument is straightforward today, it
pitched into an ongoing discussion among leading scientists and engineers of
Europe. Experiments by Mariotte, Poleni and ’sGravesende confirmed that the
effect of a moving object varies in proportion to the square of velocity in the
early 18th Century. These experiments were performed with balls
accelerated to different speeds being fired into clay or wax, with the depths
of the impressions made being measured. Of these, ’sGravesende took the most
active role in the vis viva debate.
In 1720 in his book Mathematical Elements,
’sGravesende had taken up the question of how to measure the ability of an
“action of power” to overcome obstacles.[164]
This is the Leibnizian concern with the effect
of a force or “power”. In 1722, ’sGravesende proceeded to defend Leibniz’s idea
of vis viva.[165]
In 1733, Poleni had published an article reporting the results of an experiment
in which balls were dropped onto tallow to compare the depths of the
impressions made. His analysis showed that the “force in motion” was
proportional to the square of the velocity. A Swiss mathematician Calandrin
published an anonymous article in the same issue of the same journal saying
that since the resistance of the tallow was constant, equal force should be
consumed in equal times. Feeling compelled to respond to the Calandrin article,
’sGravesende replied disagreeing that equal amounts of “force” are consumed in
equal times. The question of why the impressions left in clay by a cylinder in
motion were not proportional to vis viva
was taken up, as this seemed to contradict the theory. ’sGravesende argued that
the early deceleration split seconds after striking the clay and making the
initial impression reduced the ability of the cylinder to drive deeper into the
clay. This was confirmed by experiments with a series of parallel adjacent taut
strings that are struck by a projectile. More strings are broken per unit time
while the object is moving rapidly than when it has slowed down.[166]
’sGravesende noted that collisions are not
instantaneous but proceed in a continuous fashion causing the gradual
deformation and deceleration of the bodies upon and after collision or, during collision. ’sGravesende explained
that, likewise, in a simple machine a small mass may counterbalance a larger
provided it is moving proportionately faster at the instant the machine starts.
According to Hankins, the most aggressive detractors of Leibniz’s concept of vis viva d’Alembert and Boscovich missed
key points and, in retrospect, made little progress.[167]
The purpose here is to introduce the vis viva concept and suggest how it fit
in Leibniz’s overall interests and unfolding agenda.
Machines increasingly were being invented and used during Leibniz’s
lifetime. Essentially, control of man over nature was on the rise. A machine
systematises a physical process like a Universal Characteristic would
systematise a mental or intellectual process, and Leibniz was interested in
both. The two intersected in the calculating machine that Leibniz designed. In
a sense, the calculus is an example of a specific kind of Universal
Characteristic because a mechanical algorithm is availed to answer questions
about a huge class of (non-linear) functions.
Leibniz collaborated with
Denis Papin on his invention of the steam engine, providing input including
relating Papin’s work back to Leibniz’s concept of vis viva and also providing pragmatic mechanical suggestions.
Christian Huygens had
mentored Papin[168] and
Leibniz.[169]
Papin had worked as Huygens’ ammenuensis
(“scribe”) and had built machines under his direction.[170] Huygens
had invented a steam pressure cooker powered by gunpowder. Leibniz then worked
with Denis Papin to invent the steam engine. This may have helped Leibniz
formulate and crystallise his idea of vis
viva. Though there seemed later to have been a falling out between Papin
and Leibniz, apparently due to resentment on Papin’s side for unknown reasons.[171] Leibniz’s
participation represents his involvement in systematising methods for modifying
or improving the human condition by effecting change in the physical world,
whether by hauling earth or ore from a mine shaft, pumping water from a mine
shaft, or accelerating a transportation vessel rapidly over the sea.
The dawn of modernity was characterised by the rise of human commandeering of machine power, which was a result of contemplation and experiment to uncover physical principles on the one hand, and deliberate design as well as trial-and-error with machines on the other. The power of machines would hardly exceed what Archimedes’ machines were able to do without the vis viva or concentrated, violent or explosive force which Leibniz’s advocacy forces us to consider as distinct from passive force.[172]
The association between modernity and the rise of
machines is also represented in Leibniz. Leibniz’s role in designing machines receives little
mention in the Leibniz literature, in favour of his mathematical work,
especially the calculus, and his metaphysical thought, especially monads. Yet,
Leibniz himself wrote that if he could spend time on whatever he wished, then
he would design machines full-time.
Papin’s steam engine design
was published in the Leipzig Acta Eruditorum, a journal edited by
Leibniz, of August 1690 with the diagram in Figure 1.
Papin was using the idea of
explosive force as a continuation of Huygens’ experiments with gunpowder.
Leibniz concept of vis viva was
developed after Leibniz was aware of Huygens’ work, and was an effort to give
the different kinds of force a theoretical basis. There are letters from
Leibniz to Papin on the vis viva
idea.[173] The question of priority is not of interest, only the
fact that Leibniz was involved in the earliest experiments with steam power.
Figure 1: Denis Papin’s Atmospheric Steam Engine from Acta Eruditorum August 1690
with caption from Klemm, p.221
Leibniz’s mathematical work is usually if not always
considered separately from his interest in mechanics. Leibniz was interested in
the problems of mechanics (time, distance and motion) precisely due to his
interest in machines. He had pragmatic ends in mind. In fact, the demarcation
between mathematics and mechanics is artificial and probably was not known to
Leibniz in the way that it is presumed by modern readers.
The idea that Leibniz started with abstract curves and
then found these to be helpful in problems in the domain of mechanics has some
truth, [174] but for the most part Leibniz was thinking about
mechanics all the time and from the start. Leibniz from the outset regarded
geometry and his calculus as worthwhile pursuits because of what they add to
human capability in society and in the physical world. Grosholz refers to
Leibniz’s synthesis of different domains, “Because his [Leibniz’s] analysis
deals with magnitude in general, it can also apply to mechanics – to distance,
time velocity, and force; because it does not eschew the infinitary, it can
apply to continuous motion.”[175]
Later on the same page, Grosholz says, “A fuller integration of mechanics and
mathematics depends on precisely the reorganization of mathematical domains”.
Further, “Leibniz’s synthesis does not take place merely because he employs
abstract algorithms that can be instantiated in different domains, but also
because the synthesis engenders hybrids that exist simultaneously at the
overlap of different domains.”[176]
In fact, Leibniz is always working in both domains, and does not really see the
difference between them. Grosholz says, “Transcendental curves like the
tractrix and the catenary, and the ellipse re-imagined as a trajectory, will
illustrate my point.”[177]
However, these curves also illustrate the opposite point – that Leibniz’s work
in mathematics was work in mechanics, which was ultimately motivated by his
interest in machines, at the same time.
What are regarded as different domains in the late
20th and early 21st Centuries may not have been in Leibniz’s time, when it was
natural for thinkers to be across multiple “domains” most of the time, as
indeed most were.[178]
In describing the
constellation of scientists influenced by Mersenne, which included Rene
Descartes, Pierre Gassendi, Gerard Desargues, Pierre de Fermat, Gilles Personne
de Roberval and Galileo Galilei, Cohen wrote that they were trained in a “liberal approach to study that embraced many fields”
and “were, in fact, philosopher-scientists whose attitudes pervaded the times
and helped provide a very special basis for the establishment of the Paris
Academy”.[179] In a similar vein, Bell
describes the intellectual environment of Christian Huygens,[180] the first
president of the (French) Royal Academy.
The breadth of scientists in
Leibniz’s time went even further than Cohen and Bell have said. Scientific and
philosophical thought was connected with the necessities and potentialities of
social and political life. Indeed, Mercer constructed the theory of
“conciliatory eclecticism” to explain the philosophy of both Leibniz and his
teacher Jakob Thomasius as emanating from the religious conflict that gave rise
to the Thirty Years War (1618-1648) which Thomasius lived through. Leibniz was born towards the end of
that war, in 1646.[181]
In a sense, the Thirty Years War was a
continuation of European religious strife in the vein of the foregoing century.
That religious conflict affected intellectuals throughout Europe and gave a
greater sense of mission and urgency to their deliberations and activism. It
certainly was a motivation for Bruno in the 16th Century, many
decades even before the Thirty Years War. “Wherever Bruno travelled within
Europe, doctrinal intolerance and endemic slaughter in the name of God
reassured him that only a spiritual and intellectual revolution could ever
disassociate religion from murder, horror, and endless pain.”[182]
Yates has written extensively on Catherine de Medici’s use of talismans and
astrological rituals for the “pacification of the wars of religion”.[183]
In any case, as far as Leibniz’s contemporaries were concerned, the Thirty
Years War had a major influence on the political and theological thinking of
the time, much as WW II influenced political and economic thought for the early
post-WW II decades.
In fact, Leibniz in Theodicy did
not refrain from addressing the protestant side of the Thirty Years conflict:[184]
Luther’s book against Erasmus is full
of vigorous comments hostile to those who desire to submit revealed truths to
the tribunal of our reason. Calvin often speaks in the same tone, against the
inquisitive daring of those who seek to penetrate into the counsels of God. He
declares in his treatise on predestination that God had just causes for damning
some men, but causes unknown to us. Finally M. Bayle [Pierre Bayle, 1647-1706] quotes
sundry modern writers who have spoken to the same effect (Reply to the
Questions of a Provincial, ch. 161 et seq.).
Leibniz refers to Pierre Bayle (1647-1706). Bayle
typified some aspects of the time and is regarded as one of the most prominent
men of letters of the 17th Century.[185]
As a Protestant in France, Bayle suffered and fled religious persecution living
much of his life as a refugee in Holland.[186]
Bayle’s best-known work, the Dictionnarie
historique et critique (“Historical and Critical Dictionary”), published in
1696, comprises over 2500 articles on people from before Christ to Zeno to
Hobbes. The Dictionnarie is sometimes
called the “arsenal of the Enlightenment” because it was used by activists and
writers for material for their respective arguments. It is estimated to be the
“single most popular work of the eighteenth century”. The issues of religious
tolerance, respect for the conclusions of the consciences of others, and the
problem of evil were central concerns in Bayle’s writings.[187]
Unfortunately, we do not have the space or time to examine Bayle’s ideas
further here.
Johnson had this same thought:[188]
Leibniz lived during a period of
intense crisis. Dynamic individualism, in Church and State, threatened to
reduce the life of Europe to chaos. There was a desperate need to achieve a
harmony which would not destroy the fruitful forces of individualism. Leibniz
set himself to formulate a comprehensive philosophy which would serve as the
intellectual foundation for a new age which would facilitate fulfilment of the
best dreams of scientists and practical men, saints and sages. Leibniz did not
philosophize in a logical vacuum.
Loemker says that Leibniz had hoped that his
metaphysics “would be adopted and made a blueprint, so to speak, by men of good
will (honestas) for the restoration of European order” and it may “be
regarded as the intellectual high point of the century’s efforts toward a
renovation of Europe through the ideal of loyalty and obedience to a universal
nature and moral order.”[189]
Loemker points out that Descartes and Spinoza, like Locke, saw no imperative to
particular social rules. Descartes separated the objective from the subjective
in his two-level order of, on the one hand, creator and, on the other hand,
created “which must interact, yet cannot interact since they have no common
nature which could make this possible”. Spinoza separated God from “the
everyday evaluations of man” and is “accessible only to the mind which is
completely disciplined”. However, life in society “is recommended on the
commonsense ground of utility and conservation [preservation?], and such active
affections which contribute to them. There is no great overarching order of
commands whereby the social order is to be regulated”.[190]
We will see below that Leibniz distinguished salutary social influences from
detrimental ones not purely on the grounds of utility and preservation. This
was bound up with Leibniz’s conception of the nature of Man and the unique role
and position in the universe allocated to humanity by God.
In discussing the analysis of Grosholz, we review the
thread from the pre-moderns to modernity.
Grosholz finds it impressive
that Leibniz’s calculus could be applied to enrich Kepler’s work on the solar
system. Yet, it would make sense for Leibniz to have considered the problems
involved in Kepler’s grappling with the structure and harmony of the solar
system as a proxy for other problems in science and machine design. Planetary
motion was one of the active “research areas” of the period, a bit like
“climate change” was in the early 21st Century.
Thus, it is unlikely that Leibniz simply stumbled on the fact that his calculus
was useful for questions relating to planetary motion; Leibniz would have been
aware of Kepler’s contribution and of open problems left by Kepler. Huygens
would surely have mentioned the problem of planetary motion and Kepler’s work
to Leibniz. Further, the fact that Leibniz read so many of the Greek classics
in his teens makes it almost certain that he’d have been aware of the problems
in understanding the solar system which could have drawn him to Kepler who
represented the state of the art on the question of planetary motion in Leibniz’s
time.
Grosholz raises a supposed contradiction at the end of a paper on Leibniz[191] as a conundrum. Grosholz says that modern problems in physics such as relativity theory and quantum theory, pose grave philosophical problems because if we call them part of mathematics, and so mere patterns and relational structures, we cannot explain how they can play the role of the furniture of the universe, as in contemporary physical theory they surely do. If we call them part of the description of nature, we must wonder how nature has come to look so exact and immaterial. If we call them hybrids, they seem to contain an internal contradiction.[192]
Grosholz has raised nothing
less than the ancient problem of duality. That is, what is the relationship
between abstract mathematical structure which is outside and separate from the
physical universe, and the physical universe which seems to realise or embody
abstract mathematical structures. This problem was first dealt with no later
than Plato (c.427-c.347 BCE). According to Plato, arithmetic comes first whence plane geometry is
derived, whence solid geometry. On these foundations sit astronomy and harmonics.[193] The
fourfold structure of arithmetic, geometry, astronomy and music is known as the
quadrivium.
The primacy of the
quadrivium was maintained by Neoplatonists Iamblichus (c.242-327 CE) and Proclus
of Athens (412–485 CE). While the quadrivium was part of the curriculum of
medieval universities, it should not be regarded as naïve, arcane or quaint for
that reason. Given the sequence set forth by the quadrivium, Kepler’s
hypotheses regarding the relationship between the Platonic solids and the
structure of the solar system, and between musical harmony and the design of
the solar system, are natural and even canonical.
The quadrivium resolves the
problem of duality within itself. In the transition from geometry to astronomy,
we cross from the abstract to the physical, but all within the framework of the
quadrivium. In short, it need only be said that the quadrivium defines
structure and structure comes first. Thus, the quadrivium precedes anything
perceived, including by measurement, to be in the physical universe. It forms
part of the (imperceptible) rules governing the universe. It is connected with
the best of all possible worlds doctrine, because to adhere to the structure of
the quadrivium provides the best design for, at least, the non-living part of
the universe. We can see the effects of the quadrivium not the quadrivium
itself. Is it “part of” the universe? In the same way, the rules of cricket are
part of a cricket game, though those abstract rules are not perceptible in the
very physical acts of catching, throwing and hitting a cricket ball. Unlike
cricket, the universe has no definite rule book we can open as the final word
or arbiter; we have to work it out for ourselves. Within the confines of the
quadrivium, “Geometry is prior to astronomy,” as Proclus said. Failure to note
Kepler’s Neoplatonic mindset results in a misunderstanding of his method.[194] After the
quadrivium comes the hybrid between mathematical reasoning and observation.
Last and least is pure observation.
The quadrivium was core to
learning at least as late as the 17th Century. [195] Leibniz, at least earlier in his career, said that most
problems in physics can be resolved into problems of pure geometry.[196],[197]
However, Johnson interprets Leibniz as having gone farther than he did, in
regarding Leibniz’s ten maxims on the Art of Discovery as being tantamount to
rules of geometrical reasoning.[198]
Much of 20th Century, and 21st Century,
science has a tendency to start with the observations. Further, there is a pre-occupation
with things that we can physically observe and visualise in the mind. There are exceptions, such as string
theory. As a civilisation, we have forgotten how we reached the starting line
of 20th Century science. Franklin and
Newstead’s defence of the reality of indefinable real numbers is a sally
against “empiricist idealism”.[199] Of
course, empiricist idealism did not begin in the Twentieth Century and perhaps
existed in more strident form in certain circles in earlier centuries.
To Kepler, the beauty and
perfection of mathematics – especially geometry – reflected the perfection of
God’s methods. Astronomy, being representative of Creation on the larger scale,
had to reflect this beauty and perfection.[200] Such a
contention tends to be associated with Renaissance Platonism and its supposed
commitment to vitalism and natural magic. While Stuart Brown has done much to
dispel this misconception,[201] it
persists in abundance. For example, Lawrenz wrote that the Mysterium
Cosmographicum and Harmonice Mundi were “luxuriant
cosmological fantasias integrating mysticism, music, mathematics, Pythagorean
number lore, the doctrines of the Timaeus”.[202] Lawrenz
then says that Leibniz distanced himself
from Kepler’s work, “having recognised during his Paris sojourn that its
mathematics was woefully inadequate to the task”. On the contrary, we can see
Kepler in many places in Leibniz’s metaphysics including in the “best of all
possible worlds” doctrine. Also, the inadequacy of Kepler’s mathematics had nothing
to do with his selection of the Platonic solids hypothesis. Rather, that
hypothesis was a natural result of a priorist thinking. It would not
have been possible to be an a priorist without a familiarity with
standard Pythagorean and Platonic ideas such as the quadrivium and platonic
solids.
Let us try to draw an analogy close to the heart of 21st
Century readers. The art of ordering human affairs is increasingly being
addressed through the formal sciences, “‘the sciences of complexity’ or
‘sciences of the artificial’” – typified by optimisation – amongst which
Franklin has attempted to find a unifying thread.[203]
Belief in a Creator of the universe who has certain qualities of beneficence
leads one to conclude that there must be structure underlying the physical
universe whose creation was itself an “artificial” act. If well-organised
humans structure their activities using formal sciences, what science of the
artificial did the Creator use? The first principles of structure, the
quadrivium, must be at or near the foundation, or so it would have been
reasonable for the Renaissance and early post-Renaissance scientist to believe.
The burgeoning of modernity,
as we argue here that modernity should be defined, is seen in the
correspondence between Leibniz and Papin. Their collaboration did not arise in
a vacuum but in the drive towards national development for the benefit of the
General Welfare of the nation of France. As minister of the young French King
Louis XIV, Jean Baptiste Colbert initiated a project to discover and make
effective a source of power capable of enabling a dramatic human advance. The
project was put into motion as a national effort.
The project was conceived when the drive for development had outstripped the power sources available. For example, Klemm explains that the:[204]
...vast
and costly establishment at Marly [14 water wheels in the river Seine, each 40
feet in diameter driving 442 pumps], and its relatively small output [80
horsepower]-shows us clearly the limits of the old power-machine technology. To
seek a more efficient and reliable prime mover than the traditional wind and
water wheels was in fact to become the urgent task of the technology of that
period.
In
1666, Colbert established the Academy of Sciences in Paris to further this
purpose, and Christian Huygens was recruited as the first president of the new
Academy. Thus, Colbert was following the prompting of the ‘useful science’
idea. We do not know whether Colbert acquired this idea from Francis Bacon.[205]
Huygens proposed a program including research into the power of gunpowder of
which a small portion is enclosed in a very thick iron or copper case, research
into the power of water converted by fire into steam, and experiments with
vacuum pumps, wind-powered engines, and the communication of force by the
collision of bodies.[206]
The experiments carried out on the basis of these suggestions were the
prelude to the development of a new power-engine. Huygens’ suggestions followed
the work of Guericke who had shown in 1661 that a piston, forced by air
pressure into an evacuated cylinder, can be utilized to perform work. Guericke’s
cylindrical vessel was 15 inches in diameter and just over 21 inches long; it
was evacuated by an air-pump invented by Guericke. In 1664 Caspar Schott
announced Guericke’s experiments on obtaining power by pistons forced through a
metal cylinder by air pressure.
In 1672, Huygens acquired two students: Leibniz the 26-year old diplomat, and Denis Papin a 25-year old French medical doctor who was introduced by Madame Colbert. Within 12 months, Huygens, Leibniz and Papin had modified the Guericke air pump into an engine that could transform the force of exploding gunpowder into useful work.[207] Huygens demonstrated a model “gunpowder engine” to Colbert. In 1673 Huygens wrote this about the model:[208]
The violent action of the powder is by this discovery
restricted to a movement which limits itself as does that of a great weight.
And not only can it serve all purposes to which weight is applied but also in
most cases where man or animal power is needed, such as that it could be
applied to raise great stones for building, to erect obelisks, to raise water
for fountains or to work mills to grind grain …
It can also be used as a
very powerful projector of such a nature that it would be possible by this
means to construct weapons which would discharge cannon balls, great arrows,
and bomb shells … And, unlike the artillery of today these engines would be
easy to transport, because in this discovery lightness is combined with power.
This last characteristic is very important and by this means permits the
discovery of new kinds of vehicles on land and water.
And although
it may sound contradictory it seems not impossible to devise some vehicle to
move through the air …
Leibniz had invented a mechanical
computer in 1671, a year before Leibniz was introduced to Huygens. The computer
was built in 1673.[209] Working alongside
Huygens and Papin, Leibniz was active as an inventor of machines that could
perform physical work. Brown and Calvör describe Leibniz’s work between 1678
and 1679 on his own design for a wind-powered pump to keep the mines in the
Harz mountains dry.[210],[211] Leibniz
personally supervised construction of his mine pump and personally paid for part
of it.[212]
Klemm speculates that the failure of the pump was possibly due to Leibniz’s
frequent absence from the site and the poor skills of the workers. Despite the
difficulties, Leibniz then designed a machine which sought to overcome the
limitations that had plagued earlier
attempts. Leibniz’s hand drawn diagram is included here as Figure 2. In the
end, the Harz mine pump project was unsuccessful and very expensive to his
employer in lost potential income.[213]
Figure 2: Leibniz’s design for a wind-driven water pump (Klemm, p.211)
An anonymous
article from the late 17th Century entitled “To the greater glory of
God” refers to a letter by Huygens on the potential of gunpowder. The promise
and danger in the use of gunpowder in that period mirrors today’s debate about nuclear energy. The writer is on the “pro” side, and he cites
Renaissance-like arguments similar to those used by today’s proponents of nuclear energy. The letter defends
Huygens’ experiments with gunpowder.[214]
The letter illustrates the optimistic spirit of technological progress
for the benefit of humankind which defines Leibniz’s outlook. This optimism, in
spite of the Thirty Years War, characterized the period of Leibniz’s career.
The letter makes the transition from the philosophical regard for the primacy
of Man to its practical implications and realization in machines. The letter
also relies most heavily on the concept of vis
viva without using the term. The letter is anonymous, and given Leibniz’s
diplomatic employment, young age (late 20s) at the time the letter was
published and association with Huygens, one wonders whether it was written by
Leibniz himself or one of his friends perhaps at Leibniz’s instigation.[215]
Leibniz’s optimism about human nature and rejection of the contention
that humans cannot be trusted is more explicit when he explains why the health
of the mind and body are too often neglected. He says, “those who reflect find
more reason to admire the excellence of human nature than to despise it.”[216] Leibniz
writes that “empirical physics is useful for human life and should be
cultivated in the state” and that “new experiments are to be undertaken at
public expense, and only men outstanding not merely in science but in virtue
are to be placed in charge.”[217]
Despite the failure of Leibniz’s water pump
initiatives, his collaboration with Huygens and Papin continued. Leibniz
supplied a number of practical suggestions, right up to Papin’s invention of
steam locomotion and Papin’s proposal to use it to power a steamboat.
Klemm indicates that Thomas
Newcomen, often credited with the invention of steam-powered locomotion,[218] was
really riding on the back of Papin and Leibniz’s work.[219] Arguably,
Newcomen’s engine was a scaled-up version of Papin’s atmospheric steam pump
based on a combination of two of Papin’s earlier ideas: to use a lever to
transmit power from one pump to another (from 1687) and to use steam to create
a vacuum and drive a piston (from 1690).
Leibniz ties physical action
to the mind of God, analogously to how he participates in the design of
machines to the mind of humans. “For we have now seen, from the pre-established
harmony, that God has ordered all things so wonderfully that corporeal machines
serve minds and that what is providence in a mind is fate in a body.”[220] That is, minds are in the driver’s seat. This has the
end-implication that laziness of mind in an individual, a nation or a
civilisation is self-correcting for it leads, inevitably, to conditions which
will force that mind of the individual or the collective, perhaps in a future
generation, to reconsider.[221]
Arguably, the power of Man
over nature defines modernity. Man could not have this power were the design of
the universe not rational for otherwise there would not be sufficient order and
predictability in the universe for Man to effect change in it. A rational
universe, in turn, requires a rational conception of the Creator of the
universe. The power of Man to effect change in the universe is summed up at a
higher order of magnitude in Leibniz’s conception of dynamics. The concept is
addressed in the correspondence between Leibniz and Papin on the power of steam
to lift a column of air and more. It is indirectly brought out by Leibniz in
his Specimen Dynamicum (1695).
However, the argument in Specimen
Dynamicum mainly goes to why the ancients only had half the story with dead
force, versus vis viva, and why Leibniz believed Cartesian doctrines on
matter, force and velocity to be wrong.
We begin with the nature of
the pursuit of truth, which touches on mathematics, the limitations that
Leibniz found in mathematics, and the relationship of mathematics to
metaphysics. This will take us into the domain of Neoplatonic exactitude which
anticipates the next chapter. We will be required to introduce Paolo Sarpi and
the limitation on deductive reasoning when it is constrained by the empiricist
ideology. This will take us into a bridge from Paolo Sarpi to the English Royal
Society. Both Galileo and Newton had a doctrine of how discovery takes place,
and we see the continuum on increasing empiricist constraint stepping upwards
from Galileo to Sarpi to Newton. Relevant to this is a correspondence between
Sanderson’s 17th Century textbook on logic and Newton’s doctrine of
the discovery process, and especially Newton’s bias against hypothesis, both
explored in the next chapter. Contrary to a desire to make science appear to be
“objective”, we argue that pure positivism is not possible in either science or
mathematics, nor can metaphysics be removed from science. While having a Neoplatonist
orientation in many matters of science and philosophy, Leibniz saw the necessity
for formal reasoning in science. Rather than being an innovation that Leibniz
added to Neoplatonism, rigor is found in Platonism itself.
In the previous chapter, we
explained that modernity was about the accessibility of scientific principles
crystallising in pragmatic tools or machinery for physical influence by humans
in the universe. In this chapter, we attempt to lift these considerations to
the concept of truth itself.
That truth is discoverable
via the exercise of reason was a presumption maintained by Leibniz. This
includes physical and metaphysical principles, and principles of human purpose
and conduct. At least, if truth does not exist or is not discoverable then
Leibniz makes no sense. More importantly, since we never fully know truth, it
is the process to reach towards truth that is of greatest concern and, when it
comes to science, the only concern.
The ability of the human mind to access truth or parts of truth gives the mind special place in the universe. Leibniz saw a relationship between the human mind and the divine, which appears to be akin to how Nicolaus of Cusa understood the relation between the mind and the divine. Leibniz begins this passage in a letter with reference to the Neoplatonist Plotinus:[222]
as Plotinus rightly said, every mind
contains a kind of intelligible world within itself; indeed, in my opinion it
also represents this sensible world to itself. But there is an infinite
difference between our intellect and the divine, for God sees all things
adequately and at once, while very few things are known distinctly by us; the
rest lie hidden confusedly, as it were, in the chaos of our perceptions. Yet
the seeds of the things we learn are within us [doctrine of Reminiscence] - the
ideas and the eternal truths which arise from them. … Although our mind depends
continuously on God in its existence and action, as does every other creature,
I do not think that it needs his particular concourse over and above the laws
of nature for its perceptions, but rather it deduces its later thoughts from
its earlier ones by its internal force and in an order prescribed by God, as
Roelius, whom you quote, rightly says.
The passage raises the
question of the relationship of the finite to the infinite, which Nicolaus of
Cusa referred to time and again, which we will discuss further in Chapter 4
“Discovery and Deduction”. Leibniz also reiterates that human minds can delve
into the universe and make discoveries about it without intervention by God.
Rather, minds have their own “internal force”. The “chaos of our perceptions”
implies that our perceptions do not get us very far. What we “learn” are ideas,
and eternal truths arise from those.
Clearly, the mind’s access to what is universal and eternal deserves special
consideration. In one of his essays, Leibniz refers to the mind as the divine
in miniature.[223] Therefore,
it is not surprising that Leibniz is more Rationalist than Empiricist. He
observed that a geometrical argument made in a dream is as accurate as one made
when awake, while a perception in a dream is illusory. Of course, our
perceptions usually dominate our thoughts and there is no thought so abstract
that it does not involve the object of some perception.[224]
However, just as no-one would deny that air is essential to life, yet life is
something different from and higher than air.
For Leibniz, the nature of transcendental curves is part of the domain of truth, but transcendental curves are real not only because they or their effects are realisable. Of course, they are of immediate interest because their effects are real and realisable. As he says, “we must free the human mind from arbitrary contingencies, in order to bring out the underlying nature of the thing [the curve] itself.”[225] It is difficult to say whether Leibniz would go further than the structuralist school, which requires a mathematical to be able to manifest in the physical universe for it to be called real.[226] In any case, the point is moot because Leibniz never spent time on anything for which he could not see significance or repercussions for the physical universe, or even potential useful to humankind, and regarded such pursuits as a frivolous waste of time. The point would not be moot if the discussion led to an overturning of our conception of how God thinks or from where he got the raw ideas from which to design the universe.
Truths such as the nature of
the catenary curve can be discovered systematically. “That is, the case
of developing methods is always more crucial, than particular problems, although
it is the latter which usually bring applause.”[227] Further, such methods lead
to real benefits. It extends the science of discovery, “in other words the science of
Analysis, which up to now has been incapable of tackling such questions.
Second, it extends the progress of construction techniques. In point of fact, I
have come to realize that the resourcefulness of this curve is only equal to
the simplicity of its construction, which makes it the primary one among all
the transcendental curves.”[228]
In The Republic,
Plato said that all sense perception is opinion and thus is not of great
assistance in discovering truth.[229]
Leibniz largely concurred. Today, to the layman and scientist alike, it is
common for the testimony of the senses is regarded as the final or, at least,
the most reliable arbiter.
Yet sense perception had already gained primacy
through circles associated with Galileo, and had gained a foothold in the
British Isles, especially in the circles of what became the Royal Society. Fulgenzio
Micanzio was the closest friend of Galileo’s patron, Paolo Sarpi. Micanzio
corresponded with William Cavendish the English scientist. Those letters were
translated by the secretary of Francis Bacon, Thomas Hobbes, for wider circulation.[230]
It is possible that when Hobbes and Cavendish visited Venice, they met with
Sarpi and there is evidence in Hobbes’ translations that he was familiar with
the personality of Sarpi’s personality.[231]
Sarpi advocated a presumption that what is perceived with the senses is real,
as we will discuss later in more detail. In fact, Sarpi may have believed that
excessive reflection was a weakness and instinctive action superior leading to
the conclusion that in some ways Man is weaker than the animals which primarily
act on instinct based on their sense perception.[232]
Contrary to these conclusions reached by Sarpi, he was a regular attendee in
meetings of occult circles in Venice. Giordano Bruno was warmly welcomed by
these circles when he was in Venice, and through this connection Michael White
says that Sarpi “knew Bruno well”.[233]
It would seem that Sarpi was not persuaded by Bruno. Returning to the British
Isles, the primacy of sense perception became much more firmly entrenched after
Newton had passed away and the Royal Society had passed into the hands of its
second-generation membership.
Another name of the ideology that places sense
perception first is Empiricism or a
posteriorism as against a priorism.
Of course, even a priori thinking uses the testimony of the senses but
only to confirm or deny hypotheses formulated by the use of reason.
To maintain that the testimony of the senses is the
only thing that has validity in science removes the role of thought except for
undertaking deductive exercises from what is observed. The dangers of
empiricist idealism are pointed out by Franklin and Newstead when discussing
Ormell’s desire to remove indefinable real numbers from mathematics.[234]
First of all, what we observe with our senses is subjective. The testimony of
one’s own senses can change; the use of different measuring instruments causes
observations to differ. Observational data requires thought to give it meaning
and order; for example, there was nothing directly discernible in the
observational data that Kepler pored over that indicated elliptical orbits. To
give the senses sole authority is also to maintain that the senses are the only
reliable reference in determining what we can be sure of, that is, in
determining what we know. Such authority would be misplaced since, as explained
just now, the senses are not reliable determinants of anything.
As sense perception is mere opinion, to demand that
only the senses be relied upon is to desire that nothing be known. The opponent
of this who uses Platonic dialogue to help discover truth, however, also denies
knowability. If there is one thing that Plato succeeds in doing, it is that
answers given always lead to new questions.
Whether we use sense perception or intellectual
enquiry, we get nowhere. This catch-22 might drive anyone with an interest in
science to pessimism as to their prospects for success and depression, for it
seems that humans are excluded from science.
Humans are excluded from science in another way. Burtt’s celebrated thesis The Metaphysical Foundations of Modern Physical Science considered the human-free conception of the universe adopted by positivist physics. That is, the laws which physics seeks to understand are unaffected by humans and impact humans only in a physical kinetic sense. This stands in contrast to the Renaissance view of Dante – for example – that humans are not only part of Creation but are at its pinnacle. Burtt knew that he did not like the cold Newtonian-Russellian clockwork universe in which Man is irrelevant, but he purported to supplant it by substituting an extreme empiricism. That is, Burtt sought to bring Man back into science by declaring that perceptions of the universe are close enough to reality and therefore should be taken to be correct, in preference to a mathematical model, say. Burtt seems to think that the only way humans can be brought back into the universe is through sensory experience, neglecting the role of mind:[235]
...the space of perception is too much
like the space of real objects to reveal any essential difference from it. All
it needs is to be freed from illusions, private images, and other experiences
lacking social objectivity, to function quite acceptably as real space. And
once this point has been reached there seems no longer any excuse for
maintaining the distinction between sensed qualities and the real characters to
which they correspond.
…
There is simply no science possible of
the realm of sensible phenomena unless the trustworthiness of our immediate
perception of spatial directions and relations be taken for granted.
Not detracting from Burtt’s many and deep insights
into the evolution of science, his empiricist proposal was only possible
because he did not give due consideration to the idea of truth. Leibniz in
§§27-29 of The Monadology addressed this directly by contrasting
empirics with reason calling Reason an act of “the rational soul or mind [esprit]”
whereas empirics are a result of perception which even the beasts have.[236]
As we explained above, to maintain that the senses, including
deductions from their testimony, are the only thing we can be sure of is to say
that we can be sure of nothing. To Cusa, humans could never know the truth as
to know it would be to know God and, metaphorically, such a light would be too
bright for any human intellect to gaze upon directly. Leibniz agreed that there
was nothing but a theoretical possibility of anything but that “the number of
all truths which all men together can know is quite mediocre, even if there
were an infinity of men who for all eternity should exalt themselves in the
advancement of sciences.”[237]
We will refer to promoters of the Primacy of Reason as
Rationalists, which is a standard term. Upon investigations, many Rationalists,
up to and including Leibniz, turn out to be Neoplatonists and vice-versa. The
difference between the terms is in their historical baggage and connotations.
Both believe in the primacy of reason, or the inferiority of the senses to the
mind. Now, there are Aristotelian thinkers who give Reason a high position.
However, in the end, the Aristotelian does not give as much primacy as the
Neoplatonist, and certainly not relative to the sense perception, for the
Neoplatonist is almost purely Rationalist giving no value to sense perception
except to check the conclusions of Reason.
The “new mechanical philosophy” (“NMP”) of the 17th Century, partly due to thinkers like
Kepler, was an understanding that structure was discoverable in the universe,
pursuant to the belief that the universe has a rational design, and its
achievements seemed to confirm that the universe has a rational design.
However, the “mechanism” or order is not necessarily discoverable by sense
perception and certainly not by the senses unaided by artificial (human-made)
instruments. Thus, the NMP has its roots in – and so is more closely allied
with – Rationalism and, hence, is closer to Neoplatonism than it is to
Empiricism.[238]
Empiricists clearly differ from Rationalists on the
path to truth, though they agree that final truth – whatever their brand of it
might be – is not unattainable by humans. The nature of the two positions also
impacts questions relating to the existence of truth. Cusa regards truth as
having a reality, but considers access to it as being exclusive to God. Humans
can make gains in its direction and approach it, merely. The Empiricist
position allows judgement on whether truth exists at all to be reserved, for
the Empiricist does not need to answer the question, even though an individual
Empiricist may accept particular truths.
Empiricism allows – but does not imply – denial of the
existence of truth. Once the Empiricist tries to get into the question, he can
only affirm truth’s existence if truth, like observation, is subjective, which
is no truth at all. Burtt made the error of asking what truth is while wearing
an Empiricist hat. The perennial trick of the Empiricist is to not ask the
question, and to say that the question of truth is irrelevant or to say that,
as far as they are concerned, what is true is what we can perceive (see, hear,
touch, taste, smell) which can only be an appeal to a popular kind of
commonsense. This approach omits a significant role for the human mind except
in collating and making deductions from observations, which a computer can do
quite well.
If we allow their different conceptions of truth, the
Empiricists and Rationalists agree that the final truth is unattainable by
humans. However, there is a key difference. Empiricists say that all we can
know is what the senses tell us, mediated by our prejudices and what we think
we know. Rationalists led by Cusa generally say that we can know what the
senses indicate but we need to understand the very restricted meaning of
sensual testimony. However, using our reason we can come to know what are the causes
or causal principles P1 whose effects S1 the senses
detect. By understanding those causes well enough, perhaps we can design and
build an apparatus that allows us to observe them. Thus, through human
ingenuity, P1 gives rise to S2. We now perceive at a
deeper level because “higher order” causes have been flushed out to our senses.
In turn, we shall find that sensory effects S2 are caused by principles
P2. In general, our Pn gives rise to Sn+1
through an artificial human intervention to generate the sensory effect using
the principles Pn. In the process of human ingenuity generating Sn+1,
it may also find a method or technique (“C”) of controlling those principles within
certain boundaries, so we get Cn+1 also. In symbols, we could write:
Pn
→ Sn+1 or Pn → Cn+1
with each of the arrows representing an active role
played by human ingenuity. If observation were the only capability at our
disposal, then neither of these two steps could occur.
This question is all the less frivolous
when we note that, for the Empiricists, the causal connection between the
phenomena behind the perception and the perception itself:
Sn =
effects(Pn)
is restricted territory in which the human mind has no
business because any enquiry in that domain is pure speculation and perhaps
even fantasy. It is difficult to find a pure Empiricist. For example, Newton,
who will be discussed below in more detail, did permit some consideration of
the “effects( )” part of the above formula.
We now turn to the question of whether
religious or moral truth differs from anything discoverable in science through
whatever means. The current Pope suggests that there is no such difference,
while acknowledging that there have been attempts to separate religious faith
from universal truths. In September 2006, Pope Benedict XVI said:[239]
Dehellenization first emerges in
connection with the postulates of the Reformation in the sixteenth century.
Looking at the tradition of scholastic theology, the Reformers thought they
were confronted with a faith system totally conditioned by philosophy, that is
to say an articulation of the faith based on an alien system of thought. As a
result, faith no longer appeared as a living historical Word but as one element
of an overarching philosophical system. The principle of sola scriptura,
on the other hand, sought faith in its pure, primordial form, as originally
found in the biblical Word. Metaphysics appeared as a premise derived from
another source, from which faith had to be liberated in order to become once
more fully itself. When Kant stated that he needed to set thinking aside in
order to make room for faith, he carried this programme forward with a
radicalism that the Reformers could never have foreseen. He thus anchored faith
exclusively in practical reason, denying it access to reality as a whole.
Similarly, Leibniz stated that truth must be consistent
across all domains. In particular, he included the toughest example of science
and religion, saying that we should not accept for religion what we would not
accept in science. He was not referring to a standard of proof but was saying
that the actual content of ideas and doctrine should be consistent across
science and religion.
The scientific work of Kepler and Cusa was bound up
with their theology; any inconsistency between science and theology was
anathema to them. Indeed, there was not really any separation between the two
for the discoveries of science were but a manifestation of true theology. They
would have been forced to revise their science or theology or both were an
inconsistency to arise; at least, they would have known that one or the other
or both must be wrong.
The Empiricist position neither confirms nor denies
the existence of God. Thus, the Empiricist is free to choose on the basis of
personal preference whether or not they believe in the existence of God.[240]
It might not be a coincidence that Empiricism
was promoted by Venetian gentlemen, such as Paolo Sarpi, Galileo’s patron, who
were also inclined towards Atheism.
The consummate Neoplatonist is an a priorist
who strives for a “God’s eye view”. That Neoplatonist assumes his latest and best version of the
“God’s eye view” in formulating hypotheses within that pan-Creation context.
Many in the Empiricist school, even today, not only
deny the necessity of a God or Creator in science, but also deny that
metaphysics has any relevance to science and some even claim that metaphysics
is meaningless word play.[241]
This is in itself a kind of metaphysics though its advocates refuse to label it
so. To say that science is the study of things that can be observed and that
nothing outside of the field of observation is a valid domain of enquiry at its
simplest level denies the validity of any metaphysical enquiry. How does one
justify this position without something like a metaphysical argument? Burtt
points out that an attempt to exclude metaphysics and not hold any metaphysical
position itself carries a raft of embarrassing metaphysical assumptions.
If sensory observation is all that we have, then we
can only be aware of correlations and coincidence of events. No pan-Creation
view is possible; it would not make sense to ask what is happening on the other
side of the universe. Thus, the Empiricists differ from the Rationalists in the
nature of truth. For the former, truth
is an ever-expanding basket of correlations. Ultimate truth is all the correlations that exist. For the
latter, truth is a principle
which is implied and necessarily so in the context of philosophy, metaphysics
and theology. If it is a principle of natural science, then it must have been
confirmed by experiment whether by direct observation or by deduction or
calculation from observations. Ultimate
truth is the mind of God or the Creator of the universe.
Atheists had fewer difficulties in adopting an
Empiricist outlook. The theology of Cusa and Kepler did not permit an
Empiricist outlook. However, even though the British Royal Society was
generally Empiricist in its orientation, British thinkers such as Henry More,
Robert Boyle and Isaac Newton spent some effort in ensuring that their scientific
outlook had some basis in a theology they could believe in. Indeed, Newton
wrote, “One principle in Philosophy is the being of a God or Spirit infinite
eternal omniscient omnipotent.”[242]
However, McGuire also explains that the meaning Newton ascribes to the term
“principle” in this context is different from when he uses the term as having
implications or repercussions for science.[243]
Boyle in particular drew inspiration for his belief in God from his scientific
insights though in a rather gnostic way seeing the universe as divinely
animated. Leibniz regarded a universe of that kind as imperfect – and therefore
impossible – because it would require ongoing participation by its Creator. So
“there is no soul of the universe” and “I hold … the hylarchic principle of
Henry More, as being either impossible or superfluous; and it is enough for me
that the mechanism of things [the universe] is constructed with so much wisdom
that all these marvels come to pass through its very development, organised
beings being evolved, I think, according to a preconceived plan.”[244]
Leibniz explained that he did not think that proponents of the hylarchic
principle had a distinct idea of their conception.[245]
We are now considering the converse of the earlier
discussion on Empiricism neither confirming nor denying Theism. Assuming the
theology of Kepler and Leibniz, say, we have said enough to formulate this
summary:
Theism ==> ¬ Empiricism
Empiricism =/=> Theism
Empiricism =/=> ¬ Theism
Theism ==> Rationalism[246]
Rationalism ==> Theism
The diagram becomes more complicated when we consider
different kinds of Theism, i.e. different theologies. It would be interesting
to consider the God of Newton, and how it differs from the God of Neoplatonists
such as Kepler. For example, is it hylarchic like the God of Henry More? That
discussion is beyond the scope of this thesis.
We have situated the idea of truth in each of the
Rationalist, Neoplatonist and Empiricist schools. Those of the Rationalist and
Neoplatonist orientations (which are often found together in any given thinker)
have a better structured conception of the existence and attainability of
truth. This might be a result of their Theism. Empiricism does not necessitate
a developed conception of a Creator or the Creator’s relationship to the
physical universe and so does not derive clarity on the whole of Creation as a
single conception that the Rationalist and Neoplatonist do. We do not say that
such clarity would be impossible for an Empiricist to attain. However, it would
only be possible after further development of ideas that can only arise from
Rationalism and Neoplatonism which are not as tightly bound by the constraints
of sense perception or sense perception enhanced by scientific instruments. To
be sure, understanding how the Creator thinks is not easy but in making the
attempt leaps of understanding can occur. This sets forth the situation as it
was until the early 18th Century. Whether the attempt to understand
how the Creator thinks is also required today may require further enquiry. However,
since the considerations and principles raised in this chapter are universal
and timeless, we would expect the same conclusions.
In the previous chapter, the understanding that
structure is discoverable in the universe was discussed. It was called the “new
mechanical philosophy” (“NMP”)
though it was often anything but “mechanical” in the modern sense. Indeed,
Kepler epitomized the origins of NMP and his search, indeed expectation, was to
find harmony and beauty in the structure of the universe. Leibniz was a
continuation of this new mechanical philosophy. Today, the idea that the
universe has a rational design is associated with physical experiments and
observable evidence in the tradition of Baconian Empiricism. On the other hand,
Neoplatonism is regarded as a mystical quasi-religious movement that lacks
scientific credibility. The orientation and methods of Neoplatonism were not
always regarded in this way before Leibniz. It was in Leibniz’s lifetime that
the lines between Neoplatonism and Empiricism started to be drawn. Leibniz
adopted many of the methods of Neoplatonism but certainly accepted the
importance of experiment. In effect, Leibniz saw the pendulum swinging too far
towards what experiments can indicate to the senses, rather than using
intellectual tools informed but not dominated by observations. Because Leibniz
saw the fork in the road and made his position clear, much can be learned about
Leibniz by considering the Neoplatonist and Empiricist orientations and
Leibniz’s relationship to them.
The reality and intellectual accessibility of concepts
and “things” in a non-visible domain is the nub of many debates between
Neoplatonists and non-Neoplatonists. Concepts such as the infinite exemplify
what is unobservable. Friedrich Schiller (1709-1805) is one of Germany’s most
famous poets and playwrights. Schiller took the idea of the infinite seriously,
and we begin the chapter with Schiller’s thoughts on the topic as an example of
the accessibility of metaphysical ideas. Leibniz regarded mathematics as a subdomain
of a domain of calculation on metaphysical concepts. Any domain of mathematics
that relies on the infinitesimal presents itself as an example. To the layman,
it might be paradoxical that “medieval speculations” were instrumental in
bringing about the concept of the infinitesimal used by Barrow, Leibniz and
Newton in their respective and evolving versions of the calculus.
As far as calculations of any kind go, they are
tedious when they represent stepwise reasoning using symbols to ensure
precision. Leibniz saw the limitations and understood that such a mode of
reasoning was retarding the development of physics and engineering, because
available mathematical tools were time-consuming to use. Leibniz made reference
to his idea for a Universal Characteristic to help mathematics escape its
limitations, but left it to future generations to further build the idea on
which he placed so much hope. Leibniz also short-circuited stepwise reasoning
with his maxim that perfect knowledge of a domain requires us to see the whole
in a single act of the mind or in a single thought.[247]
From there, one can lever up to a higher conception that subsumes and
transcends the existing understanding in a single bound.
The distinguishing features of Neoplatonism have not been
mainstream since the 17th Century. Further, significant developments
are rare adding to the obscurity. It is true that forming hypotheses is a
distinguishing feature of Neoplatonism. However, Neoplatonists tend to
hypothesise based on their conception of the Creator’s mind in relation to the
entire universe, rather than on considerations attained through observation
alone. Once an initial hypothesis is formed and work to verify it begins, the
process of observation, calculation and number-crunching is indistinguishable
from any other mode of science. Thus, the Neoplatonic scientist must be as
concerned about precision as any other scientist. We will discuss Kepler and
his Neoplatonist orientation. Franklin notes that Kepler’s refusal to
compromise with precision is as impressive as that of any scientist of his age
and perhaps ours.[248]
The rest of the chapter will be spent discussing the
development of the Empiricist school prior to Leibniz’s lifetime. The British
Royal Society typifies the Empiricist school today but we argue that this is
not where Empiricism began. The ideological takeover of the University of Padua
– which was once a centre of Neoplatonism – by Averroism may have been
instrumental in nurturing the early stages of what became the “Empiricist
revolution”.[249] However,
we will not go back that far. We start with Paolo Sarpi of Venice, a
contemporary of Galileo, though 10 years elder to Galileo. England was heavily
influenced by Italy, and it was standard for the scions of wealthy English
families to undertake a “grand tour” of the Continent including Italy and especially
Venice. Sarpi’s writings were often translated into English and published in
England, though we have no evidence of how widely read they were. We discuss
how Sarpi’s atheism was naturally allied to his tendency to Empiricism.
Empiricism leads to rigorous deductivism, hence the
double-barrelled term “empirico-logic”. Any discussion of influences on science
in the British Isles in that period leads to Newton, so we will discuss
Mamiani’s discovery of the close parallel between Newton’s prescribed method of
discovery and a standard logic text of the period. Newton’s opposition to
forming hypotheses is a natural part of an empirico-logical stance, but it
contrasts with the approach of Neoplatonic scientists like Kepler.
Next, we shall discuss the consequences of mandating empirico-logic as a scientific method. There were members of the Republic of Letters who criticised Leibniz’s calculus on the grounds that it lacked precision. Leibniz openly disagreed with them and used the label “rigorists” for such writers. However, the fact that there were such critics is not surprising given the dominance of Averroism in such eminent institutions as the University of Padua. Finally, we close with a couple of open questions that would benefit from future research.
Friedrich von Schiller (1759 – 1805) is one of
Germany’s most famous poets and playwrights. Seventy-seven years after Leibniz’s
death, Friedrich Schiller distinguished the logical estimation of magnitude
from the aesthetic.[250]
Estimating a magnitude in a “logical fashion” is to relate it to the “cognitive
faculty”, experience something about the object and behold something outside
oneself. To estimate something aesthetically is to relate it to the faculty of “sensibility”.
Contrary to the logical fashion, in aesthetic estimation one experiences something
within themselves caused by the imagined magnitude of the object. In this case,
the thinker is neither measuring nor estimating magnitude, but themselves
become for the moment an infinite magnitude to themselves. Schiller writes that
anything that evokes the (intellectual) experience of being an infinite
magnitude to the intellect is sublime.[251]
Schiller’s distinction between logical versus
aesthetic estimation is the same as that between logical deduction and
Neoplatonic reasoning. This “aesthetic” is not merely a quality of being
subjectively pleasing to the mind, but is a universal quality. While Schiller
uses the concept of the infinite with in the individual thinker as the
yardstick, Schiller’s central point is that that measure is itself universal.[252]
While Schiller did not use the term “Neoplatonic”, the Neoplatonist encourages abstraction
“in the sphere of the infinite”.[253]
This is the sphere of truth, whether it is classified as metaphysical,
mathematical, moral, artistic, musical or otherwise; after all, these are human
categories and Schiller would argue that truth does not know the separations
between these categories. The corporeal human condition inevitably “resumes her
rights to give an imperious reality to our existence, to give it contents,
substance, knowledge, and an aim for our activity.” This “concretisation” of
realisations/discoveries brings new understanding out of the domain of the
infinite and into the temporal; the result cannot be universal or eternal
though it may be useful for a few centuries or millenia.
The Neoplatonist acknowledges that the domain of the
infinite is accessible to the human mind. The Neoplatonist goes further and,
like Schiller, says that discoveries are only worthy of the name if they do
pluck something new from that domain. Such a discovery is generally useful to
the bulk of humankind precisely because it brings a new principle into the
corporeal domain where “the rest of us” can work with it.
The calculus is founded on the concept of the infinitesimal
which might have an ontological reality but has no physical reality. Therefore,
the calculus is a tool of precision in metaphysics before it is a mathematical
tool. We may call it a mathematical tool because it brings ideas from the
metaphysical domain into a form which humans, or computers, can systematically
work with. However, the crux of the calculus is metaphysical before anything
else. It is also outside the domain of any possible empirical enquiry, for it
allows the expression of a principle about a curve which cannot be seen from
the curve itself. It embodies the idea of a hidden underlying principle and
provides a method for dealing with such principles in a systematic way. Today,
we take the idea of instantaneous rates of change and all that flows from them
for granted, but in Leibniz’s time none of it was obvious.
Leibniz regards the relations of forms and abstract
quantities as belonging to Metaphysics, which can be structured into a calculus
or Universal Characteristic. He regards Combinatorial Science as an application
of such Universal Characteristic. In turn, he says “all of Algebra is merely an
application of a Combinatorial Science of quantities.”[254]
According to Leibniz, Algebra is a particular class of methods of calculation
within Metaphysics.[255]
If Algebra is just a subdomain of a subdomain of
Metaphysics, and if all of Metaphysics is subject to structured Analysis, we
would expect physics – a consequence of Metaphysics – to be subject – or at
least amenable – to structured (a priori)
reasoning. We will see below that this is exactly what Leibniz believed. It is
not surprising that Leibniz placed experiment a long way below Reason in
importance.
Decades before Leibniz was born, Tycho Brahe and
Johannes Kepler had done significant work with mathematics in the sense of
“number-crunching” to test hypotheses. We know that Kepler was as ambivalent as
Leibniz about the infallibility or ultimacy of mathematical methods; that is,
Kepler did not come close to claiming that the entire universe could be reduced
to, or understood via, mathematical calculation. For Leibniz, the domain of
structured Reasoning was much broader than mathematics. Leibniz says that
“there is an art of Analysis which is more inclusive than Mathematics” and that
he has a proof that such an art or method exists.[256]
This method turns out to be a framework for reasoning about matters
metaphysical and physical based on premises which are, at least in part, a
result of a priori metaphysical
thought.
Prior to Brahe and Kepler, Boyer says that the
medieval period added little to the classical Greek works in geometry or
algebra. Rather, it contributed, chiefly, “speculations, largely from the
philosophical point of view, on the infinite, infinitesimal, and continuity” as
well as “new points of view with reference to the study of motion and
variability. Such disquisitions were to play a not insignificant part in the
development of the methods and concepts of the calculus”.[257]
One of the most significant of the “medieval
speculators” was Nicolaus of Cusa of whom Kepler was a disciple although a
century separated them. “The fullest expression of Nicolaus of Cusa’s
mathematical thoughts on the infinite and the infinitesimal, however, are found
in the work of Johannes Kepler, who was strongly influenced by the cardinal’s
ideas … and who was likewise deeply imbued with Platonic and Pythagorean
mysticism. It was probably the imaginative use by Cusa of the concept of
infinity which led Kepler to his principle of continuity.”[258]
Kepler wrote of Cusa others divinus mihi Cusanus, i.e. “Cusa and others seem to me divine” in drawing the analogy of
the circle compared with polygon to God compared with his creatures.[259]
It was Cusa, says Boyer – albeit himself rigorist in his sympathies –
who led Kepler to include normal and limiting forms of curves under a single
definition of continuity encompassing conic sections as a single family of
curves.[260]
To Leibniz, mathematics was the art of reasoning using
symbols to represent quantities. Symbols could be used for many other purposes.[261]
Leibniz’s recognition that attention to detail sufficed for calculations to be
carried out indicates that he did not think particularly highly of the creative
or intellectual content of the calculation process.[262] “For it
is unworthy of excellent men to lose hours like slaves in the labour of
calculation which would safely be relegated to anyone else if machines were
used.”[263]
Rather, the real achievement was in devising the systems and methods for
calculation, which could then be exercised in a relatively straightforward way
and perhaps even automated in the future. Leibniz had made a contribution to
such automation by designing the “Step Reckoner” calculating machine in 1671
which was built in 1673.[264]
Without effective reasoning, even
plentiful observations are worth little.[265]
Thus, Leibniz saw the need for an ability to comprehend “the whole” with
mathematical precision in order to derive advantage from new observational
technologies. Leibniz wrote:[266]
I have no hope that we can get very far
in physics until we have found some such method of abridgement to lighten its
burden of imagination. For example, we see that a series of geometrical
reasoning is necessary merely to explain the rainbow, one of the simplest
effects of nature; so we can infer what a chain of conclusions would be
necessary to penetrate into the inner nature of complex effects whose structure
is so subtle that the microscope, which can reveal more than the
hundred-thousandth part,[267]
does not explain it enough to help us much. Yet there would be some hope of
achieving this goal, at least in part, if this truly geometrical analysis were
established.
Leibniz is referring to the absence of a tool that
would allow us to calculate the inner workings of nature. For Leibniz, too much
imagination is required and far too many detailed experiments. Since nature was
constructed rationally, such a tool must be available. This returns us to
Leibniz’s high hopes for the concept of the Universal Characteristic. The idea
is that all concepts in science are representable by symbols which can then be
manipulated to locate new truths.[268]
The use of symbols helps enforce precision. “[T]he best advantage of algebra
are only samples of the art of characters whose use is not limited to numbers
or magnitudes.”[269]
It is clear that the emphasis is on introducing precision and adding tools to
the arsenal of methods available to the reasoning process. Leibniz’s Universal
Characteristic might today go some way towards “a unified theory that covers
mathematics, pure and applied, as well as the formal sciences.”[270]
Leibniz affirmed that ultimately the universe could be
explained through analytical methods of some kind.[271]
Yet Leibniz saw too great store being set by formal reasoning using the
inadequate tools of logic which were then available. Penetrating the “inner
nature of complex effects whose structure is so subtle that the microscope,
which can reveal more than the hundred-thousandth part, does not explain it enough to help us much” would
require something far more powerful than the syllogism. Indeed, it might need
to be a “truly geometric analysis”. These cumbersome methods were represented
by the positivism through formalist logic which had been gaining primacy since
Averroes in the 12th Century.[272]
Thomas Aquinas wrote a critique of Averroes’ interpretation of Aristotle,
disputing metaphysical fundamentals such as the relationship between the
intellect and the soul rather than logical methods. Whether Averroes sought to
expound on Aristotle faithfully or whether he deliberately added a little too
much positivism of his own is beyond the scope of this thesis.
That mathematics is an indispensable tool in science
is self-evident to 21st Century readers. But if we are saying that some
mathematical tools are really just a kind of metaphysics, then more explanation
is needed.
The prevailing view that science stands independent of
metaphysics contrasts with the outlook of Leibniz and his forebears in the
Humanist Renaissance. For Leibniz, science as it is done by humans is a
temporary understanding to be replaced by a future, richer understanding.[273]
This is somewhat like Cusa’s doctrine of docta ignorantia or “learned
ignorance”. Physical phenomena are a manifestation of an underlying
metaphysical reality that is dynamically in progress, and is only discoverable
by the mind. What is the nature of the underlying metaphysical reality? We will
never know it completely, but the little we can know about its abiding nature
flows from our reasoning about God and God’s expression in Creation. Thus, the
Neoplatonic scientist necessarily is an a
priorist.
This thesis is not alone in pointing out and
supporting Leibniz’s preference for a
priori explanation merely provides description. In the science of motion,
Leibniz regarded rational analysis as superior to experiment. Indeed, “experiment
must be eliminated from the science of the abstract reasons for motion, just as
they should be eliminated from geometrical reasonings. For they are
demonstrated not from fact and sense, but from the definition of the terms.”[274]
Brown mentions this without criticism. Indeed, he calls Leibniz’s quasi-a priori explanation of elasticity in
collisions ingenious.[275]
Much of this was said by Kepler in his introduction to
Mysterium Cosmographicum in which his
statements on science make it clear that he is the consummate Neoplatonist.
Neoplatonism is – among other things – a mindset, and there were areas with
greater concentrations of “practitioners” and sympathisers than others. Burtt
tells us that Copernicus sought the “southern” intellectual influences.[276]
Perhaps this was because he was already interested in the Neoplatonic outlook
which had many adherents in that region of Europe.
There is a misconception that Pythagoreanism and
Neoplatonism are synonymous with mysticism, particular interpretations of
Cabala and occult sects.[277]
There may have been such offshoots, but as far as leading Neoplatonists from
the 15th Century onwards are concerned, nothing could be less
correct. Their theology caused them to shun “mystical arts” as sophistry at
best and Satanic at worst.[278]
Copernicus himself was decisively Neoplatonic.[279]
In the decades after Copernicus lived, it was only arch Neoplatonists who
carried his standard. An example is Giordano Bruno whom no-one could disagree
was an arch Neoplatonist:[280]
In his work The Ash Wednesday Supper, a story of a private dinner, being
entertained by English guests, Bruno spreads the Copernican doctrine. A new
astronomy had been offered the world at which people were laughing heartily,
because it was at variance with the teachings of Aristotle. Bruno was carrying
on a spirited propaganda in a fighting mood. Between the year[s] 1582 and 1592
there was hardly a teacher in Europe who was persistently, openly and actively
spreading the news about the ‘universe which Copernicus had charted’, except
Giordano Bruno.
How could a work so precise and ultimately so
successful in describing physical phenomena as that of Copernicus have been
carried out by a Neoplatonist? This question only needs to be asked if one has
a preconception that Neoplatonism is a mystical and “unscientific” set of
doctrines. It would make no sense to even ask whether precision and calculation
figures in such a set of doctrines.[281]
As mentioned, this is a misconception.
The need for exactness is outlined by
Burtt:[282]
As with Kepler, so with Galileo, this
mathematical explanation of nature must be in exact terms; it is no vague
Pythagorean mysticism that the founder of dynamics has in mind.
As an aside, Leibniz addressed this directly in
c.1679, “Men have been convinced ever since Pythagoras that the deepest
mysteries lie concealed in numbers. It is possible that Pythagoras brought over
this opinion, like many others, from the Orient to Greece. But, because the
true key to the mystery was unknown, more inquisitive minds fell into
futilities and superstitions, from which there finally arose a kind of popular
Cabala, far removed from the true one, and that multitude of follies which is
falsely called a kind of magic and with which books have been filled.”[283]
This confirms Stuart Brown’s analysis in that Leibniz attacks what are
popularly – but incorrectly – thought to be elements of Platonism even today.
Galileo says that the peripatetics, especially, have
written great volumes on the problem of falling bodies and yet have never made
their understanding exact. Indeed, the idea that quantification is the only way
of defining a phenomenon precisely is Neoplatonic. This is not surprising
considering that the reality of numbers was maintained by the Pythagoreans
against opposition. Thus, the idea of measurement ultimately is Neoplatonic.[284]
Galileo himself, a pioneer of precise experimental physics and who is not
normally classified as a Neoplatonist, is said to have been “supported by the
onrushing Pythagorean tide”.[285],[286]
Leibniz’s search for precision and the idea of a Universal Characteristic for
precise and systematic reasoning is consistent with this.
Paradoxically, it is with the non-supporters of
Renaissance Neoplatonism that precision and numerical methods are usually associated
– that is, with the Empiricists. We introduce Paolo Sarpi, Chief Theologian to
the Republic of Venice. Paolo Sarpi is an example of the importance of the
intentional view of history: that it is difficult or impossible to make sense
of historical processes and the decisions of historical figures without having
some grasp of their intentions and the intentions of the cultural and political
framework in whose thrall they acted.
Contrasting with Kepler, in Sarpi’s statements on
science he declares himself a pure Empiricist. Sarpi sat at the centre of a
network of influence with the blessing of the Republic of Venice whose power
was waning but which was a significant financial and political force in Europe.
Galileo was personally close to both
Kepler and Sarpi. Galileo was friends with Kepler as a student and they
continued their correspondence following graduation. Eventually, Sarpi became
Galileo’s friend and patron.[287]
Rather than being a disinterested source of funds for budding scientists, Sarpi
had his own philosophy of science and made a clear statement which sounds a
little like Newton who was not to enter the scene until nearly 75 years later. Sarpi
says that potentially there are four possible ways of gaining new knowledge:
Sarpi rejects 1 (pure reason) as pure
speculation/fantasy. He rejects 2 (pure perception) as preventing the obtaining
of meaning. He rejects 4 (reason followed by perception) as prejudicing
perception with advance speculation based on air. Kepler addressed option 4 by
saying that hypotheses should not be formulated by idle speculation. Meli explains that Kepler imposed constrains on how hypotheses are
formed. In Epitome Astronomie Copernicae
Kepler wrote, “astronomers should not have absolute freedom to think up
anything they please without reason; on the contrary, you should give causae probabiles for your hypotheses
which you propose as the true cause of the appearances, and thus establish in
advance the principles of your astronomy in a higher science, namely physics or
metaphysics.”[289]
Sarpi prefers 3 (perception followed by reason)
because it starts with what is real and then allows the derivation of meaning
from it. We must ask what Sarpi meant by “reason” in 3; was it the same as
Newton’s prescription of induction from experimental observation or testimony
of the senses?
We also need to ask why Sarpi was choosing between
these four options. If he was aware of Galileo’s ongoing correspondence with
Kepler, then he would certainly have had been interested to look into the work
of the energetic and capable Johannes Kepler. Kepler’s Mysterium Cosmographicum was first published in 1596 while Paolo
Sarpi became state theologian to Venice 10 years later. Given Kepler’s tactful
questioning of the church’s orthodoxy on the movement of the sun, and Sarpi’s
own atheism (curious for a theologian), Sarpi must have been able to understand
that the Neoplatonic school itself had no respect for church dogma even with
its mindset of a divine Creator which Sarpi found necessary to reject. Sarpi
wrote of his fears for Galileo in being called to Rome by certain cardinals to
explain his support for the Copernican theory.[290]
At the same time, Venice was seeking to neutralise the influence of the Catholic
Church.[291] (Wootton
refers to Sarpi as a “Protestant conspirator”.[292])
Thus, Sarpi saw the opportunity to stoke conflict between forces that arose
from rationality (represented in part by the cultural tide that Kepler merely
typified) – which he opposed – and the worldly authority and wealth of the
Roman Catholic Church which also represented the Hellenic cultural current
albeit largely corrupted by ambitious or weak-minded sections of its officers
(i.e. clergy, etc).
We are seeing that cultural currents were at play
independent of the influence of any particular figure, but which could be
influenced or capitalised upon by shrewd, committed or capable persons. Kepler,
Sarpi and Leibniz are examples of such persons; chronologically, Sarpi sat
between the other two. Sarpi was a contemporary of, though elder to, Kepler
while Leibniz was born after Sarpi had passed on. Sarpi lived amidst the
conditions which gave rise to the Thirty Years War, while Leibniz’s educators
(particularly Jakob Thomasius), wider network and Leibniz himself dealt with that
war’s aftermath both politically and philosophically.
Sarpi corresponded with Francis Bacon who acted as an
inspiration for the founders of the English Royal Society. Indeed, “[l]etters
are still extant showing his [Sarpi’s] friendship with Lord Bacon”.[293]
Perhaps some of the lost letters would shed light on the origins of the Royal
Society, an organisation of political as well as scientific importance to this
day. The connection is worth looking into because the Venetian Party was an
established force in English politics at the time, and was instrumental in what
became the British East India Company. Paolo Sarpi was the most frequently
translated Italian writer in 17th Century England,[294]
which indicates quite an influence given that the English looked to Italians
for intellectual leadership in that period.
Given Sarpi’s relationship to Galileo and indirect
relationship to Kepler and his stance on how science should proceed, it would
be consistent that he should try to influence the course of science. It might
not be coincidental that the Baconian approach to science which came to
dominate the Royal Society is so closely allied to Sarpi’s empiricist outlook.
Leibniz was picking up the pieces of Kepler’s approach
to science and carried that standard, even while Empiricism rose in prominence
particularly in the circles of the Royal Society perhaps in the wake of the
influence of Sarpi.
We talk about Sarpi because scientific and political
battles often go together. Venice was a political and financial centre of
Europe during the 15th through 17th Centuries. Sarpi was
Galileo’s patron and was said to be the first to look through Galileo’s telescope.
Robertson says that Galileo and Sarpi actually constructed the telescope
together, and that that telescope was then presented to the Doge of Venice,
Leonardo Donato, as a gift.[295]
Indeed, Sarpi was, at least initially, a mentor to Galileo in astronomy “as he
[Sarpi] had had the start ... of his friend [Galileo] in the study of astronomy
and its cognate sciences, the advantage lay with him [Sarpi]”. However,
“Galileo, instead of being jealous of Fra Paolo, was jealous only for his
honour and pre-eminence, calling him ‘Il
mio pare e maestro’ – ‘My father and my master.’”[296]
Sarpi is said to have gathered a group of young
noblemen around him and secretly communicated with them his conception of
Atheism. Further, patrons are rarely dispassionate about or detached from the
work they finance, including Venetian patrons. The fact that Sarpi was chief
theological advisor to the Venetian republic while maintaining a secret Atheism
makes him particularly curious. Wootton argues that Sarpi did try “to give
practical implementation to ideas he had expressed in private: his political
activities were intended to subvert religious authority in general, his
published works were intended to undermine the foundations of religious
argument.”[297]
We must ask whether Sarpi was privy to any of the
correspondence between Kepler and Galileo. We suggest only a communication of
methods and mindset. Given the fundamental differences between Kepler’s outlook
on science and God, and Sarpi’s, and the Venetian Republic’s reputation for intrigue,[298]
the closeness of Sarpi to Galileo and the friendship between Galileo and Kepler
warrants more research.
Pietro Pompanazzi is an important intellectual figure
of the 16th Century. For our purposes here, Pompanazzi’s is relevant
to one particular issue. In The Treatise
on the Immortality of the Soul (1516) Pompanazzi argued that, to a degree,
the soul is inseparable from the body and to that extent the soul is mortal.[299]
This was at odds with Thomas Aquinas’ reading of Aristotle as consistent with
Christian theology.
Wootton explains that Sarpi’s Pensiero no. 4 reproduces Pompanazzi’s argument to demonstrate the
mortality of the soul.[300],[301]
Averroes argues that humans participate in an immortal reason and so the
rational was essentially, or behaved as if it was, immortal although it is
actually mortal. Pompanazzi said that man is a mean between mortal and
immaterial things. However, his essay Immortalitate
Animae refutes every argument it puts in favour of the immortality of the
soul[302]
and ends with an almost patronising sop to the Platonic and Christian belief in
the immortality of the soul.[303]
Wootton says that Sarpi’s Pensieri adopted Pompanazzi’s arguments against the immortality of
the soul, and then makes the case that Sarpi was relatively isolated in holding
his views as outlined in the Pensieri.[304]
Wootton misunderstands the meaning and role of “political views”. It is not necessary
for many/any to adopt those views overtly for them to be used and even provide
a steering role for the Council, particularly if Sarpi was important in
Venetian political circles and especially if the views expressed were of value
to Venice.
Some of Sarpi’s contemporaries were awake to what he
was doing and were unafraid to speak out. For example, Tommaso Campanella was
an anti-Venetian polemicist who argued that the inevitable outcome Venetian
policy was a society of moral atheists. Wootton explains that Campanella’s
warning against Sarpi’s methods provides a direct link between Sarpi’s Pensieri and the principles adopted
in/by the Venetian policy during the Interdict.[305]
Whatever Sarpi’s influence on Galileo may have been, we
certainly find much of Sarpi in Newton. Reading the account of Galileo’s
conception of simplicity in nature following the account of Kepler, we begin to
see that the core of Newton’s scientific method was not new. However, its
Averroism and its Atheism were new. We say Averroism because Newton required
that each conclusion be based strictly on observations without leaps of faith
or leaps of understanding even if the intention is to test that leap by
experiment. No conclusion is allowed unless it strictly and evidently follows
from what is observed. This is a kind of Averroist rigor, and it was a long way
from Galileo’s scientific method. Burtt explains that Galileo said that logic
is no tool for discovery and, indeed, Galileo is not a priori to the degree that Kepler was.[306]
Burtt explains that Galileo is close to Kepler in scientific mindset.[307]
However, there may have been differences in their conceptions of God.
David Wootton explains that Sarpi espoused an
empiricism similar to Newton’s: the best way of philosophising, Sarpi says, is
to begin with the evidence/testimony of the senses and to use reason to build
upon that. Wootton does not explain what kind of reason (clinically inductive
or otherwise) Sarpi advocates. Sarpi explicitly rejects perception following
reason because a priori reason can
only be speculation.[308]
All of this, of course, is similar to Newton’s position.
Burtt presents Galileo’s understanding that sensory
perception, including experimental observation, often delivers misleading
results.[309]
Application of reason guided Aristarchus and Copernicus to conclusions contrary
to observations. To say that it was purely “simplicity” that guided their
reason would be to ignore their wider and richer intellectual and cultural
environment. In any case, what is missing from the simplicity doctrine is the
higher hypothesis which, to be sure, is often more beautiful and harmonious
than the view “in the small” that had been being taken. That shift in thinking
may not at first sight appear to be simple. Perhaps a circle is simpler than a
polygon, but not when one has been reared on polygons and is trying to regard a
circle as an infinite-sided polygon.
In any case, it appears that Newton lifted the
“simplicity” idea from Neoplatonists such as Copernicus, and added it as a
qualification to a purist Empiricism.
Galileo follows these steps:
The Royal Society’s “British” approach
to science involves repeating experiments, finding a pattern, expressing the
pattern mathematically and calling that a law. This might be called a
“statistical” approach. It has the benefit of discouraging thinking that is
inexact or too speculative.
The more usual approach, closer to
Galileo’s three step and taught in Australian high schools in the 1980s, is to
form a hypothesis, design an experiment to test the hypothesis, and conduct the
experiment. Alternatively, one could check data from experiment or observation
against the hypothesis. Australian high school curricula omit the question of
how hypotheses are formed. Kepler’s a priori understanding of how God
thinks and acts would be relevant and useful in filling that gap.
In New Essays on
Human Understanding, Leibniz called the causes of phenomena “true
hypotheses”, and says that if the art of discovering true hypotheses is not
joined to Bacon and Boyle’s art of experimenting, “we shall never with utmost
cost attain [from experimenting] … what a man of great penetration might
discover at first sight.” Leibniz notes that Descartes made a similar remark
regarding “the method of the Chancellor of England [i.e. Sir Francis Bacon].”
He then notes Spinoza’s observation to the Secretary of the Royal Society,
Oldenburg, that Sir Robert Boyle “stops a little too long to draw from a great
number of fine experiments no other conclusion than this which he might take
for a principle, namely, that everything takes place in nature mechanically; a
principle which can be rendered certain by reason alone, and never by
experiments however numerous they may be”.[310]
Contrast this Newton, from an article
by McGuire:[311]
·
From Compounds to
Ingredients,
·
From Motions to
the Forces producing them in general, from Effects to Causes and from
particular causes to more general ones,
·
…until the
argument ends in the most general causes.
This is the method of Analysis. The Synthesis
comprises (i) assuming the causes discovered and established as principles, and
(ii) using the causes to explain the phenomena proceeding from the causes.
Further, the causes/principles are used to prove the explanations.
This is the method that Copernicus used, as Kepler
explains.[312] Newton’s
summary starting “This is the method of Analysis” is a restatement of Kepler’s
claim that Copernicus’ conclusions could be proved with Euclidean exactitude,
and a priori too. At that same page,
Kepler is clearly working entirely within the domain of hypotheses. He first
refers to the “customary hypotheses” with an excessive number of circles, and
then to Copernicus’ hypothesis with far fewer circles.
In point 7 above, in light of point 4, when Newton
refers to “causes”, it appears that he is referring to correlations. If Motion
B always follows Force A, then A causes B. What if there is an undetected
phenomenon that is causing both A and B? Newton’s concept of Cause requires
more examination.
In point 3 above, Newton says Hypotheses are not to be
regarded in experimental philosophy, i.e. we must always argue by induction
from the observational data. But does Newton posit a method other than
experimental philosophy in which Hypotheses are
permitted?
In the next section, a thesis on the origin of
Newton’s method of discovery that was first presented by Mamiani in 2001 is
described. It is argued that Newton’s source was a logic text by Robert
Sanderson published in 1618 which is essentially a Ramist logic text, meaning
in the tradition of Petrus Ramus who lived in Europe from 1515 to 1572. Ramus’
influence was largely through his logic texts rather than on logic itself.
While Ramus energetically promoted the need for reform in the Aristotelian
curriculum of the time, it was not so much that he thought that the content of
the curriculum was incorrect as the way in which it was taught. For example, he
thought that students were required to spend too many years learning material
which ultimately would not be useful to them. Also, Ramus argued, the many
years required for study meant that education was too expensive for children from
poor families.[313]
An extended discussion of Ramus is beyond the scope of this thesis.
Mamiani in Isaac Newton’s Natural Philosophy (“INNP”) describes the
origin of Newton’s regulae philosophandi
as this transformation:[314]
1. |
Robert Sanderson’s Logicae artis compendium (Oxoniae 1618) |
which Newton
read c.1661-4 and which follows the Ramists rather than the Scholastics, in
that it emphasizes the theory of method |
|
ê |
|
2. |
Newton’s Treatise
on the apocalypse |
16 rules |
|
ê |
|
3. |
Newton’s Principia
1st ed. 1687 |
2 rules |
|
ê |
|
4. |
Newton’s Principia
2nd ed. 1713 |
3 rules |
Noteworthy points regarding the
Treatise on the Apocalypse are:
Mamiani is intentionally seeking a transformation
process of these rules of philosophy, emphasizing the inductive nature of the
process.
The original of Sanderson gives the
method of invention as having no law, unlike methods of resolution and
composition which are stated as having five laws in common. Rather, the method
of invention is given as having means-cum-steps:
Sanderson says that the method of invention has
nothing in common with the method of resolution or analysis.
True to the statement of Analysis by Newton, Newton
writes to Oldenburg with a plan to simplify the rules of optics to the most
general form.
Newton lists laws which, says Mamiani,
start with these from Sanderson’s Logicae
artis compendium or Compendium of the
Art of Logic:
(a)
The law of
brevity
(b)
The law of
harmony
which are then transformed by Newton
to:
(a)
Observe the
consent of the Scriptures
(b)
Choose
constructions which reduce contemporary visions to the greatest harmony of
their parts.
Newton’s stance against Hypothesis is
affirmed and elaborated by Burtt.[315]
Koyré writes, “one can interpret the Newtonian view [of hypotheses] as the
end-product of the English tradition of empiricism, that of Bacon and of Boyle”.[316]
Newton’s four laws of scientific enquiry are then given:[317]
The fourth of these affirms that empirico-deductive
interpretation to be given to the first three. Newton goes beyond Galileo and
Sarpi’s empiricism by asserting that it is not only methodological limits that
prohibit hypotheses but also the nature of what is knowable. Indeed, the upshot
of the fourth law is that nothing is known or can be known beyond experimental
results (p.216). The exclusion of hypotheses acts as a prohibition on creative
mentation. This repeats one of Pompanazzi’s arguments against the immortality
of the soul: that “the intellect is inseparable from matter”[318]
which might imply – as Wootton says – that “there is no knowledge without
sensations”.[319] Sarpi
adopted the position that “there is no knowledge without sensations” in his Pensiero No. 4.[320]
Nor did this idea stop at Sarpi. John Locke adopted it saying, “The knowledge
of the existence of any other thing we can have only by sensation: for there
being no necessary connexion of real existence with any idea a man has in his
memory.” Further, “the having the idea of anything in our mind, no more proves
the existence of that thing, than the picture of a man evidences his being in
the world.”[321]
(emphasis in original) Leibniz argues the contrary in New Essays giving the example that we are certain that Julius
Caesar lived without having had direct sensation of the phenomenon. Leibniz
refers to knowledge arising from “immediate internal experience of an
immediateness of feeling” as “primitive truths of fact”.[322]
We add that Leibniz – knowingly or otherwise – was reiterating Hermes
Trismegistus.[323] In
Bruno’s De umbris idearum published
in Paris in 1582, the character of Hermes introduces a philosophy in which “The
intellect stands certain of what it instructs whereas the senses are falsely
moved”.[324]
For Newton, mathematics had a role in describing
observed phenomena precisely. Using the mathematical description, one then
simplifies that description as far as possible.[325]
From Newton’s laws of scientific enquiry, one’s observations in mathematical
expressions are reduced to something like a simplest “normal form”. Thus,
mathematical symbols are descriptive of observations, and do not represent
Platonic ideas or anything incorporeal.
Newton said that reducing motion to his three laws of
motion was an achievement, but if a single cause of all three can be “found
out” so that they can be reduced to a single equation then that will be an even
greater achievement. It must be asked how a “cause” can ever be found when all
one has is descriptions of observations.[326]
Only correlations are observable, not causes. Newton says that deductions from
observations are allowed, so that better experiments can be designed. Yet
one wonders how experiments could have been designed in Kepler’s time that
would have allowed Kepler to “deduce” the inverse square law of gravity.
Koyré argues that Newton’s hostile attitude to
hypotheses can be detected throughout Newton’s career and, indeed, “the
antihypothetical attitude is present – though in a much less rigid form – in
the very first works of Newton.”[327]
Mamiani says that in the 9th
rule of Newton’s Treatise on the
Apocalypse, Newton makes clear that simplicity is a consequence of the law
of harmony, and then quotes from the 9th rule:[328]
To choose those constructions which
without straining reduce things to the greatest simplicity. ... Truth is ever
to be found in simplicity, and not in the multiplicity and confusion of things.
As the world, which to the naked eye exhibits the greatest variety of objects,
appears very simple in its internal constitution when surveyed by a philosophic
understanding, and so much the simpler by how much the better it is
understood...
Kepler’s dedicatory epistle to Mysterium Cosmographicum emphasizes the beauty, brilliance and
harmony of creation more than its simplicity. Later in Mysterium Cosmographicum when considering hypotheses, Kepler
explicitly prefers the simple to the convoluted. Leibniz in 1686 reiterated
Kepler, “Reason wishes to avoid multiplicity in hypotheses or principles very
much as the simplest system is always preferred in Astronomy.”[329],[330]
Newton seeks simplicity by combining, transforming,
resolving rules generated by observations. Importantly, we see that Newton
adopts a mechanical process of discursive logic to “derive” the laws of nature
from observations. For Leibniz, this was nothing but the “art of reasoning
well”, quite distinct from the “art of discovery”.[331]
In 1677, Leibniz elucidated on his boyhood ambition of creating a
Characteristic whereby reasoning could be conducted in a symbolic language
comprehensible to people of all classes and nations so that disputes could
readily be settled.[332]
The important thing is that what for Newton was appropriate for discovery, for
Leibniz was extremely useful for reasoning and teasing out the implications of
existing knowledge.
As far as discovery is concerned, Kepler had gone the
opposite way to Newton by forming a hypothesis that is harmonious and
beautiful, and then going about calculating the connection between it and the
observations.
Regarding Newton’s approach in Treatise on the Apocalypse, which was
written prior to Newton’s Principia,
Mamiani writes:[333]
What is new in Newton’s rules? Neither
the content nor the expression. It is true that Sanderson’s laws are very
concise whereas Newton’s rules are verbose and redundant. We must wait for the
rules of the Principia in order to
find a conciseness equivalent to the laws of Sanderson’s Logic. There is,
however, a great difference among Newton’s rules of interpreting the Apocalypse
and Sanderson’s laws. Sanderson is repeating the precepts of a dead tradition
for presenting or teaching acquired knowledge. Newton [on the other hand] is
proposing rules to be used in discovering new knowledge.
This suggests the source of the current
view of:[334]
the English “Baconian” concept of how
to do science, followed by the Royal Society. You’re supposed to collect lots
of observations and reports, do experiments, then generalise to laws, then (optional)
think up causal hypotheses (e.g. atomism) that might explain them. That’s an OK
method to do science, especially if the science you want to do is very
empirical, e.g. chemistry. And very British and practical.
Leibniz was not “anti”-discursive logic or
propositional logic. Indeed, Leibniz has been called the most original logician
of the 17th century and credited with many original discoveries
including the beginnings of what is now regarded as standard classical
propositional logic decades if not centuries before others.[335]
The difference is that Leibniz would only have regarded this as a way of
interpreting existing knowledge, or to answer questions which could be answered
from the existing body of human knowledge. This is no small thing, and Leibniz
aimed with his universal/general characteristic even to allow disputes between
people and perhaps nations to be resolved. This coming from Leibniz the
diplomat was quite an ambition.[336]
However, it is different from making a new
discovery. We will see in a later chapter that when Leibniz considers the
principles and process of new discovery or “the art of discovery”, a different
field of considerations comes into play.
What if experiments cannot be designed but only
hypothesising is possible? An example is the quest to understand the solar
system using observational data collected from earth. It can surely be
demonstrated that restricting ourselves to Sanderson’s induction/deduction
precludes leaps of understanding, inspiration and anything that smacks of human
creativity under most definitions. No a
priori presumption of harmony is permitted. Newton has certainly excluded
Cusa’s “learned ignorance” in which leaps of knowledge arise from inspiration
rather than from “discursive logic”. Science and mathematics would cease. For
example, consider the famous case of Gauss answering his teacher in adding the
integers from 1 to 99. No doubt there would have been an admixture of some kind
of deduction alongside an “aha!” sense of inspiration. We are now as much in
the domain of psychology as science. Leibniz, in his essay “How to Reason
Well”, named the holding of all aspects of a body of knowledge in one’s mind at
once as a prerequisite to full understanding and formulating a hypothesis as to
the solution to a problem.
There are serious consequences of
Newton’s mandate for science:
Leibniz actually noted that an overemphasis on
amassing experimental data and an underemphasis on reasoning was retarding
scientific progress at the Royal Society. Some years after his 1673 visit to
the Royal Society in London, Leibniz wrote, “they confessed to me in England
that the great number of experiments they have amassed gives them no less
difficulty than the lack of experiment gave the ancients”.[339]
At the same time, Leibniz tried to reconcile the pragmatic need for experiment
with a priori reasoning.[340]
Burtt sums up Newton’s approach in, “Science is the exact mathematical
formulation of the processes of the natural world. Speculation is at a
discount, but motion has unconditionally surrendered to the conquering mind of
man.”[341] (emphasis
in original) Actually, the Newtonian method says that science is the exact
mathematical description of the
observable processes of the natural world, and this stems from Newton’s failure
to embrace the method of hypothesis even if Newton did not outright reject the
use of hypotheses. Further, motion has not unconditionally surrendered to the
mind but, rather, the mind of man is surrendering to the bodily senses’
observation of motion, which enslaves the mind by denying its capability to reason
or create.
Burtt rebuts the contention that Newton
is free of metaphysical concerns or crutches. In fact, the “objective rigor” of
Newtonian science is replete with metaphysical assumptions which we outline
below. Newtonian positivism is the idea that truth can be acquired without
presupposing any theory of their ultimate nature, such as of the Creator. Burtt
says that anyone who claims to be free of metaphysical assumptions actually
does adopt metaphysical notions in these ways:[342]
The way Newton said science should be done and the way
that Newton actually did science were two different things. Newton himself did
not proceed by pure induction in his own work. Burtt goes on to explain how and
in what way Newton adopts metaphysical assumptions of all of the above three kinds.
Koyré disputes that Newton followed his own prescription on hypotheses. Koyré
writes, “The expression ‘hypothesis thus seems to have become, for Newton,
toward the end of his life, one of those curious terms, such as ‘heresy,’ that
we never apply to ourselves, but only to others. As for us, we do not feign hypotheses, we are not heretics. It is they, the Baconians, the Cartesians,
Leibniz, Hooke, Cheyne, and others – they feign hypotheses and they are the heretics.”[343]
We now make some comments on logical deduction which
will usher in the next chapter.
Empirico-logic was promoted under a different guise by
the contention that a computer equipped with sensory apparatus and programmed
with the rules of logic can replicate the human mind. Thus von Neumann’s
viewpoint is at its heart the same as the rules of Newtonian science. This
dissertation will not investigate this question further.[344]
Despite the fact that he was a designer of machines and even of a calculating
machine, Leibniz could not have shared von Neumann’s view. For a start, the
human soul is a monad which is connected with every other monad in the universe
and has full knowledge of everything that is occurring in the universe, which
is quite unlike any computer. If Leibniz regarded the action of the mind of man
as God in the miniature,[345]
then there is another raft of reasons why Leibniz would disagree, since few
would argue that God can be modelled by a computer.
Euclidean geometry presents a system of formal
reasoning, essentially embodying the idea of the syllogism. Plato embraces such
reasoning with a geometrical problem in Meno. Plato also demonstrated
how reasoning can be conducted on non-mathematical ideas, such as justice and
goodness. There was a raft of mathematical thinkers in Ancient Greece who were
able to reason both in the abstract and in the concrete. Their work raises the
question of the relationship between abstract formal reasoning and the real
world. It also raises the question of the relationship between abstract ideas
and the physical world.
We become aware that there is a mode of thought that
is not formal reasoning, as such, but which is structured and is worthy of the
name reasoning. Indeed, Archimedes performed work in his own way, but when he
recorded or transmitted what he had found, he used formal reasoning of a kind
which was not what he used to reach the discovery in the first place. At the
same time, the actual method he used was not to simply “throw things together
and watch what happened” either. There was a searching for an order that was indefinable
because it had not yet been found. He would recognise it when he found it,
though he might be misled many times along the way. The search for order or harmony
was undertaken with a sense of necessity or passion and that search amounts to
the search for a viable hypothesis. Leibniz wrote, “The art of discovering the causes
of phenomena, or true hypotheses, is like the art of deciphering, where an
ingenious conjecture often shortens the road very much.”[346]
The formal reasoning was often not part of the search and was only added
afterwards. Of course, confirmation by formal reasoning or by experiment must
follow.
Boyer notes that Archimedes of Syracuse “tempered the
strong transcendental imagination of Plato with the meticulously correct
procedure of Euclid.”[347]
Boyer suggests how Archimedes did this in one case: that Archimedes used
infinitesimal considerations as an heuristic method “simply as an investigation
preliminary to the rigorous demonstration by the method of exhaustion.”[348]
It could be argued that the heuristic method represents the transcendental
imagination of Plato while the method of exhaustion represents Euclidean rigor.
However, must there be an exclusive choice between transcendental imagination
and rigor?
Platonists were never opposed to rigor, and not even
opposed to the kind of work that Euclid, for example, did. The difference was
that the Platonists insisted that there was an indefinable something that was
the quarry of any process of discovery or creativity, and it is always in that
that the crux of any question in science or philosophy lies. The Pythagoreans
insisted that such concepts were valid, and number was the key example in their
time. Such concepts should be the focus of science but, in doing so, the human
mind becomes the centre of science. We conclude that the human mind must be
intertwined with what science is. This is not as Burtt understood things in
opposing the Russellian pessimism and the Newtonian empiricism. Burtt argued
that human experience was intertwined with science. On the contrary, we
would argue that the Leibnizian view is that the pure ideations of the human
mind are where science actually takes place. These are tested in the world, as
Archimedes did in his bath tub or as Huygens and Papin did with their work towards
building powered engines. Adopting Leibniz’s view would resurrect what Burtt
instinctively knew to be right but could not quite understand – that Dante’s conception
of Man as the jewel of Creation due to the uniqueness of Man’s mind was correct
and the Newtonian-cum-Russellian notion is unsustainable. The Russellian notion
is that Man is not only insignificant in Creation but is not even part of
Creation, except for Man’s biological body. The inexorable conclusion from the
Russellian notion is that the mind has no role in science.
Burtt purports to bring humanity back into the
universe by saying that human experience is a scientifically “good enough”
representation of the physical universe. This is essentially Cartesian
subjectivism. As Plato said in The Republic, all experience is opinion,
which flies in the face of Burtt’s resolution. We prefer to solve the paradox
that Burtt correctly posed using Leibniz’s essays. Leibniz’s metaphor is that
reason is akin to life while experience is the air essential to “life” in that
it gives context and subject matter to reason about. An alternative analogy is that the mind is the potter and
the experience is the clay which gives subject matter to the mind.
Where did mathematics of the kind we now know
originate? For a history of mathematics in toto, we can refer to Ball,
Boyer and many others. Suffice it to say that the Pythagoreans and
Neoplatonists and their descendants made strides in mathematics, and it was
generally the role of the rigorists, as Leibniz called them, to complain that a
certain innovation could not be done or was not allowed. Such critics still
played the role of mastering the art of calculation within any given domain of
mathematics as it stood during any particular historical milieu. Further,
Leibniz was not opposed to rigor per se.
Who were the “rigorists”? There were and are many.
Leibniz named a number in his letters when defending his methods against their
attacks. It is too simplistic to associate such critics with any particular
philosopher though it is common to associate such attempts with adherents of Aristotle
or Averroes. Leibniz agreed with Aristotle on many points and sometimes
explicitly resorted to Aristotle to rebut Descartes. Leibniz says that
Aristotle was the first to give demonstrations in Logic, “and we may say he was
successful, but he was far from being successful in the other sciences he
treated.”[349]
Scholasticism owed its roots to Aristotle and owed something to Averroes, but
it also owed much to Thomas Aquinas’ purported defence of Aristotle against
Averroes’ alleged misinterpretation or misuse of Aristotle’s work and name. By Leibniz’s
time, mathematics included so much more than Averroes ever addressed centuries
prior. Further, there was no recognisable school of “Averroists” active in Leibniz’s
time.
Did Newton combine rigor such as that of Euclid and the
scholastics with the wave of Neoplatonic numerical mathematicisation? It might
seem that Newton had the effect of constraining the new Neoplatonic mathematics
with an incarnation of Aristotelian reasoning, namely that of Petrus Ramus as
presented in Robert Sanderson’s Compendium
of the Art of Logic. However, we must deny such credit to Newton due to
Kepler and Leibniz’s own rigor. What, then, is different about the
Neoplatonists?
Nikulin quotes Manin, “A good physicist uses formalism
as a poet uses language.”[350]
Nikulin explains, “since the language in poetry often says and suggests more
than is intended, and, in fact, negates and suspends the language by means of language
itself, it may become prescriptive, opening new possibilities for experiencing
the world.” We would amend this slightly to say that the good poet intends more
than the language denotes, and already knows what new possibilities they are
seeking to describe and prescribe for further investigation. Similarly, the
physicist is not constrained by the limitations of formalism, but is required
by pragmatic circumstances to fit their ideas in formal language. What Nikulin
touches on with respect to the physicist and formalism, we would also say about
the Neoplatonic mathematician and formalism. In particular, a Leibnizian
communicates the metaphysical/ontological using the constraints of formal
symbology.
Leibniz took exception to the rigorists demanding more
from mathematical formalism than it is fit to deliver. That is, he took
exception those who criticised his calculus because it relied on the
infinitesimal which is a thing that might not exist. However, this may have
been because the ontological concept of the infinitesimal had not as yet been
made sufficiently precise. It may also have been because the symbology was not
sufficiently advanced or appropriate.
Nikulin may provide the key to an ontologically real
mathematical object.[351]
However, Leibniz was not concerned whether the infinitesimal was even
ontologically real; to Leibniz, it was merely a thought tool. In any case, the
question of what an ontologically real mathematical object is might tell us what
is different about Neoplatonic mathematicisation versus Newtonian
mathematicisation. Nikulin was grappling with the question of how mathematical
objects which do not have physical existence can be used in physics.[352]
It is the duality issue discussed in Chapter 3 in the context of Grosholz’s
analysis of Leibniz’s mathematics. To Leibniz, this is the distinction between
the incorporeal domain of ideas and active souls, and the passive domain of
matter. Leibniz tied them together with his doctrine of pre-established
harmony.[353]
To Kielkopf who undertook a thorough review of
Wittgenstein’s foundations of mathematics, the mark of an “absolute platonist
[sic]” is that they hold that “the domain of mathematical objects themselves
exist independently of minds.”[354]
As we will see in Chapter 7, Leibniz is not this kind of platonist because Leibniz
does not consider ideas to be real and only regards them as able to exist in
minds. However, he also regards them as having an objective truth so that their
qualities and properties are independent of minds although they have no
existence independent of minds. Further, he finds an actual correspondence
between ideas and the physical universe which is neither subjective nor dependent
on sense perception implying that there is a universe parallel to the physical
universe in which ideas exist. Thus, we find a mixedquality in Leibniz’s conception
of mathematicals (a subset of “ideas”) such that in certain discussions, he is
platonist in the sense of believing there to be a non-physical universe in
which ideas exist, in other contexts he analyses and plays with ideas as if
they were real things, while in other cases still and especially in the context
of monads Leibniz regards ideas as never existing anywhere except in minds and
usually only in God’s mind.
Are group theory, category theory and other modern
forms of algebra Neoplatonic mathematics or are they better described as positivist
disciplines? Lie groups are indispensable to modern physics, so they should be
given at least as much credit as the infinitesimal.[355]
We know that group theory has utility for the study of symmetry and pattern,
and has been useful in the practical sciences such as chemistry. Leibniz would
have to regard them as useful and therefore legitimate domains of mathematics.
It is likely that Leibniz would embrace them. In fact, it is hard to see how
those who criticised the idea of the infinitesimal could not direct the same
criticism at Lie groups or any part of group theory. What would Leibniz say
about category theory which is an all-encompassing form of algebra or meta-algebra?[356]
Any discussion of the philosophy of category theory must consider Bourbaki, for
the founding fathers of category theory (Eilenberg and Moore) were among the
handful of founders of Bourbaki. The contribution of category theory to
diagrammatic representation of abstract concepts would surely be admired
alongside its contribution to the understanding of mathematical structure.
There was a raft of mathematical thinkers in Ancient
Greece who were able to reason both in the abstract and in the concrete. Their
work raises the question of the relationship between formal reasoning and the
physical world, and between abstract ideas and the physical world. We become
aware that there is a mode of thought that is not formal reasoning, but which
is structured and is worthy of the name reasoning. There was a searching for an
order, indefinable because it had not yet been found. It is recognised only
when found, though the thinker or scientist might be misled many times along
the way. Formal reasoning is often not the part of the search when
breakthroughs occur, but is often added afterwards. We argue that this searching
or exploration is what Leibniz called Reason, and it is the primary tool in
what he called the Art of Discovery.
The “from where to which” of
discursive reasoning is limited compared with learned ignorance (docta ignorantia). This larger “God’s
eye view” transcends discursive reasoning and also allows us to see reason and
harmony in the large. Because we can do this, we can approach God’s mind and
should seek to do so. It is from such a level that hypotheses are formed
ultimately to be tested by the structured observation of experiment.
Petrarca considers “learned
ignorance” in the sense that Augustine uses it. To Petrarca, “learned
ignorance” is ignorance which is blessed and to possess it is to be enlightened
by the teachings of the Holy Spirit. To Petrarca, it is the highest kind of
knowledge. To put it more accurately, since it is more a mindset or an
ideological and intellectual disposition, it is the most amenable to (a)
gaining new understanding of anything concrete and even (b) to thinking.
Nicolaus of Cusa
crystallised the Augustinian idea which had been picked up by Petrarca, and we
know that Cusa possessed a copy of Petrarca’s De Ignorantia. The book by Cusa applies specifically to science and
reason. In the debate with Wenck, Cusa sets learned ignorance against
discursive logic and explains why it is superior. Through De Docta Ignorantia, Cusa explained why number is so important and
guessed at why the Pythagoreans thought it to be so. Cusa claimed that even
comparative relation cannot be understood independently of number, which means
that number perhaps subsumes even the syllogism.
Since, relative to all that
can be known, unknowing or ignorance is our state, and since as Cusa says the
highest knowledge we can attain is that of our own ignorance, then no original
knowledge can emanate from deduction because deduction uses only what we
already know. To have knowledge of our ignorance is to enter the larger, and
vast, domain, of what we do not know. Deduction does not work in this domain,
but only inspiration and leaps of insight can make any sense.
Leibniz understood that
deduction allows us to clarify points of contention using what we already know.
He regarded this as something that the common man could do when arguing with
his colleague over any mundane matter. This is very different, however, from
the Art of Discovery as Leibniz called it.
The
Sarpian-Newtonian idea of conducting experiments and deducing from those
experiments is all very well too. This is not inconsistent with “physical
Platonism” explained by Davies:[357]
Most
theoretical physicists are by temperament Platonists. They envisage the laws of
physics too [like the principles of mathematics] as perfect idealized
mathematical relationships and operations that really exist, located in an
abstract realm transcending the physical universe. I shall call this viewpoint
physical Platonism.
We do not advocate this brand of Platonism as metaphysically correct, or even as a perspective of physics that Plato himself would have adopted. The question here is about how those laws are discovered understood. Overemphasis on experiment does not allow the investigator to address ironies or contradictions in what is observed, which can only be found through thought and by entering the domain of our ignorance in science, mathematics and metaphysics. In his learned ignorance discourse, Cusa is not describing a mystical process but carrying on a conversation about human thought. We argue that Leibniz worked in this tradition.
In a letter, Leibniz poetically prefaces his recent discoveries with a gunpowder metaphor on the light of discovery: “I am also sending you a little of the corporeal fire, which can well be called a perpetual light, for when properly protected, it lasts many years without being consumed. … It is easy to ignite gunpowder, either by the sun or through friction, after a little of this phosphorous is mixed with it.”[358]
How does Leibniz come by this “phosphorous”? He gave
us 10 maxims on the art of discovery.[359]
These are to be distinguished from the art of reasoning for which Leibniz in
the same essay gave three maxims.[360]
To discover something new, we must – broadly speaking
– follow these broad steps:[361]
Leibniz may be referring to this process when nine
years later he says that “my own views have become fixed only after my
considering all sides and weighing them well” summing it up as “I have
anticipated everything and gone through it in my mind”.[363]
According to the Platonists as Leibniz describes them,
they were “not far wrong” in listing the four kinds of cognition of the mind
as:[364]
Level |
Called |
Or, in other words... |
1 |
sense |
experience |
2 |
opinion |
conjecture |
3 |
knowledge |
demonstration which is reasoning by which some proposition is made certain, and this is nothing but the analysis of a truth into other truths which are already known[365] |
4 |
understanding |
intellection which looks into the connections of truth in a single act of the mind |
Of these, the third bears some resemblance to item 2
“prerequisites of the prerequisites” and the fourth (“intellection”) appears to
be item 3 or the goal of “perfect knowledge”
of the thing. Further, to have perfect knowledge of a thing, everything arising
from experience and that can reasonably arise from conjecture must be
consistent or explicable in terms of the knowledge that has been attained,
otherwise that knowledge is not yet perfect.
Intellection “belongs to God in all things but is
given to us in simple matters only”.
The most striking thing is how much hard work is
involved. The generality of the Leibniz’s ten points indicates that the same
rules apply whether one is designing a water pump, addressing a legal problem,
or working out the metaphysical foundations of science. Neoplatonism is about
having worked through these steps with the fundamental questions of theology
and metaphysics before working in the natural sciences. Having done that
background work in the largest context for natural science, one has a ready
source of plausible a priori hypotheses which are consistent with one’s
theology and metaphysics with which to start.
Leibniz explains in greater detail what is involved in
science in a paper An introduction on the
value and method of natural science.[366]
Section subheadings in the paper include these which are descriptive enough to
be useful to list here among a number of others:
1.
“Empirical
physics is useful for human life and should be cultivated in the state”,
2.
“A catalogue of
experiments is to be compiled”,
3.
“New experiments
are to be undertaken at public expense, and only men outstanding not merely in
science but in virtue are to be placed in charge”,
4.
“With the
experiments are to be combined accurate and thoroughly extended reasonings
after the manner of geometry, for only in this way can causes be discovered”,
5.
“The most perfect
method involves the discovery of the interior constitution of bodies a priori
from a contemplation of God, the author of things. But this method is a
difficult one and not to be undertaken by anyone whatever”,
6.
“Some hypotheses
can satisfy so many phenomena, and so easily, that they can be taken for certain.
Among other hypotheses, those are to be chosen which are the simpler; these are
to be presented, in the interim, in place of the true causes”,
7.
“Analogies are
useful in guessing at causes and in making predictions” and
8.
“The method of
reasoning from experiments resolves the phenomenon into its attributes and
seeks the causes and effects of each attributes”.
Discovery often takes place in the domain of
metaphysics, and sometimes even in theology. Leibniz wrote, “The a priori method is certain if we can
demonstrate from the known nature of God that structure of the world which is
in agreement with the divine reasons and from this structure, can finally
arrive at the principles of sensible things. This method is of all the most
excellent and hence does not seem to be entirely impossible. For our mind is
endowed with the concept of perfection, and we know that God works in the most
perfect way.” Due to the difficulty of this method, Leibniz says that we must
retain the a posteriori method, which
is the second method.[367]
The 15th to 17th Centuries saw a Neoplatonic revival in
the sciences. That revival brought with it a conviction that “mere patterns and
relational structures” (to use Grosholz’ words) are the metaphysical “furniture of the universe”. To a Neoplatonist, there
is no contradiction but, rather, it is the order of things as derivable from
Book VI of Plato’s The Republic.[368]
That book argues that the philosopher has a method for finding answers which
resolve existing contradictions, though there is no guarantee that new
contradictions will not be found. Book VI is famous for introducing the
“divided line” which is a classification of intellectual exercise from the
lowest level of conjecture and up to belief. Both conjecture and belief are
mere opinion, not knowledge. Next comes understanding and finally there is
exercise of reason. The latter two deserve the label “knowledge”. Opinions
derive from the exercise of the sense. Understanding includes mathematical
thought, but is defined by Plato as images and concepts of thought such as
ideal squares and cubes. The highest level of the exercise of reason is called
dialectical thought, and it deals with ideas or ideals such as perfect beauty,
justice and goodness.[369]
Leibniz was conscious of the metaphysical dependencies of his infinitesimal calculus and of its ancient roots. He recognised that Archimedes faced similar problems of political correctness that he himself was facing:[370]
When my infinitesimal calculus, which
includes the calculus of differences and sums, had appeared and spread, certain
over-precise veterans began to make trouble; just as once long ago the Sceptics
opposed the Dogmatics, … and such as Francisco Sanchez, the author of the book Quod
nihil scitur, brought against Clavius; and his opponents to Cavalieri, and
Thomas Hobbes to all geometers, and just lately such objections as are made
against Archimedes by that renowned man, Dethlevus Cluver.
Leibniz notes a pattern through the
republic of letters. He objects in particular to the use of a requirement of rigor
as a particular set of restrictions to criticise his method, as he appears to
regard such requirements as arbitrary or at least unjustified. Leibniz was well
aware of the opposing personalities:[371]
When then our method of infinitesimals,
which had become known by the name of the calculus of differences, began to be
spread abroad by several examples of its use … a certain erudite mathematician,
writing under an assumed name in the scientific Journal de Trevoux,
appeared to find fault with this method. But to mention one of them by name,
even before this there arose against me in Holland Bernard Nieuwentijt, one indeed
really well equipped both in learning and ability.
Leibniz discusses the concept of infinitesimals and
explains that, for example, “those things that are found to be true about a
parabola [treated in such a way] are in no way different, for any construction,
from those which can be stated by treating the parabola rigorously”.[372]
Nieuwentijt’s disagreements with Leibniz were not over rigor, but rather over
the use of the infinitesimal. First of all, Nieuwentijt believed that the
infinitesimal was zero so any quantity multiplied by it gave a product of zero.
Second of all, there could certainly be no second order differentials because
these require the multiplication of an infinitesimal with another
infinitesimal. When opposing Leibniz on second order differentials, Nieuwentijt
argues that “there are no gradations in the infinite”.[373]
Nieuwentijt’s proofs assume a mathematics that is based in what can be
visualised. The fact that an infinitesimal could have a geometrical meaning and
could even be multiplied by another infinitesimal thereby obtaining a new
geometrical meaning is beyond the pale if one is trying to visualize the
calculation. Of course, Nieuwentijt’s objections to “gradations in the
infinite” pre-empted Cantor’s theory of transfinite numbers.
Leibniz suggests that Archimedes faced similar
criticisms, “Truly it is very likely that Archimedes, and one who seems to have
surpassed him, Conon, found out their wonderfully elegant theorems by the help
of such ideas [infinitesimals]; these theorems they completed with reductio
ad absurdam proofs, by which they at the same time provided rigorous
[Euclidean] demonstrations and also concealed their methods.”[374]
It may be that Gauss did the same.[375]
Leibniz notes that Descartes too was aware of this.
“Descartes very appropriately remarked in one of his writings that Archimedes
used as it were a kind of metaphysical reasoning (Caramuel would call it
metageometry), the method being scarcely used by any of the ancients (except those
who deal with quadratrices); in our time Cavalieri has revived the method of
Archimedes, and afforded an opportunity for others to advance still further.[376] Indeed Descartes himself did so, since at one time he
imagined a circle to be a regular polygon with an infinite number of sides, and
used the same idea in treating the cycloid; and Huygens too, in his work on the
pendulum, since he was accustomed to confirm his theorems by rigorous
demonstrations; yet at other times, in order to avoid too great prolixity, he
made use of infinitesimals; as also quite lately did the renowned La Hire.”[377]
This two millenia context surpasses Grosholz’s interpretation of Leibniz.[378] For Grosholz referred to “the contradictory being of Leibniz’s infinite-sided polygons, at once continuous and discrete, geometric and combinatorial, infinitary and finite”. Yet the “infinite-sided polygons” were treated by Nicolaus of Cusa two centuries before Leibniz wrote about mathematics. Cusa explained that a circle or “infinite-sided polygon” is of a higher order than a polygon. It was relevant to the discovery process because the human mind is like a polygon while God is the circle. Kepler wrote “in this one respect Nicholas of Cusa and others seem to me divine, that they attached so much importance to the relationship between a straight and a curved line and dared to liken a curve to God, a straight line to his creatures; and those who tried to compare the Creator to his creatures, God to Man, and divine judgements to human judgements did not perform much more valuable a service than those who tried to compare a curve with a straight line, a circle with a square.”[379]
With God’s mind being of a
higher order to human understanding, as a circle compared with a square, so we
must delve into our ignorance, and understand its quality in order to step
closer to the unreachable magnitude and quality of God’s intelligence.
The contradictions of “infinite-sided polygons, at once continuous and discrete, geometric and combinatorial, infinitary and finite” were addressed by Cusa in his argument as to what God is and is not, in considering God’s enfolding and unfolding. Cusa might as well be speaking to Grosholz as “the Adversary” who was actually Johannes Wenck a leading Aristotelian cleric.[380] Cusa says that God is “absolutely, everything which is at all possible; and in this coincidence is hidden all apprehensible theology.” In particular, “in the mode of enfolding [God] is all things but that in the mode of unfolding He is not any of these things.”
In De Ignota Litteratura, Wenck responds to Cusa:[381]
…if God (as he supposes the essence of
the unqualifiedly Maximum to be) is the precise measure of every essence, then
how will it be the case that He exceeds, incomparably, every essence? And how
will the following [doctrine from] Metaphysics X [i.e. book 10 of Aristotle’s Metaphysics] remain standing?: “In each
genus a first thing is the measure (metrum
et mensura) of the subsequent things of that genus; hence, in each genus
there is a proper and precise measure.”
Cusa’s counter response is “the infinite is the most adequate measure of finite
things—even though the finite is altogether disproportional to the infinite.”[382] Though “the impossibility of there actually being an infinite
line is shown in many ways in On Learned
Ignorance” yet “by the positing of an infinite line the intellect is helped
to make headway toward the unqualifiedly Infinite, which is Absolute Necessity
of being.”
On a different track, Leibniz argues that if a
proposition is not true in the infinitely small, then it is questionable as to
whether it is true in the finite. He explains that all of the propositions that
hold for the ellipse should hold for the parabola because regarded merely as an
ellipse with one focus infinitely far from the other. Similarly, if rest is
regarded as infinitely small velocity or infinite slowness, then whatever is
true of motion should be true of rest regarded as infinite slowness. Formulated
laws starting to break down or show contradictions in the case of rest when it
is regarded as infinite slowness is a sign that those laws were wrongly
formulated.[383]
There are many relationships between Cusa and Leibniz,
which have been explained by historians of science.[384]
The purpose here is not to explore all of those relationships. However, as
regards the infinite and intellectual constructs in general, there are
relationships that are relevant here. If the infinite and the infinitesimal are
both infinites, then the method Leibniz and others are using with infinitesimal
reasoning are subsumed by Cusa’s understanding that “positing an infinite line
helps the intellect make headway” towards the truth. It is the positing of
possibilities or the musing about scenarios that pulls us forward, not
necessarily the truth or falsity of such hypotheticals.
We see that in Leibniz’s mathematics, the aim is pragmatism
in the service of truth and utility, not truth itself. For example, due to the
facility of arithmetic progressions in simplifying equations, to use them “is
an exceedingly remarkable method”.[385]
Extending this to dy as it depends on x, “by
this artifice many excellent theorems with regard to curves that are otherwise
intractable will be capable of being investigated, namely, by combining several
equations of the same kind.”[386]
At the same time, it is not a game with an unimportant outcome. If the method
of causing a secant to approach the tangential did not yield an understanding
of the tangent, then an array of contradictions would have emerged and Leibniz
would have had to find another method.
Leibniz is frank on his pragmatism in the use of the
infinitesimal itself, “whether such a state of instantaneous transition from
equality to inequality, from motion to rest, from convergence to parallelism,
or anything of the sort, can be sustained in a rigorous or metaphysical sense,
or whether infinite extensions successively greater and greater, or infinitely
small ones successively less and less, are legitimate considerations, is a
matter that I own to be possibly open to question”.[387]
He then goes on to address his “rigorist” critics directly and explain what the
infinite or infinitesimal are, as far as the question needs to be addressed,
saying that a quantity is such when it is “as great as you please, or as small
as you please, so that the error that any one may assign may be less than a
certain assigned quantity”.[388]
In technical terms, this does not contradict Cusa’s maintaining “the
impossibility of there actually being an infinite line”.
Leibniz declares that impossibility or
otherwise of the infinite or infinitesimal is not important:[389]
If any one wishes to understand these
as the ultimate things, or as truly infinite, it can be done, and that too
without falling back upon a controversy about the reality of extensions, or of
infinite continua in general or of the infinitely small, ay, even if he think
that such things are utterly impossible; it will be sufficient to make use of them
as a tool that has advantages for the purpose of calculation, just as the
algebraists retain imaginary roots with great profit.[390]
For they contain a handy means of reckoning, as can manifestly be verified in
every case in a rigorous manner by the method already stated.
Possibly in a continuation of the rigorist attacks on
the infinitesimal, many of which stem from its apparent non-existence, Boyer criticised
Leibniz’s infinitesimal as lacking a precise definition and said that it required
Cauchy to make it precise. Boyer said that both Nicolaus of Cusa and
Nieuwentijdt, a critic of Leibniz, defined the infinite and the infinitesimal
unsatisfactorily.[391] Leibniz maintained that it did not need to be any
more precise than it was. Moreover, is it possible that everything Cauchy used
to make the infinitesimal precise, purportedly, would several centuries before
have required the same “leaps of faith” that Leibniz’s calculus did?
The method stated above (of arithmetic progressions
extended to dy as it depends on x) translates into the d / e method
which was once taught in high school. Arguably, such “tools” are examples of
Platonic ideas, and Platonic ideas have given rise to much controversy. In his
statement about the calculus being “a handy means of reckoning” Leibniz stays
outside that controversy. For example, Cheyne and Pigden wrote that,
“Mathematical platonists claim that at least some of the objects which are the
subject matter of pure mathematics (e.g. numbers, sets, groups) actually
exist.” They argue, “Either the dispensability of mathematical objects to
science can be demonstrated and, hence, there is no good reason for believing
in the existence of platonic objects, or their dispensability cannot be demonstrated
and, hence, there is no good reason for believing in the existence of
mathematical objects which are genuinely platonic.”[392]
However, Leibniz makes clear that it is irrelevant whether the infinite and
infinitesimal actually exist and, indeed, science can derive great benefit from
them even if we believe that it is “utterly impossible” that they exist. In any
case, more than a century later, Riemann proposed unboundedness as a postulate
more general than infinitude.[393]
Cheyne and Pigden have implied
that there is a distinction between platonic objects and the “reality” of
science. For Leibniz there is no distinction between the status of mathematical
objects and scientific theories which are putative scientific realities. This
is not because Leibniz regards mathematical objects as real but, rather,
because scientific theories are as non-real as mathematical objects even though
scientific theories seem to the human mind to bear a closer relationship to
reality. At the same time, it must be noted that Leibniz’s conception of
platonic objects is not the same as those of mathematical Platonists because,
for Leibniz, platonic objects like all ideas are certain and inexorable but are
not real.
What if everything we
understood, or think we understood, about science were itself a platonic object
or corresponded to one? This is the obvious hypothesis for Nikulin to test in
his quest to understand how it is that, “Mathematical entities are to be
associated with empirical objects.”[394] To do so, Nikulin must bridge the divide between the “platonic domain”
and the physical universe. On the one hand, Cantor appears to have thrown a
spanner in the works because his theory of infinite sets purported to
understand the concept of an infinite God. Chaitin calls it “mathematical
theology”.[395] On the other hand, such an
investigation is perfectly natural and potentially fruitful in Leibniz’s
conception of science.
Metaphysical truths are at
the heart of final causes which Leibniz emphasized are one of the most powerful
ways of understanding efficient causes. Recall that final causes are
essentially the workings of God’s mind in designing the universe.[396] While the
metaphysical tools used to understand final causes do not exist, they are
indispensable to minds that put themselves to the task of understanding reality
including scientific reality. For example, like the Pythagoreans, Leibniz gives
number the highest place when he says, “But
there is nothing which is not subordinate to number. Number is thus a basic
metaphysical figure, as it were, and arithmetic is a kind of statics of the
universe by which the powers of things are discovered.”[397]
(emphasis added) Given
Leibniz’s outlining of the role of the concepts of the infinite and
infinitesimal, we can only conclude that he agrees completely with Cheyne and
Pigden when they say, “Numbers are needed to underwrite any conceivable causal
order but they themselves play no part in the proceedings. They provide a sort
of metaphysical framework for any possible physics — an indispensable, indeed,
a necessary backdrop for the causal show. But though there could be no causal
structure without numbers, numbers are not implicated in the causal shenanigans
described by any science whether actual or merely possible.”[398]
Thus, “the objects which are
the subject matter of pure mathematics” are just as real as the science to
which they may or may not be dispensable. That “reality” is not corporeality or
an Empirical reality. This is an easy conclusion to reach if, like Leibniz, one
is not prepared to embrace Empiricism. This does not mean that Leibniz believed
that mathematical objects actually exists as mathematical platonists claim.
Mathematical objects have the same existential status as any other idea: they
are not real and exist only as intellectual impressions.
Leibniz even grants that the rigorists have done a
useful service as they have “discovered an art of advancing and of deriving so
many things from a few. If they had tried to put off the discovery of theorems
and problems until all the axioms and postulates had been proved, we should
perhaps have no geometry today.”[399]
It is the process that they have given us, even though their axioms and
postulates, and so the conclusions, are not true. Leibniz cites Euclid’s
unproven axiom that two straight lines can meet no more than once which project
was hampered by Euclid’s lack of a good definition of a straight line.[400] Ironically, one of the fathers of rigor, Euclid,
assumed the infinitude of the line.[401] However, writes Leibniz, “I blame Euclid much less
[than Descartes] for assuming certain things without proof, for he at least
established the fact that if we assume a few hypotheses, we can be sure that
what follows is equal in certainty, at least, to the hypotheses themselves.”[402]
Leibniz sees a broader application of this lesson, for he writes, “if some
careful and meditative mind were to take the trouble to clarify and direct
their [theologians’ and scholastic philosophers’] thoughts in the manner of
analytic geometers, he would find a great treasure of very important truths,
wholly demonstrable.”[403]
Raynaud’s reading of Leibniz leads him to the same endpoint
as we are reaching here. In particular, that according to Leibniz “truth admits
of degrees”, and “its discovery passes by the way of logical analysis, and that
even in theoretical matters it may sometimes be reasonable to employ ‘indemonstrables’.
Logic thus becomes once again the model for science, because the first
principle of necessary truths is the principle of non-contradiction. … By the
same token, he establishes a new continuity between science and the active
life, between knowledge and practical judgement, and between reason and faith.
For if science may employ indemonstrables, that also means that the initial
absence of such principles does not impede one from progressing on the path of
reason, without having to ‘cast into doubt’ common beliefs.”[404]
Science and all thought worthy of the name Reason is
about hypothesization, it seems, and active use of hypotheses, until it becomes
clear that a hypothesis contradicts something that we know to be true through
other means.
We now move away from considerations of pragmatism,
and onto leaps of logic and the use of metaphysics in physics.
Franklin wrote that Kepler “aims to explain everything
about the system of the heavens, including facts of kinds that modern astronomy
makes no attempt to explain.”[405]
So did Leibniz, and like Kepler thought that it was not only perfectly
reasonable to pursue such an ambition but that ultimately that is the only
valid quest for the scientist and metaphysicist who, to Leibniz, were in almost
identical professions.
Aside from all else, a kind of universal or physical
morality, or morality of physics, is posited. In c.1696, Leibniz wrote, “those
minds in whom the imaginative faculty predominates … believe that they need to
use only mathematical principles, without having any need either for
metaphysical principles, which they treat as illusory, or for principles of the
good, which they reduce to human morals; as if perfection and the good were
only a particular result of our thinking and not to be found in universal
nature.”[406]
The expression of that perfection and
good is labelled by Leibniz as the architectonic nature of things, or the way
in which God acted in creating the universe. Leibniz is unambiguous on the
relation between the purpose of God in the very large, the mechanics of
corpuscles, and the relationship between religion and natural science. The dichotomy
is from God as architect to God as legislator, and there is really no dichotomy
because they are both the same God.[407] We quote from Leibniz’s Tentamen
anagogicum which is extracted at greater length in Appendix 2:[408]
...the
smallest parts of the universe are ruled in accordance with the order of
greatest perfection; otherwise the whole would not be so ruled. It is for this
reason that I usually say that there are, so to speak, two kingdoms even in
corporeal nature, which interpenetrate without confusing or interfering with
each other - the realm of power, according to which everything can be explained
mechanically by efficient causes when we have sufficiently penetrated into its
interior, and the realm of wisdom, according to which everything can be explained
architectonically, so to speak, or by final causes when we understand its ways
sufficiently. In this sense one can say with Lucretius not only that animals
see because they have eyes but also that eyes have been given them in order to
see...
The architectonic view is almost synonymous with a
presumption of intention in the design of everything in the physical universe.
Thus not only is the Creator most powerful and
intelligent, but he acts with intention. Since God created the universe and the
universe “runs” without divine intervention, it would be more accurate to say
that God has designed an intention capability – indeed, imperative – into the
universe. That is, the universe itself “intends” for certain outcomes or, at
least, for certain tendencies or trajectories to be followed in the unfolding
of events. At this stage, we refer to this trajectory as “towards the Best”. For
Leibniz, discoveries in natural science are bound up with this presumption.
When doing science, philosophy or even when thinking
about something to try to understand it, Leibniz advises that we are greatly
aided by first identifying whether we are considering efficient causes or final
causes. The former are the laws of nature or physics whereas the latter are the
appetition of souls. The appetition of souls are the tendencies or “desires” of
the more active part of the universe; these are the (incorporeal) monads with
clearer perceptions to be explained in detail in the next chapter. Neither the
efficient nor final cause is adequate without the other.[409]
We ultimately can better understand an efficient cause or how a process works
in the small if we also look to its final cause which, effectively, really is
the intention of the Creator in putting that process there or where that
process fits in the Creator’s big picture.
The architectonic foundations of reality manifest in the
ongoing functioning of the universe including humankind within it moreso than
in the initial act of creation. It is an anti-entropic outlook on everything in
the universe, including humanity.[410] On the contrary, the usual corollary of Clausius’
Second Law of Thermodynamics, often referred to as “the Second Law of Thermodynamics”,[411]
is that the universe tends towards increasing entropy. The Leibnizian corpus is
therefore opposed to the Second Law of Thermodynamics.[412] In any case, Clausius’ Second Law of Thermodynamics
was only formulated in the context of particular heat engines assumed to be
closed systems. Whether there is any such thing as a closed system is subject
to debate. However, this question is well beyond the scope of this thesis.
This is seen in Leibniz’s “best of all possible
worlds” doctrine – “everything in the whole wide world proceeds
mathematically, that is, infallibly” – as clearly as anywhere else in Leibniz.[413]
A mathematics MSc thesis in the early 21st
Century might seem an unlikely place in which to debate some of the considerations
addressed debated in this chapter. We are not undertaking mathematical
calculations, true. However, we are discussing in what context and to what end
mathematical work should be done. Mathematical work is always done with
philosophical and metaphysical assumptions. Scientific work, especially basic
research, can only benefit from at least being aware of what one’s assumptions
are so that they can be questioned. The entire direction of a scientific or
mathematical career could be changed by an awareness of the metaphysical
foundations on which a researcher is building their work.
It is often thought that modernity is synonymous with
an aversion to the supernatural. Further, the common stance is that belief in
God is unscientific or, at least, something that bears no connection to
science. For Cusa, Kepler and Leibniz, this was not the case. For them, doing
science was coming to understand God. This was not just a matter of labelling
their discoveries as a deepening of their understanding of God’s Reason. The
process of scientific enquiry was benefited by seeking to better understand God.
In fact, the act of scientific enquiry could not be undertaken without thinking
about God and God’s relationship to the universe. Leibniz held that sense
perception has limited usefulness in discovering truth and that truth is
discoverable only by the mind. Moreover, Platonic ideas when known can be used
as tools by the mind.
Were we only able to “know”
what our senses tell us, God would not exist for us since God is not manifest
to the senses. In that case, there could be no harmony or meaning, only
coincidences. But our reason tells us that there is harmony which cannot be
accidental. Cicero argued this in De
natura deorum with the analogy of the barbarians observing a ship of a size
and complexity they have never before seen and slowly realising that it was
made by intelligent minds. In the analogy, the barbarians represent thoughtful
humans and the ship represents the physical universe. If there are only
coincidences, we must give up hope that reason can ever be of use, and rather
deduce only from sensory observations. Platonic ideas cannot be discovered within
such a framework. Furthermore, we only discover the harmony – that is actually
there – using reason. Once we allow that reason, then we see a great deal else.
It does not make sense that
there would be harmony in some things but randomness elsewhere. From here we
get the principle that if only we could see things “in the large” we would see
the harmony everywhere. Leibniz took Kepler’s harmony further, beyond the study
of the solar system into all things, philosophical and metaphysical, and into
all fields of enquiry.
The empirico-deductive line of thought was largely
atheist, though some thinkers – like Newton adopted a conception of God
consistent with the positivist science they were constructing. Theistic
metaphysics gives rise to a scientific method that is more permissive in the
range of thought allowed and more powerful in pushing the envelope of human
understanding through progressive grasping of principles that govern the
universe. Leibniz would argue that it is not only more permissive but
necessarily more correct. As we shall see in Chapter 7, Kepler argued the same.
In this chapter we will describe and discuss Leibniz’s
framework for reality, including God, substance, the universe as a whole, the
components of the universe and how these components interact. There are many
references to Leibniz’s Monadology.
Leibniz’s writings lead us to the conclusion that all
science can be undertaken in the mind. Parkinson asks, “Is it true to say that
Leibniz, in his science, relies wholly on a
priori arguments?”[414]
answering “Yes” for the most part, but not exclusively as Leibniz leaves room
for observation and induction from observations.[415]
We hope to flesh out Parkinson’s answer. Science is done by investigation and
reasoning about metaphysics, mathematical rules and other ideals. Science done
in this way seeks to understand the mind of God, ultimately. Thus, theology is prominent
in Leibniz’s framework.[416]
Farrer says, “If we condemn Leibniz for writing physical theology, we condemn
not him but his age.”[417]
However, Leibniz makes it clear that it is a conscious contention of his own that
God must be a central consideration, rather than something he has imbibed from
the zeitgeist.
It has already been mentioned that in Theodicy
Leibniz defended the daring endeavour of seeking to understand the mind of God
against detractors, and supported others who encouraged and joined the quest.
While Leibniz accepted Cusa’s doctrine that in ignorance the soul finds its
repose, nonetheless what Christian faith commands does have reason at the back
of it which is comprehensible by men.[418]
While experiment is useful in checking our reasoning,
it is not necessary by force of reason. It is practically necessary because our
conceptions and thoughts are generally indistinct and confused. Occasionally, a
human mind comes along which is clear and distinct, and teaches us how to think
and in so doing gives an entirely new set of ideas. Plato and Jesus Christ
would be examples. The “believe it when I see it” mentality is nothing but a popular
choice for perception over thought.[419]
Indeed, the soul itself and its function of thought are not furnished by the
external senses.[420]
That Einstein was thinking in this vein would be evidenced by the quote
anecdotally attributed to Einstein, “I want to understand his [God’s] thoughts.
The rest are details.”[421]
The power of a
priori reasoning is why the Universal Characteristic, or calculus of ideas
or Lingua Philosophica (“Language of
Philosophy/Logic”), is the one of the most important undertakings.[422]
While Brown notes that Leibniz made little apparent progress towards it,[423]
Leibniz had a clear conception of its nature, what it would be able to achieve
and why it was possible. However, it may be that Leibniz’s work on the calculus
and especially in logic were themselves strides achieved toward the universal
characteristic.
Antognazza says that the characteristica universalis was a part – though a large part – of a
scheme for all areas of human thought and endeavour from science and technology
to law and religion which Leibniz called the scientia generalis. Leibniz wrote an array of articles towards this
goal. For example, in a series of logical texts written over 1678-1684, Leibniz
outlined a logical calculus whose structure remains valid independent of the
contents assigned to the letters used in the argument such as, to use Antognazza’s
example, a for “animal” and b for “rational”.[424]
With it, as well as many day-to-day uses, we could conduct a priori science. This is not “discovering the mind of God” per se, which is impossible according to
Cusa, since truths, physics and all else but God are distinct and independent
from God, as Leibniz explains in “Reflections on the doctrine of a single
universal spirit”.[425]
It is also impossible according to Leibniz.[426]
Nonetheless, the more clarity we can achieve, the better off we are; in
particular, the more useful our hypotheses will be.
Leibniz lamented that the art of discovery was little
known outside of mathematics and said that it should exist in systematic form
for all domains of knowledge.[427]
Developing a calculus of ideas might start with the Platonic dialogues. Leibniz’s
own development of the infinitesimal calculus might provide ideas on how to
start, albeit there would necessarily be important differences.
Leibniz distinguishes body from a phenomenon in a
coherent dream.[428]
On the contrary, Leibniz also said there is value in the Platonic perspective
that we wake up on death implying that death is more real than life.[429]
Confirming Leibniz’s dim view of how much truth is directly discernible from what
is experienced in life, he explains that he can demonstrate experimentally that
all the perceivable properties of bodies are apparent only. This does not mean
that there are no real bodies. It is just that what we perceive about them is
misleading. Bodies are real when they have substance. Such bodies may act or be
acted upon, whereas bodies that are not real cannot act on other bodies and
cannot be acted upon. Bodies that are not substantial are merely phenomena or, at most, are aggregates of
true bodies. A coherent dream has phenomena but not substantial bodies, and a
phenomenon is defined by the act of observing it not by the substance that it
may or may not include. A substance has active metaphysical power insofar as it
expresses something distinctly, and passive power – i.e. the ability to suffer being
acted upon – insofar as it merely expresses something confusedly. There are
infinite degrees between. Here we see that Leibniz has defined the power of a
thing by its clarity of expression. To Leibniz, as we will see in his
definition of ideas, clarity includes consistency with all else that is real in
the universe.
The mind is part of reality, but it also sees reality
imperfectly. By contrast, God’s mind perfectly sees all of reality, i.e. all
soul and body monads, and all ideas and their implications on reality.
The “really real” are the simple substances or created
monads which once existing endure always, as well as God himself.[430]
So: (i) God and (ii) simple substances also known as created monads also known
as souls comprise all that is real. These do not exist in the mind of God. That
is, the “verities” (universal truths or ideas) are not the result of God’s will,
contrary to Descartes.[431]
Harmony itself is a thing separate from God, but harmony is perceived – albeit
that it is an idea – so it is not really real either. God creates things in
conformity with such because to do so is the highest and best way of doing
things. Beauty is something separate from God but only God makes it possible because
everything that gives rise to it in reality (viz. monads) comes from God.
With the monad concept, Leibniz is following his own
advice in the principles of discovery in breaking things down to their most
fundamental parts. However, monads are not just atoms or quarks, for they have
knowledge of one another and of the entire universe. Thus, they each express the
whole as well as being the parts. If we regard the entire universe as an
organism, then monads are like cells in that each cell has DNA with an encoding
of the entire organism (i.e. the entire universe) while also “knowing” its role
in the organism.
A “created thing” is a created monad.[432]
Monads are actual things, i.e. they are “existing”.[433]
The monad is nothing but a simple substance, or a substance without parts, which
enters into compounds.[434]
The soul of humans is simple.[435]
The rational soul is the “substantial” part of man.[436]
The physical form of man comprises trillions or more of body monads. Those body
monads exist and are substantial, but the physical form is passing, temporary
and, in any state, is an idea only. Thus, it is insubstantial. Even as an idea,
it can accurately only be conceived in the infinitesimal and this is only an
idea. In fact, it is never existent or being, and is only ever becoming.
In explaining monads in Monadology, Leibniz referred to substance in a particular way
suited to his conception of monads. This is sensible because, as Leibniz
explains it, all substance comprises monads; there is no substance that is not
“made” of monads, and this includes substance that is not intangible such as
the soul and even such as God. Passive substance, which includes the
constituent simple body monads of a human’s physical form, is simply in a
prolonged unconsciousness.[437]
Yes, the soul is a substance and no
substance is perishable in nature.[438]
Most monads come in pairs – body and soul – wherein either
or both elements of the pair may be a compound monad.[439]
They cannot be the same monad because one is a compound monad and thus
destructible whereas the other is not. One is relatively passive, the other
active. Yet they act in harmony. This is a “pre-established harmony”.[440]
Each monad is aware of every other. There is an
inter-communication between all monads and thus between all things, so each body
feels the effect of all that takes place in the universe. Thus, it would be
possible to “read in each what is happening everywhere, and what has happened
and shall happen, observing in the present that which is far off in time as
well as in place.”[441]
Latta writes that this inter-communication is a symbol of pre-established harmony.[442]
It is not clear why Latta uses the term “symbol”. Arguably this is not what
Leibniz means. In formulating the pre-established harmony, we do not know
whether Leibniz had any purpose other than to bridge the gap between the realm
of final causes and that of efficient causes, or between the immaterial and the
material. The universal awareness of each monad is a property of monads. A
harmony, on the other hand, is an idea which governs things from outside of
themselves.
There must be body monads without a soul which are entirely
passive, and spirit monads without a body. However, the question of being
soulless or bodyless is one of degree for Leibniz posited a continuum in most
metaphysical matters he considered. Neither generally exists in the natural order
of things and Leibniz does not discuss them in detail.
The soul (part) is not divisible into an active
eternal part and a passive temporal part as Averroes believed.[443]
Leibniz calls Epicurus and Hobbes’ doctrine of
material souls an evil one. Leibniz also wrote that they have extended to
humans what the Cartesians have held for animals, which is also incorrect.[444]
Leibniz also says that it is pernicious to deny the immortality of souls.[445]
This brings Leibniz into opposition with Pompanazzi and Sarpi.
We will see later that active or more perfect monads
(souls) and passive or more imperfect monads (bodies) are all that exist aside from God. It is consistent with this that Leibniz
often wrote that the purpose of the universe is to help intelligent souls get
closer to perfection and felicity. According to Leibniz, this is the paramount
goal of God.
Ideas exist in our memory, indistinctly, and can be
brought to the fore as Plato showed in Meno.
God too has many ideas, but perceives/thinks them distinctly and all at once as
Leibniz encourages humans to do in “the art of reasoning well”.[446]
God |
Body monad |
Soul monad |
|
Perfect |
Imperfect |
Less imperfect |
|
Active |
Passive |
Less passive |
|
Immaterial |
Material |
Immaterial |
|
Memory |
No memory |
Memory |
Together, Memory and Perception are Sentiment. |
Knowledge |
Perception |
Perception |
|
Will |
Appetite |
Appetite |
|
Figure 3: Some qualities of God and substances
Time, distance or length (i.e. “extension”), motion,
the continuum and numbers express only possibilities.[447]
They are ideas at the simplest level; they are primary. Leibniz recalls that
Hobbes described space as a “phantasm of the existent”[448]
without disagreeing. A phantasm is an image as if in a dream. Dreams do not
exist anywhere except in the mind of the dreamer; dreams are not real. The
definition of real is that which conforms to metaphysical principles and
mathematical rules.[449]
It is only through such that we know that we are not dreaming; that is, that
our experience is not a mere figment of (or a mere image in) our mind. This is
because our perceptions are imperfect and indistinct; if our current reality
was a creation of our minds, then there would be inconsistencies everywhere and
there would be measurable discrepancies from ideals. In any case, there is only
one reality and it is not just a “coherent dream”.[450]
More accurately and specifically, extension is the
order of possible coexistence whereas time is the order of possibilities which
are inconsistent (i.e. cannot coexist) but which have a connection through
(continuous) events. However, space and time themselves are indifferent to what
content is put into them, just as numbers are indifferent to what they
enumerate. Here, Leibniz is distinguishing the real from the non-real. An ideal
is an idea all the same.
Leibniz consistently fails to enlighten us on where ideas exist probably because he
would maintain that an idea is a potentiality at most. It does not exist
anywhere. In his characteristic style, he would likely dismiss the question as
nonsensical. For a truth does not exist anywhere, it simply has the potential
to be used in governing the behaviour of passive substance because it is the
best pattern of behaviour. An idea is not real. For Leibniz, the non-reality of
concepts and ideas dates back to his bachelor’s dissertation published when he
was 17 and written under the direction of Thomasius.[451]
While the intelligible world in the divine mind is the region of ideas, the
ideas are not real. The divine mind knows them, and it does not need to work
ideas out (i.e. find them) or general them for it sees all at once.
In a book entitled Strict
Finitism, Kielkopf undertakes a thorough analysis of Wittgenstein’s
foundations of mathematics. There, Kielkopf says that an absolute platonist (sic,
small “p”) believes that mathematical objects have their own independent
existence. Strictly speaking, Leibniz is not platonist according to this definition
because, to Leibniz, ideas are not real and do not “exist” in any domain nor
even in God’s mind or our own minds. Ideas might be perceived in our minds, but
they do not exist there. An idea does not exist to be perceived but, rather,
the idea is itself a perception. When the mind perceives an idea, it is, in
part, creating an impression on itself and, in part, receiving impressions from
the physical universe. Note, however, that the only mind which ever perceives
an idea is God’s mind. Humans can only ever approximate ideas.
To Leibniz, “real” includes physical and metaphysical.
Though he says ideas are not real, it is clear that ideas played a role in God’s
mind during the design of the universe and they certainly occupy the activities
of human minds. This does not mean that ideas exist in God’s mind or human
minds. Concepts, too, play a role in human minds, though concepts have internal
inconsistencies causing them to fail to qualify as ideas. Nor do concepts exist
in human minds. This is because to be a mental impression alone is not to
exist.
There will now be some discussion involving ideas,
hypotheticals and hypotheses. Hypotheticals are treated as atomistic
propositions or concepts, such as the existence of a perfect square or just the
concept of a perfect square. A hypothesis is a set of concepts that are more
widely encompassing of a process, formulated with the intent of explaining that
process. The usual meaning of hypothesis as a precursor to a theory of some
physical process, force or phenomena is useful too. There is usually some
unifying thread between the concepts in the hypothesis. This high level of
abstraction is sufficient for what follows.
An idea has the same reality as a hypothetical or a
concept. None of them exists in the physical universe. The geometrical concept
of a perfect square is a hypothetical. Yet a perfect square can never be
realised in the physical universe and so it is not an idea. Nonetheless, it is
useful in understanding the universe because the universe is designed in the
best possible way and so there is an order to it. Thus, understanding the best
possible involves the use of ideals such as geometrical ideals and other
mathematical ideals. Mathematics as the study of structure presumes that there
is structure in the universe or underlying the universe. Really, a hypothetical
can be regarded simply as a concept formulated with the purpose of aiding in
understanding the universe and therefore it probably should and usually does
have some structure to it.
A hypothetical is just an intellectual impression in
the mind, and the mind is capable of formulating nonsense which could never be
realised in the physical universe as well as concepts which can never be
realised but which are nonetheless useful to us. A thought of a thing that can
be realised is no more real than a thought of an impossible thing – both are
merely a thought or mental perception. A concept is promoted to the status of an
idea if it is internally consistent and consistent with all else that must be realised
in the universe for the universe to be the best possible. Ideas do not exist in
the immaterial domain of souls, but they do define the best possible – indeed,
the only – way of organising the universe. So where do ideas exist? Again, ideas
are only perceived by minds, yet they are perceptions that can be manifested in
the physical universe. Were hypotheticals and thus hypotheses prohibited in the
conduct of science, we would not be able to attempt to work towards ideas and
thus predict and control how the physical universe works, except in a haphazard
and accidental or serendipitous way. The benefits of mathematics and physics
would be denied to us.
We will use the term ideals and ideas interchangeably.
However, the term “ideals” will tend to emphasise metaphysical principles and
mathematical rules all of which are ideas themselves. Leibniz sometimes regards
the laws of physics as comprising metaphysical principles and mathematical
rules. On the other hand, the term “ideas” encompasses all truths including
ideals.
Ideas or ideal things – since an idea only qualifies
with that name if it is clear and non-contradictory and is therefore a
derivative of ideals – express possibilities. They are not actually real.[452]
However, they are imprinted on or remembered by our soul’s memory which may or
may not be infinite though it can comprehend the infinite though in fact it
only ever does so in an indistinct way.[453]
X being remembered by the memory does not mean that X exists there. “Something
mental” only designates “the possibility of parts, not something actual”.[454]
Considering numbers, an idea is different from a
possibility but it is a characteristic or denotation of an aspect of a
possibility since no number can ever really exist. Yet ideals govern passive
substance and the passive part of substance that has active and passive parts,
for everything is composite and nothing actual is purely active or passive. The
most active souls and God choose to create in conformity with ideals because
that leads to the greatest harmony and leads to the greatest felicity for all
souls. Deluded souls (including humans limited in understanding) create in a
way that contradicts ideals and, thus, in a temporary way. However, God has
designed things so that something good comes from perceptions and appetite
based on delusion or incomplete understanding (or “silliness”). Relatively
passive souls like those of ants are mostly governed by laws and cannot choose.
Humans with understanding can choose even better than nature can because nature
passively unfolds matter in accord with ideals, which is different from the
proactive creation undertaken by an active or less passive monad.
It is only through the impact of principles/ideals on
passive substance that we can ever really “see” them. Yet the
active/experienced mind can think things through and “see” what is right
without it having to manifest in the world of substance. Note that the “world
of forms” is nothing but mental images or perceptions in the interior of
monads. Thus, ideals are significant although they can never be touched even by
God. Ideals can only subsist as images/impressions in the mind, including in
God’s mind. At the same time, they only have any effect due to the universe of
passive substance. The world provides the opportunity for minds to test their
notions and to draw ideas out.
How can it be that a geometrical straight line or
triangle do not exist anywhere? They are impressions in the mind which we can
describe unambiguously and so can convey to another mind which will then also
have the impression in their mind. The impressions will likely not be identical
though they will be very close, particularly between the minds of, say,
geometers who conduct proofs with them and so force themselves to coincide
almost entirely in their conceptions
or notions of these ideas. Testing our notions in the
physical world also aids.
The Aristotelian view of ideals seems to agree with
the Leibnizian view because (Platonic) ideas are apparent only in the created
(physical) universe. However, a mind can find and “know” ideas whether or not
they seem to be available to sense perception by their correspondence to
phenomena in the universe.[455]
God does not need to “see” ideas and probably does not
perceive them one at a time like human minds do. God can grasp not only the
entire actual universe in one act of his mind, but also all possible universes
in one act of the mind. His “perception” is indistinguishable from his mind in
perspicacity. The concept of sense perception makes little sense for God.
For humans to perceive something via the senses
usually generates a more distinct mental impression than does conceiving
something purely through an act of the mind. For humans, no act of the mind is
pure but is always bound up with a memory of a sense perception to a greater or
lesser degree. But God has created souls anyway from which the physical
universe results because it is his nature to create and to do so in a way that
is the highest and best and, to a mind that can grasp it (and only God can do
so fully), the most beautiful.
If ideas required the world to be perceived, by being
realised in objects, substance, radiation, etc. then God would need the world
to guide divine decisions which is patently absurd since the Creator of the
very laws of the universe does not need to refer to the Creation that results
from his thoughts. For example, harmony is an idea. The fact that there may be harmony
among ideas or a “harmony of ideas” suggests that ideas are independent of the
world. However, on the other hand, the meaning of the Best is intertwined with
the universe which God has designed to be
the Best.
There is no domain of principles by which, if we were
to grasp it, we would understand the universe. The universe is what it is, and
our minds’ development of ideas is nothing but us (humans) trying to comprehend
it. Ideas are indispensable as intellectual tools because the universe is perfect
and ideas have the necessary condition of perfection that they are internally consistent
with themselves and externally consistent with all other ideas and, indeed,
with all of reality. It could almost be said that ideas describe the real
universe. There is a correspondence can be seen between mathematical frameworks
and the physical universe, which Figure 4 presents in schematic form.
The utility of mathematical thinking is not diminished
by the fact that it is not real. This is because actual things cannot escape
its rules as Leibniz put it.[456]
While ideas allow us to understand how phenomena must have happened and how
phenomena will happen, they are only useful because they are non-contradictory.
Where a conception is confused then its power is merely of passive or primary
matter, otherwise it is active.
Distinctness of perceptions gives active power, and
confusedness gives passive power which is attributed to metaphysical matter or materia prima.[457]
So distinctness of ideas is paramount in monads. The universe comprises only
monads, but most are effectively asleep, indeed comatose, or passive.
For example, though extension is an idea, it is of
great importance in physics. Leibniz’s entire discussion of Aristotle’s primary
matter takes place in the realm of ideas not once referring to monads (which
alone are real, aside from God) or their equivalent.[458]
Figure 4: Ascendence in the geometrical part of idea systems of improving perfection towards one that corresponds to “the Best”
Rather than saying that actual things cannot escape
the rules of mathematical thinking, it would be more accurate to say that minds
attempting to understand the universe cannot escape its perfection, and so will
be drawn to mathematics. So the converse is true, that the rules of mathematics
cannot escape actual things.
Creation of additional mathematical structures that
bear no relation to observed phenomena in the physical universe is useful and
beneficial. In particular, if the conceptions so developed bear no
self-contradictions or contradictions with other truths – i.e. are ideas – then
they furnish readymade hypotheticals which may help close gaps in understanding
that may arise from paradoxes in future empirical observations.
If Leibniz is so adamant that ideas are
not real, then why do we need them? After all, in becoming a well-known
mathematician, Leibniz spent a great deal of time with ideas, or putative
ideas. It is certain that the human race would little more than an advanced
mammal if we were to abandon our quest for ideas. Ideas are an intellectual
abstraction and resolution that are essential for these among many other
reasons:
1. Prediction of future events/phenomena
2. Committing to a record and communicating from one
person or group of people to another, and from one generation to another
3. Connections between ideas connect seemingly unrelated
phenomena
4. Machine-building
5.
Computer programming,
which is a specific case of machine-building.
Indeed, this discussion could quickly expand beyond
ideas into concepts and into thought itself. In fact, the above five points
relate primarily to concepts because most or all or what we think we know to be
ideas will in future generations be found to be incorrect and to be merely
concepts. A successful hypothesis is an idea whereas an unsuccessful hypothesis
is a conception. Both ideas and conceptions are examples of thoughts.
Necessity arises in deductive processes from
presumption, axioms or postulates. Necessity in what is manifested in the
physical universe is a result of something being part of “the Best”. The
concept of the Best does not arise merely as a thing in isolation (like a
triangle with angle sum 180°) but as a process of events over time which – by
the definition of an idea – connects with everything else in the universe. It
is hard to see how an idea could ever be conceived without having the ability
to perceive the entire universe and all of its processes over all of time in a
single thought. This is not even equal to the ability to perceive all ideas in
one’s mind at once in a single thought, and it is this which would be needed
for the design of the universe as a whole. One would also need to know what the
Best is, and either create that (in one’s mind, first) or choose it from
multiple possibilities or options. On the other hand, rather than this “brute
force processing power”, which would require no effort for God in any case, it
may be that the Keplerian search for harmony would be more accessible and more
fruitful. That is, start with what we know has harmony and beauty, and expect
to find that in the physical universe.
It is in these considerations that the relation
between God’s mental universe, or the domain of ideas, and the physical
universe is found. The connection with our intellectual/mental world is
similar, but is complicated by the fact that we do not know any ideas only
concepts and those confusedly and indistinctly. Nonetheless, over time, we can
get closer to ideas and thus to an understanding of the physical universe.
Leibniz does not regard
ideas as things that exist. Nonetheless, he sees a certain necessity to them.
He regards ideas as being able to be instantiated in the physical universe and,
indeed, as the only things that can be instantiated in phenomena and processes
in the physical universe. Richard Brown writes, “To Leibniz nothing exists
independently of individuals. Common properties shared by individuals do not
actually exist in re; they are purely
creations of the mind.”[459] Once an
idea has been instantiated, the instance exists in re, but the idea does not exist in re.
Having regard to the standard
definition of Nominalism,[460] Leibniz
denies both the existence of universals and also of abstract objects.
For Leibniz, not all
universals are abstract objects. An abstract object is a mental artifice, which
is either a concept or an idea. A concept is a confused intellectual creation
which might be useful notwithstanding its confusedness. An idea is a refined
concept, that is consistent with every other idea. A concept that is not an
idea cannot be instantiated in the physical universe but an idea must
instantiate at some point.
“Universals” like whiteness
(the example at the Stanford Encyclopedia of Philosophy), or strength or
humanity (the examples from the less formal and more popular Wikipedia), are nothing
but a concept laced with opinion derived from sense perception and useful for
day-to-day conversation. It is a particularly vague kind of concept, and a kind
of concept which we – at least in Western culture at the time of writing – are not
motivated to evolve upwards into an idea because it a serves
a role in its vague form. Other kinds of concepts, like mathematical and
metaphysical ones, should be developed into ideas so that they may be of use –
or of greater use – in physics and in helping to understand the universe as a
whole. The evolution of Euclidean geometry to non-Euclidean geometry is an
example.
Thus, we have the paradoxical position that ideas are
discoverable a priori, but anything
found a priori does not exist and is
only a thought with some status or other, whether pure idea, pure concept or
something between. At the same time, a
priori discoveries are useful in understanding that which has been
instantiated in the universe because the universe adheres to the concept of the
Best, and we know that God uses ideas to design and choose the Best for the
design and unfolding of the universe. Hence, a priorism – particularly with a Nominalist foundation – can only
make sense when we are consciously attempting to retrace God’s thoughts. Were
one not Nominalist, a priorism could
conceivably be used to directly discover the universe. As a Nominalist,
however, thoughts are useful in understanding the physical universe to the
extent that they are similar to – or help us understand – the modus operandi of the Creator.
If you grasp one idea, you grasp them all. But you
cannot grasp any one idea without grasping them all. We explain why.
The conclusion is consistent with Leibniz’s view that
the entire universe is contained in a single grain of sand. This is just a
physical analogy of the metaphysical fact that every monad perceives every
other monad albeit with varying degrees of distinctness.
The corollary of the above conclusion is that humans
do not understand any single idea and never will. Even now, for example, all
the consequences of the Euclidean concept that the angle sum of a triangle is
180° are not known. Of course, that concept is not even an idea because it is
not consistent with every other idea. It acted as an hypothesis for many
centuries, but ultimately was done away with.
Do we know
any idea? We know some such as “God is creative” though few humans ever to have
lived would know the meaning that would need to be ascribed to “God” and to
“creative” in order for that statement to be correct, i.e. for it to be an idea
and so consistent with every other idea. Similarly with the statement “God is
good”.
The large class of statements we think we know, the
mathematicals, are not as evident as they first seem to be. For example, the
statement A Ù B Þ A Ú B requires qualification to avoid nonsense, such as when A º T and B º ^. The statement A Ù A Û A does not hold in linear logic in which propositions act like
resources.
Another word for monads is “unities”, and the words
are similarly connected to the idea of an indivisible unit or a whole or a One.
Leibniz explains that monas means
unity or one, in his preface to Monadology.[463]
Physical bodies or objects as we experience them, on the other hand, are
multitudes/compound monads which can be destroyed through dissolution of their
units. By inference, this answers the question of how compound monads are
formed. Monads aggregate to form a compound as part of the divine choices made
for the best in the way the universe unfolds from the infinity of possibilities
in conformity with ideals.
Leibniz’s concept of the pre-established harmony ties
the incorporeal part of the universe where souls “reside” with the corporeal. A
single soul monad in the incorporeal domain can control many monads in the
corporeal domain at once. The example of the human body was just what I thought
was an easy way of showing how a single soul monad can neatly be tied to a multiplicity
of monads in the corporeal domain – which I am calling “body monads” for
convenience – through the pre-established harmony.
The soul monad is simple in that it is indivisible,
has independent perceptions and exists in incorporeality. Any given soul monad
is tied to a body monad through the pre-established harmony.[464]
A single soul monad in the incorporeal domain can control many monads in the
corporeal domain at once. We will use the example of the human body to
demonstrate how a single soul monad can be tied to a multiplicity of monads in
the corporeal domain through the pre-established harmony. Since Leibniz writes
that every soul monad is tied to a body monad through the pre-established
harmony, it must be said that, for all intents and purposes, a soul monad only
consciously acts on compound monads. For example, a human mind which Leibniz
largely conflates with the soul can move its little finger or their whole body.
However, even the little finger comprises trillions of independent body monads.
The soul does not have any business with each of those trillions of individual
body monads but only with the whole that we call “the finger”. We cannot rule
out the possibility that a soul monad has a pre-established harmonic
individually with each of the trillions of independent body monads that make up
the little finger compound monad.
Thus, we make Leibniz’s framework more precise by
adding that the soul monad is tied to a definable boundary compound body monad through the pre-established harmony.
For a person, that boundary compound body monad is commonly called the “human
body”. This is the largest compound monad with which the soul has a
pre-established harmony. Alternatively, we could say that the human body is the
largest compound monad over which the human soul (which is itself a monad) has
direct “control” though that control is really just a manifestation of the
pre-established harmony. The soul monad is also tied to a multiplicity of smaller
compound body monads that comprise the boundary compound monad. For example,
the left leg forms a compound monad that is part of the boundary monad. Though
Leibniz did not extend the idea much further, we can say that that the boundary
of control can be extended indefinitely by the building of machines and other,
wider forms of influence of humans over the physical universe. However, when
humans influence vast areas through projects such as dam-building, railway
construction, farming and forestation, it is not only human souls that are at
work. Rather, the decisions of human souls are influencing processes that will
function in a new or altered way – at least for a time – even without ongoing
human intervention.
Another extension we can make is that soul monads usually
work as compounds too. In the case of people, most people take their perception
and appetitions from others or from their perception of the group’s perception
and appetitions. Still, even an individual soul monad’s perception of the group
is its own. Each of two monads in a mob who are each blindly following their
own perception of the mob has a radically different perception of the mob. The
difference between those two may not appear to be very great judging from the
actions/phenomena that arise from those monads’ “membership” of the mob.
Figure 5: How a soul monad typically communicates with another
Figure 6: More general schematic of soul monads and body monads
We now return to elucidating on Leibniz’s conception
of monads. As Figure 5 indicates, there is no direct communication between the
two body monads except via ideals manifested in the laws of physics. Figure 5
also helps conceptualise how a soul monad can be tied to a set of compound monads
whose simple monad constituents are all members of some fixed set.
The theory of pre-established harmony does not imply
that the body is a ship without captain or crew which somehow reaches its
destination regardless (as Bayle complained).[465]
This is because God mediates between the two to make it work. The mediation is
in conformity with “pre-established harmony” which is like the law of gravity.
God would never decide to act against this law any more than he would suspend
gravity for a day, as there would not be a sufficient reason to do so and it
would diminish the harmony and grandeur of the universe.
It is significant that Leibniz notes that everything
is a monad and that monads are substance which would cease to exist without
ongoing perception and activity. In other writings, Leibniz agreed with
Aristotle that there are whorls in every part of matter, which never cease in
their motion. These two positions are consistent.[466]
If an imperfect monad can only be acted upon, i.e. can
only suffer,[467] how
can we distinguish an extremely imperfect and thus passive created monad from
an asteroid which is a physical body and thus a compound monad? The difference
is two-fold. First, the monad albeit imperfect is eternal whereas the asteroid
is not. Second, the monad has some perfection however minute whereas the
asteroid (qua asteroid, i.e. rock) is
entirely passive albeit that it comprises relatively passive monads with
unceasing internal activity.
There are two extreme kinds of monad: body and soul. In
fact, monads are on a continuum between these extremes. The former kind operates
in the domain of efficient causes or physics, which is nothing but metaphysical
principles and mathematical rules – i.e. ideals. Thus, body monads are passively
subject to those “laws of nature”. The latter kind is driven by perceptions and
appetite, which do not always correspond since the passage “from mind to heart
is so long.”[468] Thus,
soul monads can make decisions. These are known as final causes. Body monads
are controlled by the laws of physics and by the decisions of corresponding
soul monads. The clearer and more distinct the perceptions of a soul monad, the
more power it has to affect other monads. (On the ground of ability to affect
other monads, God has infinite power.) Leibniz’s theory of intention subsists
in the power and tendency of monads to affect other monads which will not be
investigated further in this thesis.[469]
The theory of pre-established harmony is a harmony
between efficient and final causes. Accordingly, Leibniz says that neither
efficient nor final causes are adequate without the other.[470]
This is Leibniz’s resolution to the problem of dualism and his answer to
Descartes’ occasionalism.[471]
Nothing can be found in a simple substance or created
monad but perceptions and their changes.[472]
However, the ability to perceive does not imply consciousness.[473]
So not all created monads are conscious.
Latta writes that a created thing can
act outwardly to the degree of its perfection, and can but be acted upon to the
degree of its imperfections.[474]
No monad can act outside of itself.[475]
However, it would appear for all intents and purposes that it can act:
(a)
On the body via
pre-established harmony, albeit that that harmony was put there or implemented
by God.
(b)
On other monads
via the intermediation of God albeit this is always in conformity with ideals.
At a precise technical level, however, Latta is
correct.
A created monad, due to confused understanding (i.e. “perceptions”),
can cause its corresponding body to ingest beer so causing further confusion
and leading to a nobbling of the mind/soul monad’s ability to control the body.
That is, the pre-established harmony reduces the refinement in the physical
motion of the body in line with the confusion in the perception of the soul
monad which the alcohol somehow caused. In short, pre-established harmony works
both ways. The alcohol also nobbles the ability to reason and remember. Platonically,
referring to Meno, reasoning and
remembering are the same thing noting that the memory activated in Meno is of thoughts that have always
been with the soul, i.e. since the beginning of time, albeit in indistinct form.
Thus, the physical action of the chemical of alcohol on the brain is affecting
the perceptions of the created monad or soul.[476]
It is not just pairs of monads, simple and/or
compound, that are in harmony with one another. The entire realm of final
causes (souls) is in harmony with that of efficient causes (bodies).[477]
This does not mean that they move in lockstep and are copies of one another.
However, it must mean that they are derivable each from the other because harmony
implies that the “connection of ideas” is indubitable when properly worked out.[478]
Souls are themselves vital principles with perception
and appetite. The appetite means that they must also have intention provided
they are not passive.[479]
If they have memory, then they are intelligent to varying degrees.[480]
Substantial forms include rational souls, but not in
an organic body such as a stone. Further, there is no part of matter that does
not include an infinity of organic animated bodies. Thus, even the stone does.
However, there is matter which nevertheless is not animated in spite of what is
in it. So a stone is not animated though it has in it an infinity of animated
bodies. A pond is not animated even though it has fish in it.
In order to clarify some of the
concepts raised, we segue to a question and answer format with these five
bullet points:
·
What are such
non-animated bodies? Compound monads
·
Are they merely
ideas? Yes. Are they actual but non-permanent and without memory. No
·
Since a thing
without motion – though its parts have motion – does not exist, then such
non-animated bodies do not exist. Correct
·
Does this include
all compound monads? Yes
·
If they are
ideas, then are they not provably and permanently true? Yes, but they are
implied by the entirety of the rest of the universe. When the rest of the
universe changes, including their constituent monads, they are no longer
implied and thus are not necessary.
A vital principle is a substantial form. Only animated
matter, animated bodies and conscious monads are substantial forms. Thus, a
vital principle is “life” and not all monads have life. Vital principles are
not eternal for a living thing may die.[481]
Sentiment is perception accompanied by memory, i.e.
there is an echo of the perception for some time afterwards. A living being
with sentiments is an animal, its monad is a soul. When the soul has reason, it is a spirit.
Since everything comprises monads and monads are with
perception and thus animated, there is “life” everywhere, even in the coldest
asteroid.[482] For
Leibniz, every cubic millimetre of the universe is packed full of substance;
that is, the universe is a plenum.[483]
For Leibniz, God would not create a universe in which there was any empty space
or in which two things were alike because either would amount to redundancy,
violating the principle of sufficient reason.
Since the asteroid qua
an asteroid is inanimate, it is only an idea – i.e. it does not really exist.
This is even though the compound monad must conform to metaphysical principles
and mathematical rules, as must its behaviour and that of its constituent
created monads. This is consistent with the consequence of Leibniz’s theory that
only God and simple substances exist. Leibniz affirms this when he writes,
“Sensible things, however, and composite things in general, or the
substantiated things, so to speak, are in flux and become rather than exist.”[484] For example, the asteroid is undergoing a constant
exchange of simple monads with its environment. As far as biological organisms
on earth are concerned, such exchange with the environment and ingested
nutrients is well-established and confirmed by experiment. Leibniz is saying
that this is true of all composite monads.
In a discourse of Hermes to Tat, Hermes
explained that God sent down to the physical universe a great mixing bowl which
was “Mind”, inviting all human hearts to immerse themselves in it. Those who
did, and who do, participate in knowledge and become “perfect people because
they received mind”.[485]
The mixing bowl or “Mind” is the monad of Hermes. Hermes says that the monad is
in all things as a root and
beginning, presumably because all things have mind to some degree however
small. Nicolaus of Cusa’s conception of mind is similar to that of Hermes, for
tells a story similar to Hermes’ mixing bowl whence all souls may sup of Mind
and so acquire their own mind.[486]
Things without any mind, in Leibnizian terms, would be
pure body and for Leibniz this is extremely rare and does not get much
elucidation from Leibniz. For Leibniz, each thing that is real – except God –
is a monad whether or not it has developed an intellectual faculty worthy of
the name “mind” (small “m”).
Since souls are what the universe is made of, and
since humans have incorporeal souls (which, popularly, is the only kind of
soul), let us examine Leibniz’s theory of death.
Since souls are eternal, what happens after that which
we call “death”? Leibniz addressed Pythagoras’ conception of transmigration of
souls[487] and wrote
that his own theory implies even more. It is not only the soul that is
indestructible.[488]
Not only does the animal’s soul subsist after death but the entire animal does.
So there is no metempsychosis but there is metamorphosis.[489]
This is clarified when Leibniz explains that an animal might change its
“organic slough” as its “machine may often perish in part”.[490]
The personality of beasts is not preserved on death,
though the personality of humans is preserved. While beasts’ souls are merely
indestructible, human souls are immortal. In the case of humans, “not only the
soul but also the personality subsists. In saying that the soul of man is
immortal one implies the subsistence of what makes the identity of the person,
something which retains its moral qualities, conserving the consciousness, or the reflective inward feeling,
of what it is”. However, “this conservation of personality does not occur in
the souls of beasts: that is why I prefer to say that they are imperishable
rather than to call them immortal.” (emphasis in original)[491]
Leibniz is emphatic that animals have souls too, in
opposition to the Cartesians. However, he is also emphatic that the
intelligence of humans is such that humans must and should rule over the
beasts.[492]
Descartes argued that beasts are mere machines, and Bayle took Descartes’ side
against Leibniz.[493]
Presumably, no soul monad is a mere machine, though soul monads – especially
those which correspond to passive matter – may often approximate mere
mechanism. Indeed, from the superficial perspective of textbook physics, this
is what makes the behaviour of inanimate matter amenable to human study and
human control.
Leibniz muses, “It is certain that God sets greater
store by a man than a lion; nevertheless it can hardly be said with certainty
that God prefers a single man in all respects to the whole of lion-kind.”[494]
Later, he says the maxim “that all is made solely for man” is “old and somewhat
discredited”.[495] Far
from making Leibniz an environmentalist or animal-lover, we can really only
conclude that Leibniz is opposed to gratuitous mistreatment of animals. Leibniz
says that punishing and even killing animals would be justified if it
effectively prevented beasts from causing disorder or other harm in human
society.[496],[497]
Animals have rational souls, as do humans. However,
human souls are “raised to the rank of reason and to the prerogative of minds”.
Minds are “images of the Deity” whereas ordinary souls are not.[498],[499]
All that exists must be interrelated, and therefore
everything must express the same nature but in a different way. All minds have
intercourse with each other and express the same nature.[500]
Consistent with this is that every monad contains
within it complete information on every other monad and, thus, complete
information on the entire universe. However, such perceptions are confused and
indistinct.[501] Because
all monads are on this equal footing, Leibniz says that all of history, both
sacred and profane, is confirmed.[502]
(We also discern from this that no monad – much less a soul monad – is a mere
machine but can potentially manifest complex behaviour. The internal state of
such entities would be orders of magnitude greater in richness than their overt
behaviour may appear to be, though not prohibitive of being understood.)
Since simple monads’ perceptions are “indeterminate”,
i.e. non-ideal, the choice of the best possible universe is a choice leading to
superior guidance of our and other simple monads’ internal perceptions and so
affect their “appetitions” and thence their decisions. Indeed, even galactic
events are impinging on the perceptions of humanity, and so are affecting our
decisions and behaviour.[503],[504]
Leibniz formulated a coherent system which embraces
material reality, immaterial reality, God, ideas and thought. There are many
exciting directions to take from here, not least the one which Leibniz
prescribed: creating a Universal Characteristic.
The Leibnizian conception of the physical universe as
outlined in this chapter shows Leibniz to be Nominalist. However, Leibnizian
Nominalism is qualified by the necessity of the Best, since only the best may
ever manifest in the physical universe. Things can be the Best only when they
correspond to ideas, though we do not know whether correspondence with ideas is
sufficient – it probably is. Thus, we can increase our understanding of the
physical universe by studying ideas because, of all possible thoughts and
concepts, only ideas can be instantiated in the physical universe. Since ideas
have structure and clarity, they can often be given mathematical form and are
subject to calculation. This is why mathematics is indispensable to physics,
and why mathematics sometimes precedes physics.
Perhaps the most important idea of all is that of the
Best, since it subsumes all other ideas. Arguably, Leibniz’s “the Best” is what
Plato referred to in The Republic as
“the good”. An idea is only instantiated at a particular point in any series of
events if it is consistent with the Best. It is the need to understand “the
Best” which often places metaphysics, aesthetics and art, and theology prior to
mathematics and physics. The first three situate, contextualise and bound how
close human thought can approach “the Best” in their endeavours through mathematics
and physics.
The Best is made more complicated – or richer, or more
harmonious and beautiful – by the proactive and wilful role played by thought
in the running or “unfolding” of the physical universe. These are the thoughts
of souls – in particular, human souls and other souls equally capable as human
souls though we presently do not have direct evidence of any of the latter –
which are in symbiosis with the appetitions of, or the choices made by, souls.
Choices are also made by God in the interaction between souls and the physical
universe. Physics cannot be complete without subsuming the role of these
thoughts, appetitions and choices. A calculus or some kind of mathematics is
needed to help our human minds cope with it in a structured way. These thoughts
and choices are indispensable to the “onward and upward” unfolding of the
universe in the Best possible way. Without these thoughts, appetitions and
choices, the universe would be inferior to what it is, it would certainly not
be the Best, and it – for reasons not yet understood by this writer – might not
even be possible.
There is an apparent contradiction in the fact that
God makes choices from moment-to-moment in the unfolding of the universe, and
the fact that Leibniz held that a universe that requires intervention by God
must be imperfect. This contradiction vanishes when we understand that the
universe only appears to be running or operating from moment-to-moment. It is
actually being re-created anew from moment-to-moment per the perceptions or
thoughts of soul monads which, to varying degrees, are changing in tandem with
the unfolding of the physical universe. The relationship involved in that
tandem-ness requires further study, and it essentially is nothing but Leibniz’s concept of the
pre-established harmony.
In this chapter, these matters are addressed: the
choices God makes, the ongoing creation of the universe, the role of the
thoughts of souls, the role of free will and the necessity of including
intellectual intent in physics.
While the physical universe functions without divine
intervention, a contradiction in Leibniz is that the Monadology has God making choices continuously.[505]
The “choices” are “made” as a result of the universe and the intentionality
built into its design. Thus, the choices are not directly of God but are of the
universe. The universe functions almost as an intelligent entity or system that
makes choices continuously. We now explain Leibniz’s theory using his
terminology.
No monad can affect another of its own accord. This is
natural since monads are all self-sufficient. However, monads act on one
another through the mediation of God. We remind the reader that we are forced
to read Leibniz as meaning “through the intentionality of the universe which
God created”. Such mediation is only ever in accord with ideals.[506]
Thus, there is not really a “choice” for God, because he always finds and
chooses the best possibility though he has the power not to so choose. Whence,
we understand why Leibniz says that (the functioning of?) the entire universe
relies on “God” for its existence each and every moment.
Grace cannot be understood by reason but only by
revelation. We leave this to one side since the natural workings of the
physical universe are untouched by grace, as they are governed by metaphysical
principles and mathematical rules. For God to arbitrarily modify these would be
an imperfection in the order of things.[507]
Thus, the reliance on God’s mediation is not through grace as such, for God
would not choose to act in any way other than the best and that, in turn, is in accord with ideals.[508]
Perhaps the pre-established harmony is a subclass of
the interaction between monads in conformity with ideals through the mediation
of God. Since it adheres to ideals, it is in that sense pre-established.
However, it is not pre-established in the sense that the perceptions of the soul
are indistinct and so the choices unpredictable or, at least, non-ideal. Or, to
God, they are largely predictable because he knows the internal perceptions,
and their changes, of simple substances or created monads.[509]
Nicolas Malebranche was a philosopher who was a
contemporary of Leibniz. Malebranche lived in Paris from 1638 to 1715.
Malebranche concluded that “there is only one true cause because there is only
one true God; …the nature or power of each thing is nothing but the will of
God; … all natural causes are not true causes but only occasional causes”.[510]
This is now known as occasionalism. Leibniz’s thesis that God mediates between
monads seems to verge on occasionalism, it has very different conclusions.
While occasionalism denies any causation aside from God, monads by contrast determine
their own internal states. Thence, interactions between monads are affected. That
is, the internal states of monads are independently driving unfolding order of
the universe. Leibniz’s doctrine appears similar to occasionalism, but it leads
to a conclusion contrary to occasionalism.
There is an infinite number of possible universes[511]
all in accord with ideals or they would not be possible. God chooses the best
of all that are possible.[512]
Ideas are as they are, and ideas cannot be mutually contradictory or some of
them would not be ideas. Therefore, the difference between possible universes can
only be our and all monads’ perceptions. Another way of considering this is to
“distinguish metaphysical necessity from moral necessity”.[513]
Furthermore, God does not act purely “according to mathematical laws, following
an absolute necessity”.[514]
In short, “nature leads to grace and grace perfects
nature by using it.” Here, “nature” means metaphysical principles and
mathematical rules represented by the laws of physics.[515]
This use of the word “grace” does not mean “grace” in the usual sense of
miraculous divine intervention, for a propulsion towards the best possible
outcome is subject to an ideal or rule.
Can a calculus of such “best outcomes” be produced?
However, we pre-empt ourselves for we need a calculus of metaphysical
principles first. Indeed, we are a long way even from an overarching calculus
of mathematical rules. However, it might make sense to start on all three
projects simultaneously as they might assist each other.
God must be making a choice continuously based on
sufficient reasons for the best of all possibilities which God perceives in his
mind all at once. This is not Zeusian arbitrariness in playing with the
universe and humans. If we had the all-encompassing perceptive ability of God,
we would know what he will choose.[516]
Leibniz suspends judgement on the extent of the role of grace in God’s making
the choices,[517] though
we do know that Leibniz denied the necessity of grace to this universe being
the best possible one.
Were it not for this selection undertaken by God, the
principle of plenitude would imply that the domain of ideas bears a one-to-one
correspondence with the physical universe. However, this cannot be true because
there is an infinite number of possible universes which conform to metaphysical
principles and mathematical rules, when in fact only one universe exists.[518]
When does God need to mediate or communicate between
soul monads, albeit he always does so in conformity with ideals? Alternatively,
when do two soul monads ever communicate or impinge upon each other except via
the laws of physics via corresponding body monads? Leibniz does not provide an
example. It is unlikely that God ever does. However, Leibniz probably does not
wish to contradict the Bible by excluding the possibility of miracles. Most
likely, Leibniz believes that there is a scientific explanation for what
apparently are miracles.[519]
Ideals matter in the outcome of interactions between
monads. They also happen to produce maximum harmony and beauty, and the best
possible outcome. They govern choices by God “on behalf of” passive substance
(e.g. planets) in order that our perception may grasp this harmony and be
improved by it. God makes such choices because they please him.[520]
So, in any case, God is extremely active (infinitely
so) in producing “the most beautiful combination of justice and goodness which
could be wished”.[521]
It is through this continuous selection process that God creates a repayment of
good from evil plus interest.
Clear and distinct perceptions in a monad are
different from appetition. Leibniz says little about what determines appetite.
Presumably, clarity on certain ideals (such as justice and goodness) will
generally create concomitant appetite, whereas clarity on other ideals (such as
plenitude) without clarity on, say, justice and goodness might lead to evil
appetite. However, this is not necessarily the case. The passage “from mind to
heart” is long.[522]
Such monads might have power for good or evil in the universe, respectively,
depending on the combination of ideals and conceptions perceived and the degree
of clarity with which they are understood by the monad. Monads capable of any
of these were made by God in his image. Leibniz also calls such monads
“spirits”.[523] “The
mind is not a part but an image of divinity, a being which represents the
universe, a citizen of the divine kingdom.”[524]
Leibniz draws parallels and distinctions between God’s
knowledge and a created monad’s perceptions, and between God’s will and a
created monad’s appetite.[525]
The latter qualities (of the monad) are only imitations of the former qualities
(those of God), with a quality corresponding to the degree of perfection of the
monad.
What is an example of where there might be a choice,
both options in conformity with the laws of physics, one tending to the better
and the other to the worse – where God of course chooses the former?
One is reminded of the postmodern “chance event” films
where the falling of a cigarette butt in a particular square centimetre of
floor triggers a course of events which changes a person’s day or life. But
even in this case there is no place for a choice
to be made by God, for the coordinates of the location where the cigarette butt
falls is determined by the laws of physics and since people have free will God
cannot intervene in their perceptions, decisions or reactions.
The overriding contradiction is that Leibniz said in
several places that a universe which required intervention or involvement by
God at all, much less continuously, would not be a perfect universe and so not
the kind of universe that God would have created. We cannot think that Leibniz did
not realise this, and so do not find it a stretch to say that, in this context,
by “God” Leibniz meant “the universe that God created”. Indeed, since God would
in any case choose the best, why would he not create a universe that does so
“on autopilot”? Autopilot programmed or built in to the universe by God is many
(perhaps infinitely-many) orders of magnitude beyond ad lib “thinking on your feet” human creativity.
In short, God has to maintain the correspondence
between souls or active substance, that is incorporeal, and body which is
passive substance and corporeal.
Leibniz objects to God being required to act at every
moment, and thus explains that his theory of pre-established harmony is
superior to Descartes’ theory of occasional causes. Pierre Bayle agrees. Yet Leibniz’s
theory of ongoing choice of the best possible universe[526]
also requires ongoing intervention by God though we have explained above how we
prefer to understand this. The difference is that pre-established harmony seeks
only to explain the movement of body under the action of soul/mind. God should
not have to get involved in mere body.
Leibniz’s explanation of the ongoing choosing of the
best possible universe must leave the laws of physics to work as they do, while
allowing God to make the “choice” that remains of the still-infinite
possibilities even with the universe constrained by the laws of physics. The
only intervention needed is to maintain the pre-established harmony. That is,
“to change the natural course of the thoughts of the soul to adapt them to the
impressions of the body” and vice-versa.[527]
God could choose to tamper with the pre-established harmony, and presumably not
impact the functioning of the laws of physics at all, but chooses not to
because it would derogate from the universe functioning in the best possible
way.
This is an enormously significant point that cuts
across theology, morality, psychology and physics. If a human (or animal or
bacterial) soul had a thought or intention that was not given effect to in the
corporeal realm, then not a scientist in the world would notice. Yet it would
make a very real difference to phenomena in the corporeal world. What, indeed,
should that correspondence be between a thought and an action? Is it possible
to define the correspondence as a mathematical relation? Leibniz argues that
God links thoughts with actions in the best possible way.
Arguably, Leibniz remains true to his dictum that a
universe that requires God’s ongoing intervention would be imperfect, while
allowing that God does have an ongoing influence over creation.[528]
The ongoing intervention is not of God but of soul monads. soul monads have
thoughts which collectively form a space or a field which is incorporeal. Each
soul monad’s thoughts impact the physical universe in the best possible way,
indeed, causing change in or of the physical universe. A different set or field
of thoughts of soul monads corresponds not only to a different universe because
soul monads and their thoughts are part of the universe, but also results in a
different physical universe.
An entirely different universe would result if such an
entirely new set of thoughts were possible and if the best possible
manifestation of those thoughts were an entirely different universe. Of course,
such an entirely different universe would need to be in conformity with ideas.
However, soul monads’ thoughts do not change very often and often move in
lockstep with one another – at least, those of human soul monads do – and thus
we see correspondingly little change in the physical universe.
There is ongoing change in soul monads’ thoughts
creating a new space or “thought field” causing change in the physical
universe. We say “change” because it manifests as an incremental “delta”.
However, for Leibniz to be correct it cannot be mere change because “change”
implies a running or functioning or ongoing operation, which cannot require
God’s intervention and therefore cannot have a pre-established harmony since
God has a hand in the pre-established harmony. Rather, the universe is being
re-created every moment according to the thoughts of soul monads which act as a
diktat on the physical universe in
the best possible way via the pre-established harmony.
Soul monads individually are eternal and
indestructible while their perceptions change as a result of experience of each
other and of the physical universe. Again, these perception-change or thought-changing
experiences are preceded by the action of God between body (passive substance
or physical matter, which includes manifestations of the thoughts of other
souls which is the only way we can experience or know the thoughts of other
souls) and the soul. While each monad (active and passive alike) is itself
indestructible and eternal, the nature of the universe and the way monads
interact with one another is transitory and dynamic. It is these changing
relationships along with the dynamicism in the internal perceptions of soul
monads that characterise each new incarnation of the universe as a whole. The
way such “characterising” works is a function of the pre-established harmony.
The pre-established harmony needs to be investigated as a serious branch of
metaphysics. Ultimately, when our understanding is sufficiently clear, the
pre-established harmony would become a branch of physics. An analogy is that
while H2O molecules do not change, the way in which they
relate to one another as well as their internal (energy) state can differ
greatly. This gives us the difference between ice, water and steam. Thus,
individual monads can change greatly in their interrelationships and internal
state creating greatly different universes. The differences of concern in this
analysis is in the incorporeal (active or soul) part of the universe as well as
in the corporeal/physical (passive or body) part of the universe which is
controlled by the incorporeal part, as well as in the entire universe regarded
holistically.
An open question is what soul monads are there aside
from those of simple living organisms, animals and humans? It may be that other
soul monads are very close to being passive (i.e. body monads) but because in
their dullness they work in such large numbers and with such uniformity (far
moreso than the soul monads of, say, humans do) they have enormous influence in
the physical universe via the pre-established harmony. Such would be the source
of the “controlling” field of creation of stars, galaxies, their motion and
evolution, and the behaviour of cosmic radiation. All cosmic phenomena
apparently involving passive (i.e. non-living, inanimate) matter would fall in
this category. Consistent with this is that, according to Leibniz, there is no
matter that is entirely inanimate. The identity and nature of these soul monads
– which correspond to non-living (in the conventional biological sense) body
monads – requires further investigation.
If God controls in the best possible way the relations
between monads, what actual role is there for free will? It seems that God via
the Best actually controls everything. The central factor that God does not
control is the internal perceptions and appetitions of monads. It is only when
these are clear and distinct that the relations between monads and thus
considerations of the Best come into play.
Our free will, and that of all soul monads to the
extent that they are active/soul,
consists in our ability to choose what we focus on, our physical experiences
(to the extent that we can), our appetitions, our use of rationality in making
interpretations and our intellectual input. Of course, the extent to which any
individual can make those choices is limited too, but that very limitation is
itself a function of the pre-established harmony because the scope of those
choices is a function of interactions between our soul monad and other monads.
Perhaps this is why Leibniz advocated top-down policy towards structure society
in such a way as to maximise the positive intellectual and cultural potential
of the common mass of people to enable them to make superior choices for
ongoing betterment of themselves and of humanity as a whole.
Leibniz’s essay “On the true theologia mystica”[529]
is notable for what it does not say about God. However, it is very clear about
what is not real. “Most knowledge and invention [Tichten] belongs to the shadow way.”[530]
He says that, “Corporeal things are but shadows which flow away, glimpses,
shapes, truly dreams.”[531]
Leibniz later restates this in a letter written in Hanover on 30 June 1704.[532]
He goes on, “Essential truth is in the spirit alone. But inexperienced men take
the spiritual for a dream and what is tangible for the truth.”[533]
Leibniz is neither Gnostic nor a mystic. In another
place, he wrote that witchcraft and sorcery are based on delusion or fraud. He
wrote of what sounds like a description of transcendental meditation, “There
are some who imagine a world of light in their brains.” However, “this is not
the light but only a heating of their blood.”[534]
He says that those who claim to experience mystical phenomena are only in an
aberrant psychological state. We should try to “conserve them in this beautiful
frame of mind, just as one preserves a curiosity or a cabinet-piece.”[535]
Leibniz cannot even be said to be a Cabbalist, for he explains why the idea of
the “universal soul” is untenable.[536]
Leibniz is amending and extending the actual Theologia Mystica[537]
by Dionysius the Areopagite (“Dionysius”). Leibniz’s title suggests that his
intention is to supplant it. Due to the status and institutional theological
significance of Dionysius, it is likely that Leibniz is being diplomatic and
agrees with very little of it. It is too “mystical” for Leibniz. Dionysius is
emphatic about what we cannot know, and the uselessness of mind. It reads
almost like a manual in transcendental meditation in that it instructs the
seeker to abandon mind, body and everything that he has.[538]
However, in “On the true theologia
mystica” Leibniz criticises “the denial of self” saying that the only
denial of self we should consider is hatred of our non-being which is the
source of sin. Sin does not come from God; rather, “original sin has arisen in
some creatures from their nonbeing and hence out of nothingness”.
For Leibniz, the incorporeal is subject to rational
inquiry just as much as the physical universe is amenable to experiment.
However, Dionysius seems to deny all kinds of knowledge and inquiry when he
writes, “We pray to enter within the super-bright gloom, and through not seeing
and not knowing, to see and to know that not to see nor to know is itself the
above sight and knowledge.”[539]
Thus, Dionysius advocates removal of mind, which would certainly have played a
role in rousing Leibniz to write a response to counter Dionysius. On the
contrary, not only does Leibniz say that a human is something, but he also
explains the nature of self-knowledge: “The only self-knowledge is to
distinguish well between our self-being and our nonbeing.” Furthermore, “one
must make use of sensual things and must view the shadow pictures only as an
aid or a tool and not rest in them.”[540]
For Leibniz, it makes sense to inquire into the nature
of God and, indeed, it is necessary to do so to progress in science. However,
Dionysius has made the concept of God entirely mystical. Leibniz also writes
that, “Essential truth is in the spirit alone. But inexperienced men take the
spiritual for a dream and what is tangible for the truth.”[541]
Dionysius does not agree that truth is in the spirit but has detached the
concept of truth from rationality. Indeed, it may be that Dionysius is denying
the concept of truth altogether though he does not explicitly say so; he is
certainly instructing the “adept” to abandon the pursuit of truth.
While, for Leibniz, “Corporeal things are but
shadows,”[542] he
repeatedly says that what is physically tangible or “body” must be used by
humans as a tool to further advance our soul which is to progress in
understanding and intellectual capability. However, “God belongs to me more
intimately than my body”. Analysing body is effective in helping us to come up
with useful ways of doing things, which can help us in day-to-day life.[543]
In doing so, we can follow Leibniz’s art of reasoning approach, which is not
far from Newton’s prescription. Likewise, in his concept of metaphysical
substance or materia prima, Leibniz
is using his reductionist principle of his pragmatic “art of reasoning”.[544]
Leibniz finds it useful to distinguish primary from derivative when in
discussing forces[545]
and in discussing phenomena.[546]
Using this method of reduction, Leibniz explains why the Cartesian concept of
extension is not fundamental. He explains that extension or distance and time
are merely relative ideas, in some ways anticipating Einstein and Einstein’s forebears
in relativity theory.[547]
Yet 14 years later Leibniz refers to “phenomena” and
“actual events”.[548]
This seems to be shorthand for constituents of the physical universe as far as
we can ever vouch for it. More specifically, it seems to be the impact of
ideals on substance. More accurately, it is the adherence of the universe to
the best possible because God has designed it that way. The “best possible”
happens to be in conformity with ideals. Why
this is the best possible is a question of huge importance, which is beyond the
scope of this thesis.
Leibniz restates conformity of the laws
of physics with ideals by saying that it cannot be otherwise for nothing ever
happens which violates any of the most exact rules of mathematics.[549]
Paradoxically Leibniz does not think that mathematical determinism suffices for
an all-encompassing corpus of physics or of the unfolding of the physical
universe. This is due to:
(a)
possibly, the
self-affecting dynamical nature of the universe;
(b)
the imperfect character
of perception and the non-ideal decisions by monads that flow from the appetite
resulting from those imperfect perceptions;
(c)
the creative
ability of some monads which surpass other monads in their clarity and
distinctness of perception, and purity of their appetition; and
(d) the fact that God does not act purely according to
mathematical rules.[550]
If we restrict consideration in (a) to passive
substance, then its behaviour is likely to be calculable provided we knew the
exact location of every atom in the universe, and full knowledge of every
possible metaphysical and mathematical rule. Of course, this will never be
possible for humans, but we can comprehend that it might theoretically be
possible.
An example of (b) is the result of geopolitical
manipulations leading to humans blowing each other up in a senseless war. An
example of (c) is humans improving upon nature by, for example, redirecting a
river to bring life to a portion of desert. These are by definition outside
what can be addressed by ideals, and metaphysics and mathematics are restricted
to ideals.[551]
In other words, perception and all that depends on it
is inexplicable on mechanical grounds.[552]
Perceptions are no small consideration even though they are outside the corpus
of deterministic textbook physics, which only even attempts to address
efficient causes.[553]
Indeed, only perceptions can be found
in a simple substance[554]
which is a created monad.[555]
This class of influences on the universe is called final causes.[556]
All internal activities of created monads are a result of perceptions which are
also a kind of thought or mental image or notion. Each created monad is
independent of everything as if there were nothing in the universe but itself
and God. This is notwithstanding the fact that every monad perceives every
other monad, since perception is not the same thing as direct influence.
Apparent “influence” of a monad on another (e.g. of a person holding another in
their thrall) is actually the latter monad voluntarily pursuing the perceptions
of the former or having their perceptions manipulated by the former; in common
parlance, the latter has ceded its will or personal sovereignty to the former.
Another way of considering (b) and (c) are that God’s
raw material is us, along with monads below and above us. He needs to wait especially
for us and monads above us to improve, and gives us every conceivable opportunity
to do so without violating our free will. He would not force us because that
would not be the best way. The way this unfolds leading to our realising our
co-creative ability is outside metaphysics and mathematics, but is a function
of the “thought process” of a monad.
How can this thought process be improved to help along
God’s intention for the universe? Of course, the Universal Characteristic as
Leibniz conceived it would aid thought. The Universal Characteristic can now be
regarded as a means whereby some of the most powerful fundamental constituents
of reality – active monads or minds – can empower themselves to fulfil their
function even more effectively. First, the Universal Characteristic must encompass
the discovery and use of metaphysical principles and mathematical rules. To be
complete it must be able to assist or systematise calculations with both (b)
and (c), and accommodate problems with respect to those. Somehow, (d) needs to
be taken into account too. To understand (d), we need to understand the
question under the above subheading “Unanswered question re second kind of
choice”.
That creation should unfold in the best possible way
is pre-ordained because this is built into the design of the universe. At the
same time, there cannot anyway be sudden and bizarre changes due to the
principle of sufficient reason. Human free choice adds a certain amount of
variability, within a particular range. If we had perfect knowledge, then we
could forecast how things are going to work, subject to (or modulo) the free
choice of human souls and all the other soul monads in the universe. With any
given amount of knowledge, the best choice in a given scenario is deterministic.
Thus, if we knew all that was important to a person and what their inclinations
were, then we would know in advance what they were going to decide. As
mentioned in the above section “Creating the universe anew in each moment”,
soul monads do not change their thoughts very significantly very often though
over time there can be huge changes. Of course, there are decisions that can be
made randomly or without reason. However, these have little impact on the
overall course of history.[557]
The large changes on history are caused by changes in the “thought field” of
large numbers of monads, which of course are sparked by insights had by
particularly active soul monads such as that of Christ.
In the sense that the architectonic design allows us
to foresee what will happen provided we have perfect knowledge of all facts and
all possible ideals, yes.[558]
But note that this is much harder than it might sound. We would need to know
the internal perceptions of all created monads, and know which outcome will
produce the best possible conditions for improving/clarifying their
perceptions. We will also need to know the impact certain experiences will have
on these perceptions, and their perceptions of their own and of other monads’
perceptions, the best possible choice from that point, and so on. In short, we
would need to be God.
Leibniz makes an art and a science of reading God’s
mind. This is because it is rational to do so since God is the Creator, and
because it is possible to do so as God’s mind is rational. “Reason is also
choice” says Milton in Paradise Lost.[559]
Aquinas also makes an art and a science of reading God’s mind.[560]
Leibniz says that the best science is done by attempting
to read God’s mind like Cusa and Einstein did.[561]
Experimental results are useful to confirm or deny what we think that God would
have done, or to check the accuracy of our reasoning, much as we would check
our accuracy in carrying out a long and complicated piece of arithmetic.
Science is a result only of our ability to grasp
demonstrable truths such as in logic, number and geometry for they “make the
connection of ideas indubitable and their conclusions infallible”.[562]
“The mathematical sciences, moreover, which deal with eternal truths rooted in
the divine mind, prepare us for the knowledge of substances.”[563]
Yes, when it is available to a human mind.[564]
However, it is rare that any human’s “intellectual perceptions” (i.e. the
internals of any created monad belonging to a human) are clear and distinct in
anything. All created monads are nearly always confused and indistinct in their
perceptions with respect to all things. Even in the angels and blessed “there
is always some confused perception mingled with distinct knowledge”.[565]
How could innate truths regarding topics other than
physics – i.e. outside of metaphysics and mathematics – ever be found through
empirics?[566]
Leibniz’s “empiricism” is simply that sensory
observation prompts thoughts and checks reasoning, including the soundness of
reasoning. “The external senses,
properly speaking, do not deceive us. It is our inner sense which often makes
us go too fast.”[567]
Empiricism says that the senses give us knowledge and that thoughts are only
needed to organise the knowledge that the senses give us. This is at odds with Leibniz’s
position as explained in New Essays.[568]
Similarly, entirely detached from sense perception,
ideas are not possible. At least, we never have ideas that are so abstract that
they are entirely disconnected from sense perception.[569],[570]
Is this effectively saying that we can never hold a pure idea in our mind, but only
something mixed with empirics? Perhaps not. Leibniz says in New Essays
that we already know a great deal. It is sense perception or general experience
that draws it out of us or which triggers thoughts from our memory. In this,
Leibniz follows the doctrine of reminiscence from Plato’s Meno.[571]
As we will explain below, if we have one Idea within us, then we probably have
all ideas within us, with varying degrees of distinctness. This is not
surprising because “each distinct perception of the soul includes an infinity
of confused perceptions which envelop the entire universe.”[572]
Similarly, “Each soul knows the infinite, knows everything, but confusedly.”[573]
If one rejects the idea of remembering things that we
already knew, then the need for experience to trigger thoughts which lead to
ideas might mean that the domain of ideas is actually bound up with the
impressions on our senses that, by definition, can only occur in the physical
universe. This leads to the “sufficient reason” for the existence of the
physical universe.
That no human thought is ever entirely abstract might
encompass the fact that humans “experience” or “conceive” ideas in a cultural
context. Perhaps it is merely the perceived context, whereas the idea itself is
universal and objective. Perfect objectivity is not possible for humans, and different
cultures facilitate the conception of the universe in different ways.[574]
Gose in his discussion of right reason refers to the
removal of choice by addictions to pastimes such as drinking and smoking.[575]
This is how Gose explains that the corollary of Milton’s concept of “right
reason” is that only the wise can be free. “With his accent on liberty and
freedom, Milton is quick to affirm that liberty is not license. … Milton
asserts that with the ‘dignity and freedom of individual man’ comes a
responsibility for individual discipline.” The addict cannot exercise that
discipline.
Leibniz would agree that only the wise are truly free.
However, this is because they have a greater appreciation of what will lead to
true happiness, which is pursuit of God’s mind. Thus, we would say that Milton
meant that “To exercise Reason is
also choice”[576] and
the wise tend to choose to.
Gose makes a couple of other useful remarks. He says
that “The classicists as well as other non-Christian thinkers” do not deny that
Reason is a source of (i.e. a source of knowledge of?) God. But, Gose says, “it
is limited by human imperfection.” This echoes Nicolaus of Cusa. Next, Gose
says, “Milton says that the unwritten law of God: is no other than that law of
nature given originally to Adam, and of which a certain remnant, or imperfect
illumination, still dwells in the hearts of all mankind; which, in the regenerate,
under the influence of the holy spirit, is daily tending towards a renewal of
its primitive brightness.” Here we see the Platonic-cum-Leibnizian memory of
what the soul already knows, but by a different method.
Gose says, “The classicists as well as other
non-Christian thinkers … are in need of divine revelation and love to complete
their search for truth.” One could be forgiven for reading Cusa in this way,
and perhaps this is what Cusa meant. It is certain that Cusa believed we could
never equal God. However, contrary to Cusa,[577]
Leibniz is clear that divine revelation in the sense of a mystical enlightening
or a rapturous experience is no way to achieve knowledge. We might sometimes
feel that that is how we are discovering – or recalling – something, but in
fact it is a process of Reason, and not necessarily a deductive one, that gets
us there. On the other hand, if we are remembering, then perhaps there is a
place for revelation though Leibniz never allowed for it.
To Leibniz, there is no such thing as freedom in the
sense of liberalism or of “licence”.[578]
Leibniz often used the pejorative barbarism. However, he also noted that men
from all cultures have a sense of justice even if they do not apply it to all
things and times or to the extent that Christians do or would if they followed
Christ’s teachings all the time. In New
Essays he gave the example of the barbarisms of the native Indians of
America and of some native tribes of Peru, but noted also that in some matters
those peoples adhere to the same standards of justice as Christians of Europe.
An “evil person” is a soul monad with perceptions that
are contradictory to ideas. One may ask, “Contradictory to which ideas?” The
answer is, “To all ideas.” Remember that “idea” has a specific meaning for
Leibniz. They are not merely thoughts or concepts. An idea is correct by
definition, and to be “correct” it must be consistent with everything that
exists and has ever existed. An evil person is further away from ideas than a
good one.
This definition encompasses moral wrong, and
deliberately emphasizes that moral truths are as much subject to rational proof
as a mathematical principle may appear to be. Perceptions contradictory to
ideas may have a degree of clarity and distinctness greater than many “good”
soul monads have of concepts that bear some resemblance to ideas. To anticipate
how Leibniz deals with the problem of: It is the clarity with which the evil
soul perceives what is contradictory to ideas that forces other soul monads to
clarity and drive their own conceptions closer to clear and distinct
perceptions of ideas.
“Evil” soul monads are active just as are “good” soul
monads, and so are superior in to and able commandeer more passive monads and
passive matter just as well as the good souls are, if the more good souls allow them to. The definition of an “evil”
soul monad is that it works in contradiction to ideas or relatively so in
comparison to “good” soul monads. Thus, nothing that evil soul monads do can
last, thought they have can cause destruction indefinitely as long as they are
allowed to before that uplift themselves, or are uplifted by more correct
perceptions, i.e. by perceptions that are closer to ideas. Since only ideas can
exist, evil works in conformity with non-being or nothingness, which is
precisely what Leibniz wrote in “On the true theologia mystica”.[579]
That is, sin is not from God but from nothingness or the non-being of certain
creatures.
In order to wreak havoc, evil soul monads make use of
discoveries that have been made in conformity with ideas, which permit machines
of control and weapons of destruction all of which arise from engineering and
science. Leibniz wrote that a bad European is much worse than a bad tribal
native because the bad European has at his disposal all the power arising from
the positive scientific culture of European civilisation.
One of the most frequent criticisms of Leibniz’s
system of metaphysics is his “best of all possible worlds” doctrine which is
actually a consequence of the principle of sufficient reason. If this is the
best of all possible worlds, then why does evil exist? Davies raises this
objection to the “Leibnizian optimism” but leaves it aside as an “ethical
issue”.[580] Yet it
is clear that question of justice, goodness and morals generally are as much a
part of ideas and the workings of “the mind of God” as are metaphysical
principles and mathematical rules. Davies then considers “maximum variety” as a
substitute for “the best”.[581]
This might better have been considered under Leibniz’s doctrine of plenitude.[582]
Nonetheless, Davies pursues a promising line of enquiry when he considers
beauty as a guide to truth.[583]
Leibniz explained that the truest and most enduring
happiness comes from pursuing the understanding of the whole which ultimately
means the mind of God.[584]
However, this pursuit must be free in order to be fruitful. Thus, evil is expected
in this pursuit but those minds which know better are likewise free to best the
proponents of evil conclusions reached through misguided reasoning, and to struggle
against those forces and attain supremacy over them.
Erasmus is close to Milton on free will, e.g.
“exhortations, commands, choice, reward and punishment, all present in
scriptures with respect to salvation, would be meaningless in the absence of
free will.”[585] This is precisely what Milton has God say to his Son
in Paradise Lost.[586]
Leibniz’s explanation of free will befits his rationalism and best of all
possible worlds doctrine, whose corollary is that people tend to make the best
decision they can at any given juncture, and the outcome however good or bad
leads us to a wiser or otherwise better state in the end. Leibniz even says
that a murder will ultimately lead to something better than had the murder not
taken place. However, he also takes pains to emphasise that this does not mean
that those who do wrong should not be punished – they should. However, he says,
ultimately sin punishes itself anyway. The past has occurred for the best but
we must not acquiesce in the present for the future as the quietists advocate;
we should always do our best.[587]
Also see the effect of evil and what God permits.[588]
God does not accidentally allow evil to occur; he does provide the tools or
circumstances which can be used for evil undoubtedly knowing that this “risk”
exists.[589] God
permits evil because he can bring a greater good out of it. The whole of
creation actually derives a net gain from evil.[590]
Moreover, Leibniz seems to be saying that certain kinds of good would not be
possible at all were it not for evil. Leibniz does not see the relationship
between good and evil as a balance sheet, where we allow evil to debit us one
unit so we can be credited two units worth of good. Rather, evil can trigger
entirely qualitatively new categories of good and catalyse possibilities for
good that would not otherwise have been considered or embraced.
We can hearken back to Milton’s Paradise Lost,
where God says to his Son that Satan should not be stopped from carrying out
his plan. Humans need to be left free to choose whether they shall yield to
temptation or not. Given the reality of free choice, the inevitable result is
that some humans will yield to temptation while some will not and innocents
will necessarily suffer. This evil is not merely accidental or part of an
optimisation but is an intended and, in view of the whole, a good result
which works for the betterment of the innocents as well as those who yielded.
Perhaps innocents are not innocent, for all must take a responsibility for the
whole.
Leibniz writes that he cannot “approve the opinion of
certain moderns who maintain boldly that what God has done is not supremely
perfect but that he could have done much better.” Rather, he holds that, “God
does nothing for which he does not deserve to be praised.”[591]
So does God deserve to be praised for the starving children and the murderers
and rapists? Leibniz addresses this when he asks, “Why does such a Judas, who
is merely possible in the idea of God, actually exist?” about which he says,
“no answer can be expected here on earth, except the general one that since God
has found it good that he should exist in spite of the sin which God foresaw, this
evil must be compensated for with interest in the universe and that God will
draw a greater good from it and that it will turn out finally that this
sequence of events, including the existence of this sinner, is the most perfect
among all other possible kinds.”[592]
Evil plays the role of revealing the good and, over time, we step back in order
to leap forward better.[593]
Leibniz’s conception of evil is more as natural
imperfections that are an inevitable part of the process of upwards evolution
or moral upshift, ultimately leading to every kind of improvement, enrichment
and increased power. Ultimately, when realised in human endeavour, such
upshifts includes kinds that are perceptible in the physical universe through
projects such as those discussed in Chapter 3. So for Leibniz the defining
characteristic of evil is not that which is judged as such by God, but it is
more closely linked with that which has a role built into the design of the
universe, just as that which is good has such a role.
While we are identifying Leibniz as a Neoplatonist –
as he says “I tend more towards Plato, Mr Locke to Aristotle” but on some
things we differ from both of these great ancient writers – we must note that
Leibniz is more rationalist than anyone before him.[594]
If there are two things we can take away from New Essays, the first is that all is
comprehensible. The universe was made in such a way that if a mind knows all or
is admitted into the secrets of things, then there will be reasonable
explanations for things and all will be understood. Second, the discovery
process is a remembering of what the soul already knows.
Hill argues that “Leibniz’s views of intentionality
are closely bound up with key elements of his metaphysics and epistemology –
especially his understanding of relations, his commitment to the explanatory
power of theism and the role of the divine ideas”.[595]
The objective way in which God works embodied in the
architectonic universe is probably Leibniz’s most significant contribution, and
we would argue that this is rationalism. What Kepler discovered and described
for the solar system, Leibniz explained for the universe in toto. In fact, Leibniz continued Kepler’s programme of, in
Franklin’s words, constructing a theory of everything.[596]
In New Essays
it is very clear that God would not create the universe in any way other than
the best. Thus, the universe could not be otherwise than as it is. Therefore,
we study the universe as it is (i.e. do science) by reference to that, not to
God per se. However, we can come up with hypotheses by considering (the)
perfection which is God’s nature – and all the (other) characteristics that God
necessarily must have. Writers like Wenck would say this is “constraining God”.
God’s nature, however, is to act for the best. There is a mathematical kind of
certainty to how God would act, but our reason is so clouded and our knowledge
so miniscule that we are infinitely far from knowing or understanding “the
best”. The concept of “the best” is an objective one and is just as
discoverable as any of the ideas. We are still using the strict definition of
“idea”.
While ideas are perceived in the mind of God, it is
likely that everything we think is an
idea, in fact, is not. Indeed, each idea is bound up with the entire universe
and cannot exist or be understood except as part of the entire universe. Only
God can hold the entire universe in mind in a single thought. Do ideas actually
exist? As discussed above, according to Leibniz, ideas so do not exist even
though they are perceived by the mind of God or held in God’s mind. Ideas are
realized or manifested in the physical universe. Indeed, ideas are all that is
manifested in the physical universe: this is one of the definitions of an idea.
The closest we get to a Platonic world of ideas in Leibniz
is the “intelligible world in the divine mind, which I also usually call the
region of ideas”.[597] We know that they have the effect of defining the
best. However, it is only a “world” in the sense that God has a lot of ideas in
his mind at once. A capable human mind could also have in it a world of ideas.
Hill argues that the scholastics
believed in the existence of such a domain of ideas:[598]
the scholastic philosophers developed a
“package” of metaphysical claims which underlay a common approach to
intentionality. These claims were that relations have reality outside the mind;
that God’s ideas function as exemplars of their objects; and that the mind
abstracts “intelligible species” from the objects of perception. These views
allowed them to defend the intuition that a thought of X is linked to X in some
way by the relations of both similarity and causation. … Leibniz held views
that were structurally similar to the [scholastic] “package”, which allowed him
to hold a similar approach to intentionality.
According to Hill,[599]
the scholastic theory of intentionality was supported by three primary
metaphysical doctrines:
Each of these contradicts Leibnizian ideas in various
ways. Each of these contradicts Leibnizian ideas in various ways. Taking each
of the points in turn:
1.
For Leibniz, an
idea is a concept that is consistent with the entire universe and so is an
expression of the entire universe. An idea is not located in the divine
understanding or anywhere. An idea has no location. It is true that only the
divine understanding can ever grasp an idea because only the divine
understanding can grasp the entire universe in a single thought. Human
understanding can approximate ideas but never fully.
2.
The intellect,
active or otherwise, never takes on the form as a thing that it contemplates.
The intellect is an active monad or a thing in its own right. An active monad –
meaning, a mind – can have impressions of things upon it which are nothing but
perceptions.
3.
A relation is
itself a concept that might approximate reality to the extent that its
predictions are confirmed by goings-on in the physical universe. A concept only
exists as a mental perception. If a relation did exist outside the mind, like
anything that exists outside the mind, it would be part of the physical
universe. Once we accept that something is part of the physical universe, it is
no longer a mere concept and therefore cannot be a relation. The concept of
relation is merely a thought tool to help us advance our concepts closer to the
likeness of ideas.
Hill’s project of demonstrating Leibniz’s use of or,
at least, agreement with scholastic ideas is not isolated in the literature. In
his 1686 “Discourse on metaphysics” Leibniz wrote, “the opinions of the
Scholastic philosophers and theologians are much sounder than has been
imagined, provided that they are used appropriately and in their proper place.
I am even convinced that if some exact and thoughtful mind were to take pains
to clarify and assimilate their thoughts after the manner of the analytic
method of geometricians, he would find a great treasure of very important and strictly
demonstrative truths.”[600]
Farrer says that Leibniz was a
scholastic but was bent on reforming if not rewriting scholasticism.[601]
Leibniz notes with approval that Scholastics at one
time expressed that God is the light of the soul. Thus, Leibniz is actively
looking for areas to agree with the Scholastics.[602]
As far as intention goes, it seems that the ultimate
or umbrella intention of all is God’s. We only have ideas of things in our soul
because of God’s continuous action on
it. Further, as already discussed, ideas are all that we have that can cause
phenomena (in fact, to which phenomena correspond because phenomena are the
best possible, to be explained further below under the final heading in this
chapter before the conclusion) as notions are too indistinct.[603]
When we say ideas “cause” phenomena we
mean, in fact, to which phenomena correspond because phenomena are the best
possible. This is explained above under the headings:
·
Ideas and “the
Best” are outside all that exists
·
The apparent
ruling of actual things by ideas
and below under the heading:
·
Foundations of a
programme for discovery.
The apparent physical world as far as we can tell has
been provided as a way to help our indistinct and confused intellect come up
with ideas or remember ideas that it
has always had within it, but which in some cases have never been brought to
the conscious fore of the mind. As we know from our experience, real ideas are
usually bound up with moral purpose and a kind of detached, i.e. not heatedly emotional,
passion.
Regarding relations, Hill says that Leibniz’s
“apparently contradictory statements about relations can best be understood by
distinguishing between two kinds of relations in his thought. He thinks that
relational properties have extra-mental reality, but ‘inter-substantial’
relations do not, and are mere abstracta. In this, Leibniz is very similar to
his scholastic forebears.” It would appear plain that relational properties
exist beyond the mind due to the reality of the things of which they are
properties. On the other hand, “inter-substantial” relations do not have a
reality beyond the mind because they are not real since all substances, i.e.
all monads, are independent (despite their being the same and being in constant
inter-communication).
Hill writes:
Leibniz speaks about God’s ideas a great deal – not
just those of human beings – and assigns them a part-cause in creation, with a
deliberate appeal to Augustine. These ideas are conceived in a fundamentally
relational way. Leibniz rejects Malebranche’s identification of our ideas with
God’s, but he does think there is some important connection between them; for
us to have an idea is to exist in some relation to God’s ideas.
However, that relation does not seem to be any
different from the relation between two souls having the same idea. For
example, if Jack and John both prove that the angle sum of a triangle is 180°
then there is a relation between their minds when both have the image of that
idea in their mind.
Hill writes:
I argue that an interpretation of Leibniz as a
nominalist about ideas, which has been defended by some recent commentators, is
not accurate.
Leibniz himself seems to be in support, “As long as we
have only a nominal definition, we cannot be sure of the consequences drawn
from it, for if it concealed some contradiction or impossibility, we could draw
conflicting conclusions. This is why truths do not depend on names and are not
arbitrary, as some modern philosophers have thought.”[604]
However, recall that to not be a nominalist is to believe either universals
exist or abstract objects exist in a Platonic sense.[605]
Leibniz believes neither of these and therefore is not a nominalist. Believing
we are forced inexorably to conclusions that are correct or true is different
from believing that truths exist as real objects. Rather, as explained when
Leibnizian ideas were introduced, what is an idea is bound up and ultimately
defined by the totality of the best of all possible universes.
Continuing with Hill:[606]
According to this [nominalist] interpretation, Leibniz
believes that to have an idea of something is simply to have a disposition to
think about it. I argue, however, that this is only Leibniz’s understanding of
ideas when they are not being thought of; when they are being thought of, they
are objects of thought.
It is unclear what an “object of thought” is. However,
Leibniz does not take an “idea to be an immediate object of thought or for some
permanent form”.[607]
Leibniz says that our mind can manifest whatever it desires to itself, “As a
matter of fact, our soul always does have within it the disposition to
represent to itself any nature or form whatever, when an occasion arises for
thinking about it.” This makes thought akin to a “phantasm”.
Hill writes:[608]
Moreover, the divine ideas certainly
cannot be understood dispositionally at all. In fact, the divine ideas are
identical with their objects, a claim which is fundamental to Leibniz’s
overarching argument in the Theodicy.
Here again, there are strong similarities between Leibniz’s position and that of
the scholastics.
…
Leibniz distinguishes between “a”
concept of something and “the” concept of the same thing; the latter is
identical with God’s idea of that thing, which is also identical with the thing
itself.
We would disagree that in Leibniz’s analysis God’s
idea or concept of a thing is any more real than a human mind’s idea or
concept. An idea is not a real thing but a mental perception of real or
potentially real things. Leibniz distinguishes mental images, even God’s mental
images of ideas, from real things. Thus, divine ideas cannot be identical in
the sense of being the same as their objects, for a mental image cannot be
real. Leibniz distinguishes a mental image, even God’s mental images of ideas,
and real things.
Leibniz is strict about what it means to have an idea.
“One can boast of having an idea of a thing only when one is assured of its
[the thing’s] possibility.”[609]
Leibniz also writes, “it is obvious we have no idea of a concept when it is
impossible.” Further, “in the case of merely suppositive knowledge, even if we
may have an idea, we do not grasp that idea, for such a concept is known only
in the same way as are those concepts which
involve a hidden impossibility; even if it is possible, we cannot learn of its
possibility by this way of knowing [i.e. by supposing it is so].”[610]
Overall, Hill is looking at the pieces of Leibniz’s
philosophy and letting us know where it seems to be scholastic. The overall
intention of Leibniz differs from the overall intention of most scholastics. Leibniz
as a self-described “more Platonic than Aristotelian” thinker undoubtedly adopted
some ideas similar to the scholastics and in some cases fully agreed with the
scholastics.
Hill discusses the kinds of ideas that Aquinas says
that God can have - i.e. T-ideas (ideas as types) versus TE-ideas (ideas as
types and as exemplars).[611]
Hill’s discussion around Aquinas misses the fact that God acts for the best
because it is in God’s nature to do so. God is not impelled by nature but by
his own nature, albeit that his
nature is to act for the best; thus, we see ideas in what becomes the laws of
physics (i.e. nature) only due to God’s nature.
Upon thinking of something, for God to will it is for
it to become real. Thus, intention for God is a very different matter from
intention for humans. At the same, while action not will alone is needed for
humans, action is a result of will. As far as the soul is concerned, will is
action for the soul. Pre-established harmony causes the will of the soul, or a
particular kind of will of the soul, to invoke changes in the compound body
monad which effects physical changes in the universe.
God only chooses to will into reality thoughts that
are in conformity with ideas. Could humans act to bring into reality plans that
are not in conformity with ideas? Of course, though action by humans is much
slower and more painstaking that will is for God.
We must leave open the possibility that there are
ideas accessible by God that require faculties that God has which humans do not
and never will have. For example, God’s mind can comprehend the entire universe
and all of its possibilities in a single thought.[612]
This may well open up an entire class of ideas which are of no practical use to
humans without the capability of instantaneous pan-universal comprehension.
However, humans could presumably still grasp and reason about the process,
structure and harmony in the ideas used; it is just that we could never make
practical use of those ideas in the way that God does. We might still be able
to make practical use of those ideas for smaller matters. However, it might be
possible that there are some ideas that only make sense when the entire
universe and all possibilities are under consideration.
Ideas are the foundation of the unfolding of the
universe even though ideas are not of the universe or in the universe, nor of
the immaterial realm of souls (which too is part of reality) nor even part of
God. Thoughts too are merely images or perceptions. Nevertheless, both ideas
and thoughts follow an order, can be structured and demand study. Indeed, they
motivate the Universal Characteristic.
How can we speak of “them” if they do not exist?
Language and speech are nothing but vehicles for representing thought, and a
thought may be about “any nature or form whatever”, thing, notion or idea. In
any case, all we are doing is progressively bringing out of our memory in
clearer and more distinct form ideas that have always been with us. They would
have to have been, since our soul – like all monads – perceives all other
monads and therefore has knowledge of the entire universe. However, the
perception is unclear and indistinct.
A mind/soul can represent to itself a thing that is
not real. However, the universe was (in the design phase) constructed in God’s
mind. The perfect universe (of the many so conceived by God’s mind) was willed
into existence. That universe is in conformity with ideas, which only means
that the universe is consistent with itself and is good, i.e. is a reflection
of God himself. So by understanding the universe we understand God. The
universe, in turn, manifests every possible good. Ideas are simply mental
constructs for understanding the universe and hence the good (God) as well as
vice-versa, i.e. ideas are mental constructs for understanding the good (God)
and hence the universe.[613]
There is a paradox in that what is most evanescent
offers the greatest power, perhaps the only power, to affect the physical
universe. Three concepts are considered: ideas, which do not exist; souls,
which are immaterial; and perceptions within souls which are merely mental
impressions. Ideas are not even real and so are “less” material than the
immateriality of souls and are also less material than the images which
comprise the perceptions within souls. Yet it is from ideas that souls derive
their power. In particular, it is by the distinctness with which ideas are
perceived that their power is measured. This is only because a thought can only
correspond to physical reality to the extent that it represents an idea. Riemann
formulated a theory of thought that appears
to be a maturation and consolidation of the mechanism and role of Leibniz’s
“perceptions”. Riemann used the term Geistesmassen
which can be translated as thought masses, spirit masses or mind masses.[614]
The term “mass” could mislead. Riemann’s emphasis is on the fact that Geistesmassen are the locus of activity
of intelligent souls. Mixing in Leibniz’s terminology, when a thought mass is
sufficiently clear and distinct, it can give rise to action by the
corresponding body monad. The closer the thought mass is to representing an
idea, the more effective the consequences of its guidance of the corresponding
body monad in the physical world will be. For example, if a scientific concept
is incorrect, then it is not an idea and physical structures or machines will
not work if their design is based on that concept. If a scientific concept is
correct, even if only within a particular domain, i.e. it is closer to being an
idea, then it can be used for new engineering possibilities.
When monad X distinctly and unconfusedly perceives the
indistinct perceptions in monad Y, are the perceptions of monad X ideas? Yes,
because such distinct perception would understand the long chains of causes and
implications of the indistinct perceptions of Y. Such perceptions by X give it indirect
power over Y because it understands Y.
Due to the relationship of God to Man, since God is
the starting point for a priori
scientific thought, it is necessary to understand Man in the a priori fashion too. Indeed, just as it
is only possible to theorize about God in the a priori fashion and possibly have some conclusions thereby reached
confirmed or denied by experiment or mathematics, the same is true of Man. Even
an Atheist, to whom the concepts of divinity of Man and God’s relationship to
Man are meaningless if not offensive, the fact that Man has a conscious
creative power not found elsewhere in nature begs special attention.
The difference between our ideas and our brand of
creation versus that of nature (i.e. what God chooses to allow, aside from
humans) is critical to our understanding the character of intellection beyond
nature but below God’s. God made nature, but could do much more; instead, God wants
humans to extend nature. This can only be to allow humans the opportunity to
improve themselves, since God could have done and could do anything that humans
might ever do. The upwards development of humans is part of the process of
perfecting the universe. A teacher gives its pupils opportunities to create,
and does not jump in and take over out of impatience. The teacher’s task is for
each student to successfully create autonomously.
At the same time, there is nothing in nature which
could do what Man can do. Only a self-conscious entity such as Man can
undertake creation of the kind that Man can. Thus, Man is God’s natural instrument
for that very kind of creation. Necessary to the self-consciousness that must
go hand-in-hand with the divine creativity that Man has is political conflict,
oligarchism, etc. Because we are self-conscious, we will naturally have trouble
understanding our own nature or character, just as it took thousands of years
to discover various natural physical laws. On the other hand, Man is not God’s
instrument. Man is God’s project. God wants Man to succeed in autonomously
developing or evolving the physical universe upwards but, again, for the sake
of Man not the universe per se.
Ideas, which are often presented as laws, about
ourselves and our creative capability cannot be studied empirically in the way
that some physical phenomena can be studied using physical experiment. As yet,
we have no systematic empirical way of discovering such laws. Thus, we have
been limited to the a priori or, at
least, purely intellectual method.
Sensory perception and concepts that are able to be
visualised clearly in a way that is akin to a sensory perception but held in
the mind are the locus or raw materials of the Empiricist mode of thought and
investigation. These loci are not a very effective way of discovering such laws
because there is little in the external world that is indicative of the
potential, nature or role of Man. First of all, humanity largely lives in the
realm of final causes which, by definition, does not permit the use of
empirics. Second, manifestations of humankind in the realm of efficient causes
generally require the discovery to be made first and then be implemented into human
society and human law and government. It may then take decades or centuries for
the results to manifest themselves which can then be pointed to as proof of
whatever the presumption about human nature had been. Thus, only the a priori method is of significant value
and this is based purely on thought.
There have been many political experiments in history.
There are libraries of books on the results of those experiments. Arguably,
many political systems resulted in disaster because the a priori work had not been done or had flaws: such flaws would be
regarded as moral flaws. As mentioned above under the heading “The problem of
evil”, moral ideas are as much a part of the domain of discoverable truths as
any “physical scientific” question of, say, gravity or air pressure. The point
is the moral truths are physical
scientific, and the nature of Man is a physical scientific question because Man
is a force for change in the universe. Man is a particularly interesting
question of this kind because Man intentionally effects physical change in a
way that is bounded but is not restricted, as such, by the physical laws that
govern inanimate matter or the laws that govern living biological matter.
Historically, very few people think. Thus, progress is
slow. Ideas about human nature impact human institutions which means that
oligarchical or control structures are threatened. This discourages mentation
on such ideas, further slowing progress. People generally take their signals
from empirics and can thus be corralled relatively easily, discouraging or
preventing mentation on ideas about our self-conscious selves and even more
strongly suspending their promulgation, implementation or even serious
consideration.
Thus, progress in the realm of final causes – which is
the immaterial or soul domain – is far slower than our progressive
understanding of “efficient causes”. Yet the realm of final causes is more
powerful or, at least, enormously more influential in Man’s impact on the
universe. Yet, it is certain that such ideas can be reasoned about. Plato’s The Republic is a starting point for a demonstration
of how. Leibniz explains that it is so in New
Essays and provides further guidance as to the “how” throughout his
writings.[615]
We now have established sufficient background to be
able to resolve and draw together a number of apparently conflicting threads in
Leibniz’s and Nicolaus of Cusa’s conceptions of the role of ideas and
possibilities for scientific discovery.
Leibniz is unambiguous that ideas are not real, in
that they have no metaphysical/ontological existence and certainly no physical
existence. However, ideas are a standard by which perfection is measured (at
least, sometimes) and, therefore, define components of - and totalities within
- the physical universe. To Leibniz, there is nothing “more perfect” about a
geometrical square than a roughly cut square paver in the physical world,
because the physical world is (a) real and (b) the best of all possible worlds.
The roughness of the paver is necessary, is a part of its beauty and that
beauty includes its role in the totality of the universe, and is a
characteristic of the unfolding upshifting of the universe - and the people in
it - to betterment.
It seems that there is an independent concept of “the
Best”. Or is it simply God-defined? If “Ideas” are given to us by the best
possible universe, they are given to us by “the Best” which is itself an idea.
Because we do not know what ideas are - or which concepts are ideas - unless we
see them in all the complexity of the universe, we need to define them (and the
Best) by the universe. God does not care about a perfect square’s ideality. He
cares about how to get the best possible universe, and a perfect square may or
may not be part of that.
This may be why Riemann suggested that physics (as a
study of the actual, physical universe) comes first and mathematics second. It
often happens that we find ideas in this Best universe. Thus, it is useful for
us to investigate ideas (or what we in our limited understanding think of or
perceive as ideas) so we have advanced ones “on the shelf” for when our studies
in physics need them. After all, the principle of plenitude says that
everything that is possible - is.[616]
Ultimately, we do not know whether what we think is an idea is an idea until it has been tested in
the universe or by checking against the universe (i.e. by experiment).
Theoretically, we could know without “testing” but that would require us to -
like God - hold the entire universe (and all of its possible unfoldings) in our
mind in a single thought. Similarly, as it is hard to the degree just described
which concepts are ideas, it is just as hard (perhaps by orders of magnitude)
to know what “the Best” is. Again, the complexity is such that it may be that
it can only be known by holding the entire universe in the mind at once. No doubt, however, there are principles
for the discovery of which we need to start forming hypotheses, as Leibniz has
done. They ought to be simply. Leibniz believed that the hypotheses can be
formed or worked out using his Universal Characteristic. Thus, it is possible
that it is not necessary to have to
be able to hold the entire universe and all its possible unfoldings in our mind
at once in a single thought in order to be able to determine what the Ideas are
or even what the Best is.
We are still left with the question, (a) does the Best
Universe define the Ideas (which includes the Best), or (b) is it vice-versa?
Because, from De Docta Ignorantia, we
will never equal God, assuming (a) is more correct. However, we necessarily
chip away with (b). The Universal Characteristic would provide a series of
leaps in the pursuit of (b). The Universal Characteristic itself would
necessarily have limitations or a “ceiling” per De Docta Ignorantia, and so would each success incarnation or
version of the Universal Characteristic.
We see in all this how tightly woven science
associated with experience of the universe as it is, (a), is with a priori thought and the methods of
Reason which include development of the Universal Characteristic, (b).
As a result of their perceptions, appetitions and
capacity for rational thought, human and other souls make decisions that impact
the physical universe. God plays an intermediating role, as shown in Figure 7.
Thus, the consideration of passive matter alone cannot give a complete – and
probably not even close to correct – understanding of the universe. The domain
of physics needs to be expanded to include the incorporeal, since the
incorporeal control the corporeal, as well as to include the connection between
the two since that is the only way it is possible to understand how the
incorporeal controls the corporeal. More work is needed to understand the
pre-established harmony, which is in the middle part of Figure 7, and on how
thoughts (i.e. the internal activities of soul monads) can attain more closely
to ideas which is the left node in Figure 7.
Two open questions present themselves. The first
relates to the pre-established harmony.
Is it possible to define the correspondence between
the soul or thoughts and the corresponding body monads as a mathematical
relation? Leibniz argues that God links thoughts with actions in the best
possible way. This leads us to the second. Can we discover a general definition
for “the Best” perhaps for improvement over time? Since the universe
corresponds with “the Best” possible, which is not always the same as what we
think of as ideals or mathematicals, can we formulate an overarching criterion,
or criteria, for the Best without having to hold the universe and all of its
possible unfoldings in our mind in a single thought like God can? We know from
Kepler that the reality of the universe is harmonious, beautiful and brilliant,
but those do not serve as working criteria nor were they intended to.
The ongoing re-creation of the universe as directed by
the thoughts of soul monads necessitates a design which places thinking soul
monads at the top of the universe. The Creator has given soul monads the
prerogative to create if they are able to be active (i.e. to think rationally,
in the strict Leibnizian sense) and choose to be. Indeed, even if soul monads
choose dullardism, they are by the nature of the role allotted them still the
ones to which God’s attention turns in the unfolding of creation via the
pre-established harmony. Thus, Leibniz vindicates the Renaissance view of Man.
Figure 7(a): pre-established harmony between a body and soul pair of monad counterparts
Figure 7(b): pre-established harmony at the collective level of all reality between all that is incorporeal (all active matter) and all that is corporeal (all passive or physical matter)
Leibniz sought to re-orient
and enhance the burgeoning program of experimental science. Leibniz wanted a
primary emphasis on structured thought with experiment playing a supporting
role to the reasoning process. Leibniz saw that in his lifetime, there was a
risk that experiment would become primary with reasoning relegated to a
secondary and supporting role. Indeed, some writers such as Sarpi and Newton as
discussed above sought to exclude reason, and substitute experimental evidence
or the testimony of the senses for the capabilities of the mind.
Wiener notes an oscillation
in Leibniz’s writings “between his a
priori system of irreducible real definitions and the experimental aspect
of his program”.[617] Leibniz
certainly gave attention to experimentation, even writing that he is in the
habit of writing out a catalogue of experiments to be done when he examines
some matter of physics. He will ensure that the list includes all experiments
needed to find the cause of what is in question “through demonstration and not
through Hypothesis”.[618] This
shows that Leibniz was not an a priori
fanatic, but – as in all other things – a pragmatist.
Nonetheless, a discussion of
the importance of a priorism in
science would be incomplete without giving examples of its use. Are there examples to which we can point
and say that a priori thought was
critical to a discovery or a process of discovery whose results had enduring
value? We will discuss three examples: Leibniz’s work in creating the calculus,
the discovery of non-Euclidean geometry and Kepler’s work in Mysterium Cosmographicum.
A priori thought is often equated with seeking to understand the mind of the
Creator with observed physical phenomena playing the positive role of acting as
a challenge influencing the direction of this quest by introducing new
questions or challenging existing answers.[619]
Therefore, the chapter ends with a brief section on some contemporary
references to the mind of God which are not in agreement with the Leibnizian a priori programme.
The obvious challenge is:
name one discovery that was made by observation alone. Discoveries are rarely made
by pure observation and, when they are, much more hard work that requires
thought as well as observation is required to give meaning to the initial
interesting observation. Contradictions might be found between observations.
Problems in accuracy and consistency are found with observations. Critical
information is found through observation, but observation does not give new
understanding. Typically, observations while sometimes answering questions
usually lead to more questions. Eratosthenes might have hypothesized the shape
of the earth from an observation. But the same observation had been made by
others many times before without Eratosthenes’ hypothesis springing to their
mind. Eratosthenes might then have used observations to determine the radius of
the earth, and thus answer a question, but in his mind he had already formed a
view as a result of observations. Differences in the lengths of shadows at the
same time of day exposed a contradiction which Eratosthenes attempted to
resolve by forming an hypothesis. Once he had done so, he was able to ask a
question which could then be answered by further observation, which was really
interpretation of experience through the lense of his understanding/hypothesis.
The calculus’ development
from Archimedes to Barrow to Leibniz involved little or no practical
experimentation. The same can be said of the development of Euclidean and
non-Euclidean geometry. However, the discoverers were affected by thinking that
resulted from experience. For example, Archimedes and Leibniz were engaged in
experimentation and thinking related to physics, which must have had some
impact on their treatment of the infinitesimal.
Clearly, Kepler’s
development of planetary motion and gravity was partly guided and partly set
off in new directions by empirical observation, as was Copernicus’ work. Their
development of ideas relating to properties of the universe may not have been
made had it not been for the development of precision in astronomical
observations in their time. To have discerned those ideas from principles
pertaining to how God works (“that the universe could not be other/better than
it is”) would have been too much of a leap for Copernicus’ and Kepler’s time,
and perhaps even for today. This does not mean that Kepler’s discoveries would,
in principle, have been impossible without the telescope. Leibniz’s point is
that the indistinctness and confusedness
of our reasoning leads to erroneous theory, and generally prevents us from learning
about the universe without at least some empirical information versus by way of
pure thought. Leibniz agrees that more precise observations can reveal errors
in theory. Kepler and Copernicus’s striving to correct the disagreements
between the Ptolemaic theory and increasingly precise astronomical observations
is nothing but an example of new empirical information revealing the
“confusedness and indistinctness” of our mental conceptions and reasoning.
Indeed, pure thought even with human imperfection and
our general dependence on observation maintains a power that observation cannot
have. Franklin’s paper, on two perspectives of Leibnizian optimism explains how we can, and
often do, work out mathematically – i.e. through pure thought – what is not
possible and thus what things we could never observe because they could never
occur.[620]
The paper notes that proving what must be
is a long way from proving what cannot be. Nonetheless, Franklin argues that
pure thought can be a very powerful thing and can provide results that no
experiment ever could: a truth that holds always and everywhere.
In the three domains of
discovery considered here, a priori
thinking dominated and was probably decisive. Most domains of discovery bear
fruit over several generations of thinkers who build on one another, so a
complete argument that a discovery resulted from a priori thinking requires consideration of those thinkers over
multiple generations and centuries. We do not have the space for a full study
of that kind.
Leibniz’s development of the
calculus involved much more than the infinitesimal though the infinitesimal is
a core concept. However, the great ancillary work did not involve empirical
observation though it did involve mental images and diagrams of curves to help
clarify ideas. Bagni says that the development of the calculus often assumed
“an aprioristic platonic epistemological perspective”.[621]
This perspective was not
limited to curves in the abstract, nor did it even begin with abstract curves.
In 1675, Leibniz generalized the calculation of centre of gravity of a figure,
and the calculation of the moment of a figure about any given line.[622] Leibniz
acknowledges that Cavalieri pioneered the use of the infinitesimal idea for
moments.[623]
Child points out that Cavalieri used the phrase “incrementum difforme gravitas”
to “connote a gradual increase that
follows a definite law” (emphasis in Child).[624] In 1679
in a Letter to Tschirnaus, Leibniz wrote, “Huygens, who thought me a better
geometer than I was, gave me to read the letters of Pascal, published under the
name of Dettonville; and from these I gathered the method of indivisibles and centres
of gravity, that is to say the well-known methods of Cavalieri and Guldinus.” [625] In July
1676, when investigating the “inverse method of tangents” (quadrature) Leibniz
wrote, “There is indeed another method that is more general and a priori, namely, by the intersection of
two tangents, which should always intersect between the two points at which
they touch the curve, as near one another as you can imagine”.[626]
Leibniz’s principle of
continuity was based on his conception of how God thinks, via the principle of
sufficient reason which is a consequence of the best of all possible worlds
principle. Of course, the infinitesimal is a corollary of continuity. We do not
know whether the mathematical conception of continuity or the metaphysical
principle of continuity came first, but it is difficult to believe that one did
not influence the other or perhaps the influence worked in both directions. Yet,
Cusa understood that, “As the mind considers it, the continuum is divided into
what is always further divisible and the number mounts to infinity. But in
actually dividing a line, one comes to a part that is actually indivisible. It
is this I call the atomic unit.” Yet Cusa knew that the atomic unit is
“nothing” for he quoted Boethius saying that, “If you add one point to another,
you effect no more than if you join nothing to nothing” and then wrote “if you
connect the ends of two lines, you do make a longer line, but the point of
connection constitutes no length at all”. The emphasis here is on how “the mind
considers it”.[627] It is on
this intellectual artifice that the infinitesimal is based. Katz and Sherry
argue that the artifice of the infinitesimal is not fatal to the
well-foundedness of the calculus. Indeed, they say that the artifice is
testament to the forwardness of Leibniz’s thinking, by saying, “Leibniz’s was a
remarkably modern insight that mathematical expressions need not have a
referent, empirical or otherwise, in order to be meaningful.”[628]
The presumption that
continuity has atomic components – which are metaphysical only – with which
humans can calculate is the stride that Leibniz takes with his calculus.
Indeed, the ratio Δy/Δx approaches a ratio of atomic lengths
which is called dy/dx. Had Cusa not
brought the idea of the infinitesimal out into the open, and discussed and
analysed it, future generations of thinkers might not have had the confidence
to use the “atomic unit” as Leibniz did. Without
Leibniz’s conception of the infinitesimal, there would have been no Leibnizian
calculus. Further, Cusa’s analogy of the quadrature of the circle is so central
to Leibniz’s work that Grosholtz refers to the “infinite-sided polygon”
perspective of the circle (for which Cusa is famous) as “Leibniz’s
infinite-sided polygon”.[629]
Leibniz’s a priori progress into the infinitely
small allowed him to develop a method of general applicability. Of course,
thinkers before Leibniz had ventured into the infinitely small. Child tells us
that Leibniz’s sole introduction was the infinitely small division between
ordinates in the method of Cavalieri c.1672.[630] From
there, Leibniz considered the infinitely small triangle which he called the
Characteristic Triangle. (Note the synonymous terminology with the sought-after
Universal Characteristic which Leibniz believed could be found as the basis for
a general method to solve any problem.) Leibniz knew that this triangle was
“indefinite” and so it was not possible to perform calculations on it directly.
However, “yet he perceived that it was always possible to find definite
triangles similar to it.”[631]
Incidentally, the use of of dy/dx
predominantly in reference to the Characteristic Triangle is one reason why
Richard Brown considers Leibniz’s calculus as fundamentally different from
Newton’s fluxions which assists Leibniz’s cause in the calculus priority
debate.[632]
We have just described a priori thought that is removed from
observation by two degrees of abstraction. First of all, the definite triangles
on which Leibniz performs calculations are themselves creations of the mind.
Second, he uses those calculations to make inferences about the indefinite
triangle about which he can presume to know very little directly.
The calculus is not
necessarily complete. New understanding of the infinitesimal domain might be
reached in the future causing the Leibnizian calculus to give way to a better
and more powerful calculus. For example, there may be a time in the future when
we find certain qualities of the indefinite triangle which need not be regarded
as true or which are provably false in the infinitesimal domain. That is, just
because an indefinite triangle is similar to a definite one, why must they necessarily
share certain qualities? The edifice on which the calculus is built may
collapse to give way to a more general calculus.
Leibniz reached a certain
point in his development of the calculus when the calculations were entirely
algebraic. This in itself was an achievement. However, the differentiability or
integrability of a curve then depended on the form of its algebraic equation.
As is standard for mathematicians today but was not in the 17th Century,
Leibniz considered equations of “infinite prolixity”[633] and
systematically showed that such equations did not pose a problem for his
calculus.
Leibniz proceeded to integrate
the powers of x and stopped at x3 for, as Weissenborn said,
“his soul is in the throes of creation”.[634] He
proceeds to derive the method of the integration of a square, such as f(x)2 and thence a particular
case of integration by parts xf(x).[635] Shortly
thereafter, Leibniz notes that integration has the effect of raising the
dimension of the thing integrated regardless of how large the dimension might
be.[636] For
example, integrating a line over an interval produces a 2-dimensional area,
integrating an area through some revolution produces a 3-dimensional volume,
and so on.
In 1675, Leibniz completes
working which he believes will enable him to find the quadrature (integral) of
any conic. Child notes that the effort comes to nought and, even if it had not,
he would have obtained a large quadratic whose roots would have been too
complicated to use.[637]
Nonetheless, in the process, Leibniz exhibits a number of qualities of a priori reasoning.[638]
Leibniz considers how y changes when x changes in arithmetical progression. This is an “artifice”
whereby he draws out details on the expression for the curve for the following
purpose of eliminating terms.
Given that Leibniz has
established himself in purely algebraic manipulation, he seeks “artifices”
which may simplify the working. Where there are more equations than there are
unknowns, solution is problematic. However, by taking a variable to be in
arithmetical progression, he is able to create additional equations and thus
eliminate an undesired unknown.
By employing an argument
that is partly geometrical and partly algebraic, Leibniz proceeds with an
alternate route under the heading of “moments”. That is, he is able to produce two
expressions for the same thing with one slightly different due to its being
resolved into two parts with the one being infinitely small compared with the
other.[639]
We can understand why Cauchy
complained that Leibniz’s infinitesimal lacks precision.[640] This is
because Leibniz uses “artifices” to trick his formulae into simplification so as
to “lay bare” their secrets.
In addressing claims that
indefinable real numbers cannot exist because they are not denumerable and
cannot be written down in decimal format, Franklin and Newstead argued that the
fact that we cannot see something directly does not mean that it is not there.
Leibniz has an unusual variation on this problem with his own critics. He took
the view that if we have forced something out into the open that we did not
know to be there, we can still use it even if we do not understand why it is
there.
Further, the more we appreciate
the “precision” which the limit conception (or δ/ε definition) of the
infinitesimal brings, the more it seems to be at the cost of distracting or
misdirecting us from the more exciting – and, probably, ultimately more
fruitful – hunt for what Leibniz was actually taming. There seems to be a wild
beast (or orchid) in the domain of ideas of which Leibniz had only begun to get
a smell.
Leibniz
defended and encouraged the use of “artifice” to permit calculation with the
continuous. For Leibniz, it was not necessary that his conception of the
infinitesimal actually exists because it is a useful tool. Why would he so
water it down, by disconnecting it from how he regarded the universe to
actually be? We suggest three possible reasons:
(a) He did not want society to
be denied the benefit of his calculus as a pragmatic aid to human development.
Further, he did not think that the Republic of Letters would reach agreement on
his metaphysical arguments on continuity any time soon.
(b) He knew that he was
expressing a property of metaphysical reality in precise terms, and was not so arrogant
as to think that such a profound idea could not be expressed with greater
fidelity than he had expressed it. In the meantime, he hoped his calculus to
nonetheless be able to enable a great contribution to human development.
(c) He intended his
infinitesimal calculus to be a tool of engineers and other “men of action” who
had no interest in metaphysics or theology.
Beyond Leibniz, it is by the
a priori method that Riemann
qualifies the validity of Leibniz’s infinitesimal triangles. Over II §2 and III
§1 Riemann makes it clear that infinitesimal triangles are only applicable in a
space of curvature zero and in which the angle sum of a triangle is two right
angles.[641]
The arguments surrounding
Euclid’s Fifth Postulate and tacit assumptions made by Euclid are rich with a priori reasoning with far-reaching
ramifications for geometry and for human conceptions of the structure of the
physical universe.
Euclid’s “postulates” were
in the original Greek merely “requests”.[642] Torretti
says that Euclid did not regard his “requests” as self-evident. Nonetheless,
Keyser wrote that Euclid’s Fifth Postulate (as it is called) was “perhaps the
most famous single utterance in the history of science”.[643] Euclid’s The Elements is often regarded as a
triumph of the ideal, or a testament to what the mind can create through pure
logic beginning with some basic postulates. Thus, Euclidean geometry might
suffice as a persuasive example of the power of a priori thought. It is perhaps not surprising that it should be found
wanting by a priori methods.
Nonetheless, because Euclidean geometry has been so important, we regard the
devising of non-Euclidean geometry as a legitimate example of a priori mathematical and – as we shall
see when we get to Riemann – scientific thought. The consequences of
non-Euclidean geometry are more significant for physics than its own discovery,
because non-Euclidean geometry allows physics to be done with mathematics.
The Fifth Postulate was
queried for many centuries before its true nature was found. For example, in
the 13th Century, Nasir al-Din (Eddin) al-Tusi, astronomer to Hulagu
Khan (brother of Kublai Khan and grandson of Genghis Khan) thought that it
could be proven by the first four postulates and therefore did not have to be
included as a postulate.[644],[645] Boyer and
Merzbach write that al-Tusi was the last of the Arabaic precursors to
non-Euclidean geometry, that John Wallis published al-Tusi’s work in the 17th
Century which was the starting point for the work of the Jesuit Girolamo
Saccheri in the early 18th Century.
Saccheri assumed that the two
summit angles of a quadrilateral are less than 90 degrees and tried to find a
contradiction. Due to his preconception of that the two summit angles had to be right angles, he “found” a
contradiction with his initial assumption that the upper angles are less than
90 degrees. This was published by Saccheri in a booklet entitled “Euclid
Cleared of Every Flaw”. Since Saccheri’s did not convince mathematicians, it enlivened
interest in the non-necessity of the fifth postulate even though this was not
the conclusion reached by Saccheri himself.[646]
A breakthrough came with Lobachevsky
and Bolyai simultaneously and independently. Lobachevsky motivated his
Pangeometry by the fact that the consequences of the Fifth Postulate “which,
although they appear simple, nevertheless, appear arbitrary, and consequently,
inadmissible.”[647] He begins
by defining a line by the locus of intersection of equal circles centred at two
fixed points, and a plane by the locus of intersection of equal spheres centred
at two fixed points.[648]
Lobachevsky makes it clear
that he regards his new non-Euclidean geometry, “Pangeometry”, as an a priori construction which is “a
complete geometric doctrine”.[649] It is of
interest because it provides “methods that are useful for the computation of
various geometric quantities.” He says that the idea that the angle sum of a
triangle is constant is “not a necessary consequence of our notions of space”
and then says “only experience can confirm the truth [or otherwise] of this
assumption”. So from something that is inadmissible because it appears
arbitrary, Lobachevsky has made a prediction that if we examine real
rectilinear triangles – especially large ones – we will find that their angle
sum varies.
Lobachevsky and Bolyai
independently developed non-Euclidean geometry at almost the same time.
Riemann, working in the post-Fifth Postulate environment, distinguished
“unboundedness and infinite extent”[650] which removes
the necessity for infinitude of the line and creates circumstances wherein the
Fifth Postulate fails.[651] Wolfe
pointed out that Euclid unconsciously assumed the infinitude of the line, which
is a weak point of the Fifth Postulate, in his proof of Proposition I.16.[652] This is a
case of the “‘hidden lemma’ or unacknowledged assumption” referred to by
Richard Brown, the uncovering of which is central to the dialectical
development of mathematics.[653] Joyce
points out that the first 15 propositions of Book I of The Elements hold in elliptic geometry but not Proposition 16,
saying “When a ‘straight line’ is extended, its ends eventually meet so that,
topologically, it becomes a circle. This is very different from Euclidean
geometry since here the ends of a line never meet when extended.”[654] These
considerations are all from thought alone.
It may be argued that the
realm of geometry is by its nature the realm of thought. However, geometry and
physics are intertwined. Let us consider someone relatively recent, namely,
Riemann. Riemann found the basis of geometry as inseparable from physics when
he said that if space is a discrete manifold, then its metric relations can be
discerned from within our own understanding of that discrete manifoldness. If
it is continuous, then we must understand the binding forces which act upon it
in order to understand its metric relations. These questions are asked a priori and they provide the setting
for the research programme to be undertaken in understanding the universe.[655]
There may be critical
experiments that can be designed to determine whether space is a discrete or a
continuous manifold. Such design would necessitate a priori investigation. Indeed, the work already done with Lie
groups and algebraic geometry in general can only be regarded as a priori. Once we have determined
whether space is a discrete or a continuous manifold or some third possibility,
then a priori speculations will need
to begin anew. In the end, we see that according to thinkers such as Riemann,
physics not only can but must be carried out just as a priori as can geometry. Physics since Riemann seems to have borne
this out, with most progress occurring in the mathematical domain punctuated by
experimental results, rather than vice-versa.
Riemann identifies a domain
of growing interest in which only a
priori speculation can lead to progress when he concludes in the
fourth-last paragraph that, “it seems that the empirical notions on which the
metrical determinations of space are founded, the notion of a solid body and of
a ray of light, cease to be valid for the infinitely small.” Thus, he says, we
are free to suppose the metric relations of space in the infinitely small do
not conform to the hypotheses of geometry. Moreover, he says, we should suppose this as soon as it
permits a simpler explanation of phenomena, which is nothing but a criterion
for the desirability of an a priori
speculation.[656]
Typical of the conclusion of
a priori work in physics from
Riemann’s view is the work of Archytas from the 4th Century BCE.[657],[658] The physical result of
doubling the cube arose from manipulating curves in a way that only the mind’s
eye could conceive and which is not subject to empirical observation (unless it
has already been constructed deliberately).
Wolfe points out that
intuition is not always reliable.[659] Thus,
Pasch and others introduced the Postulates of Order which may seem to be
obvious,[660]
but which do not always hold, and which Euclid assumed in proving Proposition
I.21. It would seem that such tacit assumptions of Euclid would be among the
difficulties that Riemann referred to when he said “the empirical notions on
which the metrical determinations of space are founded … cease to be valid for
the infinitely small.” Leibniz would agree that some geometry that holds for the infinitely small, for there must
be structure at all levels, but such a geometry is not something that would
correspond to intuitive conceptions resulting from our day-to-day experience.
We know what Leibniz says
about continuity: the principle of sufficient reason demands continuity always
in the universe, or no discontinuities. Dedekind provides what is generally
regarded as a satisfactory definition of continuity, but questions whether it
holds even in regard to an arbitrary straight line. He says that he cannot
prove it and nor can it be proved.[661]
Dedekind’s postulate of continuity is that if all points of a straight line
fall into two classes, such that every point of the first class lies to the
left of every point in the second class, then there is one and only one point
which produces this division of the line into two classes (or this severing of
the line into two portions).[662] For
example, one can imagine singularities in the domain of the infinitesimal in
which there is no concept of left and right, or in which one actually has a
single point which has nothing to its left and nothing to its right. This might
not be as fatal to geometry as it might seem, as Wolfe says that a large
portion of Euclidean and non-Euclidean geometry can be constructed without the
principle of continuity.[663] This
vindicates Riemann’s uncertainty as to whether space comprises a discrete or
continuous manifold.
Leibniz’s belief that the
universe must be continuous due to the principle of sufficient reason was based
on a priori reasoning regarding the
nature of God. Suppose it was found that in the realm of the infinitesimal –
which is precisely where most discussions about continuity take place – Dedekind’s
continuity criterion fails. Such an outcome would be rich with paradox, and
could itself only be reached via a priori
reasoning.
In discussing the
foundations of mathematics it could be expected that a priori thought is prominent. There is often an unavoidable segue
into metaphysics, philosophy and even theology to reach conclusions more sound
from those which were held previously. Yet the tools and presumptions of
physicists, engineers and even policymakers emerge from these foundations.
Thus, the processes followed in exploring and concluding in the foundations of
science have vast real-world consequences over centuries of human history.
Therefore, the role and method of a
priori thought is best explored, understood and refined rather than ignored
or cast aside in favour of putative pure empirical observation.
We can say that Kepler’s
wiping the slate clean of epicycles upon epicycles (i) was – arguably –
justified by “common sense” (which is easy to say in retrospect) but (ii) was
argued by Kepler as more consistent with the simplicity and beauty of how he
expects the Creator to work. With Kepler’s theological background and clear orientation
towards “understanding how God thinks” as being the purpose of science, we
would say that (ii) was more likely the critical component which persuaded
Kepler that there was something fundamentally wrong with the epicycles approach
and to “start again”. We doubt whether Kepler gave “common sense” any
importance at all.
This is diluted, certainly,
by the fact that for Kepler to add even more epicycles to refine the epicycles
theory to fit the observations a
posteriori was “getting a little bit ridiculous”. However, we should
hesitate to ascribe such vulgar late 20th Century tests on Kepler.
Kepler had to discard the
concept that the perfectness of circles makes them a necessary component of any
theory of how the heavens move. We add that Kepler was not just influenced by,
but was an intellectual disciple of, Nicolaus of Cusa. Thus, he had to have
been aware that no human presumption of truth (such as that planets must move
in circles) was necessarily permanent. Rather, more is to be gained by exploring
the domain of ignorance, since that domain is inevitable and infinite. The
question to ask would have been, “What if we’re wrong about circles being
necessary?” It is only by doing so that we may possibly come up with a better
theory. Cusa’s theory of docta ignorantia
is entirely couched in the eternal journey of the human soul towards the
understanding that God has and towards an understanding of God.
Kepler begins Chapter XI of Mysterium Cosmographicum, entitled “On
the arrangement of the solid, and the origin of the Zodiac”, with a paragraph
both characteristically Platonic and Pythagorean.[664] He
writes, “I have deduced the natural properties of the planets from immaterial
things and mathematical figures” and “I dare to investigate the origins of the
circles [of planetary orbits] from frankly imaginary cross sections.” Kepler
then justifies the view that God is a God that wills based on absolute reason,
and nothing exists except by his will. We will call this the Divine
Architectonic Principle (“DAP”). For
the scientist, the effect of DAP is that we must limit our enquiry to the
bounds of the inscrutable powers of the Founding Wisdom. In long hand, this
means that a scientist should work within the truth of the fact that God gave
effect to the mathematicals by using them as the foundation for the
architecture of the universe. Thereby, Kepler’s intention in Chapter XI is draw
“likely explanations” for the way universe is from “the quantities” meaning
numbers and which, by extension, means the Pythagorean corpus.
We must pair this with
Kepler’s earlier footnote that God, as well as a tongue, has a finger. We must
not attribute meaning to God’s words that contradict the works of God’s finger
as seen in the physical universe.
Notwithstanding Kepler’s
footnote 1 that the chapter could be omitted for it carries no weight, in
footnote 2, written apparently 25 years after he wrote Mysterium Cosmographicum, Kepler says that working within the DAP has
repaid him with interest over the last 25 years even though in this instance
(i.e., M.C. Chapter XI) it did not lead to a happy result. In particular, he is
talking about the chapter’s aim in demonstrating the arrangement of the zodiac
and numbers. He does not resile from his modus
operandi and the intellectual criteria for a useful argument that follow
from DAP.
For the scientist, the
Mathematicals are the cause of natural things and the reason for this is that
the Creator had the Mathematicals as archetypes with him from eternity in their
simplest divine state of abstraction even abstracted from numbers (when we
consider numbers in their material or empirical/intuitive aspect as quantities
of things). It may be that it is with this “shadow domain” that Leibniz was
grappling, as outlined under the previous heading “Infinitesimal”.
Kepler notes that Aristotle
carped at the theory that the mathematicals are the cause of things. Given that
Aristotle denied the existence of the archetypes, it is not surprising that he
denied the existence of a Creator and decided that the universe was eternal.[665] Kepler
admits that the archetypes would have possessed no force had God not had regard
to them in the act of Creation. This hearkens forward to Leibniz’s Theodicy (§184), where Leibniz writes, “It
is true that an atheist may be a geometrician: but if there were no God,
geometry would have no object. And without God, not only would there be nothing
existent, but there would be nothing possible.”[666]
It seems that Kepler is
saying that we find the expression of God in the universe through (and,
perhaps, only through) the subsistence of the archetypes in the
design/architecture of the universe. This reminds of Leibniz’s architectonic
conception of the universe.[667]
Ultimately, Kepler concludes
that it must be that “splendid and plainly necessary causes” for the
eccentricities of the orbits of the planets nearer the sun:[668]
Kepler sticks to this line
even though, as he says, in this chapter the application of the principle did
not have a happy result. As Leibniz would say, our perceptions are confused and
indistinct so the final arbiter on all of our a priori reasoning, however cogent it may be, is the physical
universe, i.e. the works of God’s finger.
Given that
the conception of the Creator’s mind was so important to Leibniz and Kepler
among other scientists in formulating their a priori conceptions, it is apt to
mention some contemporary views of the Creator’s mind.
Bruce Lipton
comes close to saying that a study of the universe is a study of a mind; we add
that the only mind that that could be is that of the Creator.[669] He writes that, “Physicists are being forced
to admit that the universe is a ‘mental’ construction.” Leibniz is clear that
the universe is not a mental construction in the sense of being an illusion,
and that the universe exists independently of any single mind including the
mind of God.
Richard Conn Henry writes that
the universe has a mental nature. However, he seems to be verging on Kantian
subjectivism. Elsewhere, he quotes Sir James Jeans, “The stream of knowledge is
heading toward a nonmechanical reality; the universe begins to look more like a
great thought than like a great machine. Mind no longer appears to be an
accidental intruder into the realm of matter … we ought rather hail it as the
creator and governor of the realm of matter.” However such snippets and
references to quantum physics as Henry indulges in remind of the
quasi-scientific essays of the now-discredited postmodern milieu.[670]
While Henry has only quoted
Jeans in brief, and so we cannot be sure of Jeans’ full meaning, we would not
agree that the universe is merely a thought. Leibniz made it clear that the
universe is real and is separate from God. It is true that God can create with
merely an act of will which is arguably tantamount to a mental act, but in
doing so God creates something that is separate and independent of his mind.
Superficially Henry appears to
agree with our contentions regarding Leibniz’s conception of a priorism and the power of a priorism, because by understanding
God’s mind before or without observation we can understand physics.
Nonetheless, for the reasons explained, we distance ourselves from Henry’s
position.
We have shown that a priori thought played at least a key
enabling role in three examples, namely, in the development of the calculus,
non-Euclidean geometry and the Keplerian theory of the solar system. By
becoming more aware of the a priori
foundations of other scientific domains and of science today in general, the
study of a priori foundations may
receive a fillip which will allow for greater benefit to be derived from
experimental efforts. Equally importantly, a new array of theories may arise
from contradictory or unexplained experimental data already collected.
In this short final chapter, we recap on Leibniz’s
historical role and describe Leibniz’s agenda referred to in the thesis title.
Leibniz continued an ancient “agenda” certainly of the Neoplatonists but also
pre-dating the Neoplatonists to the Egypt of Hermes Trismegistus and perhaps
before that. In Chapter 3, it was explained how Leibniz helped usher in
modernity. Chapter 4 through 9 explored how Leibniz helped define, direct and
fight for the kind of thinking – and the kind of thinking about thinking – that the onset modernity relied upon. For
modernity to shift upwards rather than collapse, it would be well to maintain
Leibniz’s methods and orientation, and undertake some of Leibniz’s projects which
have been embraced in a serious way by only a few, such as the Universal
Characteristic.
To compare Leibniz’s view of Man with that of other thinkers,
such as Kant, is beyond the scope of this thesis. However, an account of Leibniz’s
metaphysics and the underpinnings of his approach to science requires mention of his
conception of Man’s nature and Man’s role in the universe. In this, Leibniz was
in the humanist tradition of Dante and distinguished from the quasi-mystical,
albeit at the same time somewhat Christian humanist, attitude of Leibniz’s British
contemporaries such as Boyle. Leibniz might not have said like Dante that
humanity is the jewel of Creation or the highest part of Creation, but he did
say that humanity and humans can act like the Divine in the miniature. This is
not far from saying that humanity has a special place in Creation, for Leibniz
did not say that any other kind of creature can act like the divine in
miniature.
It must be asked what it means that Man is able to
reason in the way that he can. Kepler put this down to the uniqueness of Man in
all Creation and the special role that God prescribed for Man by giving Man the
power of Reason. If nothing else, it can be concluded that there is something
special about Man. From here, it is easy to enter the Dantean Humanist
Renaissance. Leibniz was the kind of humanist that Pico della Mirandola and
Giordano Bruno were, who regarded Man as magus who can arrogate and wield great
power and effect vast change in the universe like a race of gods. As mentioned
in this thesis, this is aligned with the Hermetic conception of humankind.
The intention to benefit humankind
is implicit in Leibniz’s descriptions of, and optimistic hopes for, human
knowledge and in his idea of progress. For Leibniz, progress in itself and
progress for humankind were synonymous. Leibniz wrote and promoted designs for
social organisation to promote improvements in culture and knowledge among the general population, believing that the
best of human attainments were intellectually and morally accessible to all
people.
Leibniz was at a historical juncture which he helped
create. He was a classicist and a medievalist himself. He was a conciliatory
eclectic thanks to Thomasius as Leibniz’s teacher and also thanks to the
efforts of Thomasius to ensure a lasting peace after the Thirty Years War.
Leibniz’s optimism is
inherent in his best of all possible worlds doctrine. This doctrine is the
logically necessary and otherwise natural outcome of Leibniz’s metaphysics and
its interplay with his theology. In many ways, Leibniz’s metaphysics is
contained in or is at least derivable from his theology.
Leibniz was a universalist
and “Renaissance man” who helped create modernity with all of the power of
modern rigor. No longer was it Cusa arguing against the Aristotelians on
metaphysical grounds. Rather, it was a Renaissance-type humanist arguing
against the neo-Aristotelians, Averroists and Empiricists on many grounds other
than metaphysical. Leibniz combined the power of the Platonic dialogue with the
scientific success of Kepler and Galileo,
and with his own and his associates’ achievements in mathematics, invention,
machinery, metaphysics and physics.
In Chapter 9, it was shown that working through the steps
of discovery for fundamental questions of theology and metaphysics can affect
where we end up in the natural sciences. Conversely, as Riemann indicated and as
Leibniz allowed, empirical discoveries can indicate where our a priori reasoning went wrong. To
address the basic questions in metaphysics while conducting work in the natural
sciences was always the approach of the Neoplatonists. That background work
forms the largest context for natural science. Our answers to the “big” a priori questions can be tested with
our work in the natural sciences. That background work, the “big” a priori questions, must include the
nature of humankind and the role of humankind. This is because in a universe
resulting from the Creator, minds with the capability to address such questions
first of all cannot be accidental and have a purpose different from that of,
say, the nearest passing meteor.[671]
The nature of the universe cannot be understood
without including the role of humans as physical force and as a uniquely
dynamic law of nature. Humanity’s capability and potential to exercise its
power as collective magus or as a wilful physical force must be taken into
account. Related endeavours such as investigations in mathematics are aided or
hindered by the scaffolding provided by the culture as much as anything else,
which includes the language, art, literature, music as well as philosophy, mathematics
(of whatever kind) and science of the age, which creates conditions for
particular kinds of thought to flourish. How does a culture develop conditions
conducive to particular kinds of thought and creativity? We can say that a
culture’s preoccupation with a particular class or kind of Platonic ideas,
absent interruptions caused by war, deliberate sabotage or other factors, may
lead to a raft of discoveries in that domain over successive generations. Such
discoveries can feed into economic, military and cultural breakthroughs
creating prosperity for the civilisation or culture concerned. The political
framework and conception of humanity that creates social fertile conditions for
any such flourishing of thought was of great interest to Leibniz. He promoted
related ideas through his circle of influence.
Leibniz to our knowledge did
not use Cusa’s term docta ignorantia
“learned ignorance”. He was optimistic about what was already known and not
written down or organised. He was more optimistic still, and even excited,
about what humans would come to know in the future. He sometimes seemed a
little daunted by the extent of what humanity does not know. Even though
Leibniz was aware of the rigor and power of mathematical tools, he did not
believe that everything could be explained or subsumed by mathematical methods.
In responding to Descartes’ Principia
Philosophie Part I point-by-point in 1692, Leibniz wrote about the horizons
of human knowledge:[672]
On Article 26. Even though we are finite,
we can yet know many things about the infinite: for example, about asymptotic
lines, or lines which approach each other continuously when infinitely produced
but never meet; about spaces which are infinite in length but not greater in
area than a given finite space; and about the sums of infinite series.
Otherwise we should also know nothing with certainty about God. However, it is
one thing to know something about a matter and another to comprehend the
matter, that is, to have within our power all that is hidden in the matter.
On what humans can know and what humans
should investigate:
On Article 28. As for the end God has
proposed to himself, I am fully convinced both that they can be known and that
it is of the highest value to investigate them; and that to disdain this
inquiry is not without danger or suspicion. In general, whenever we see
anything that is particularly useful, we may safely assert that one, among
others, of the ends which God has proposed to himself in creating this thing is
precisely to render these services, since he both knew and planned this use of
it. I have elsewhere pointed out, and shown by examples, that certain concealed
physical truths of great importance can be discovered by considering final
causes, which not have been discovered as easily by efficient causes.
Considering final causes is tantamount to pursuing an a priori program. Article 28 also states
that no phenomenon in the universe is accidental, and the deliberate causes of
things are knowable by humans.
We referred to Loemker in Chapter 3 who said that
Leibniz had hoped his metaphysics “would be adopted and made a blueprint, so to
speak, by men of good will (honestas) for the restoration of European
order”.[673] Counterposed
to Leibniz was an “alternative interpretation of human nature, human thought,
and the good”. Loemker places this under the subheading “The New Way of Ideas:
Locke” and that new way is based on “another theory of ideas – that of
nominalism and positivism”.[674]
Mercer’s thesis on Leibniz makes much of the concept
of “conciliatory eclecticism”.[675] To be sure, much of Leibniz’s work and thought can be
regarded in this way. Raynaud begins his commentary by saying, “In comparison
to Descartes, or Hobbes, or Locke, Leibniz presents himself as a moderate or a
conciliator: he means to rescue a portion of the heritage of Plato and
Aristotle, he rejects Hobbes’s radical nominalism as well as his legal
positivism, and he breaks with Cartesian mechanism in order to make a
significant place for natural teleology.”[676]
In Chapter 6 under the heading “Science may use
indemonstrables”, Raynaud’s conclusions were given regarding Leibniz’s
pragmatism. For Leibniz, methodological doubt was not a reason to forego deny
useful benefits derived from particular lines of enquiry.[677]
Further, the requirement to answer a priori questions before conducting was
also not absolute though it can lend great power. “For,” Raynaud says, “if
science may employ indemonstrables, that also means that the initial absence of
such principles does not impede one from progressing on the path of reason,
without having to ‘cast into doubt’ common beliefs.”[678]
For Leibniz, there are many useful and legitimate paths. For Raynaud, Leibniz
“establishes a new continuity between science and the active life, between
knowledge and practical judgement, and between reason and faith.” [679]
Benefit to humankind is but a
self-evidently good side-effect of the seeking after the kingdom of God. We
will end with Leibniz’s words in which he reiterates Kepler’s advocacy of
seeking scientific insight to improve ourselves but, unlike Kepler, he also
recognises the pragmatic benefits to the day-to-day lives of humans made
possible by scientific endeavour:[680],[681]
I have shown on several occasions that the
final analysis of the laws of nature leads us to the most sublime principles of
order and perfection, which indicate that the universe is the effect of a
universal intelligent power.[682]
As the ancients already held, this truth is the chief fruit of our investigations;
without mentioning Pythagoras and Plato, whose primary aim was such an
analysis, even Aristotle sought to demonstrate a prime mover in his works,
particularly in his Metaphysics. It
is true that these ancient thinkers were not informed about the laws of nature
as are we since they lacked many of the methods which we have and of which we
ought to take advantage. The knowledge of nature gives birth to the arts, it
gives us many means of conserving life, and it even provides us with conveniences;
but the satisfaction of spirit which comes from wisdom and virtue, in addition
to being the greatest ornament of life, raises us to what is eternal, whereas
this life, in contrast, is most brief. As a result, whatever serves to
establish maxims which locate happiness in virtue and show that everything
follows the principle of perfection is infinitely more useful to man, and even
to the state, than all that serves the arts. Discoveries useful to life,
moreover, are very often merely the corollaries of more important insights; it
is true here too that those who seek the kingdom of God find the rest on their
way.
With Leibniz, the humanist conception of Man as magus
and as the jewel of Creation was crystallised in pragmatic policy and, in
particular, in national government policy. Leibniz was born in 1647 the year
before the Treaty of Westphalia. This treaty enshrined the doctrine of national
sovereignty in the law of nations, and established the modern nation state
system, which today is known as the Westphalian system.
Leibniz was partly a product of the optimism inevitably
brought by the end of the Thirty Years War (1618-1648) as well as of the
cultural optimism of his milieu. For example, Bach lived and worked in Leipzig
at the same time as Leibniz though it is not known whether they knew each
other. The possibilities of the new era were being conceived during Leibniz’s
lifetime and Leibniz helped conceive them. Leibniz and Denis Papin were early
recruits into the Academy of Sciences in Paris under its inaugural leader
Christian Huygens. The Academy was charged by Louis XIV with developing new
power sources for France. Thus, it was a scientific academy with a national and
industrial purpose. From the start, Leibniz was forced to grapple with the idea
of power and of increasing Man’s physical power. He accepted the challenge.
Leibniz came to understand partly by exposure to Huygen’s gunpowder experiments
that controlled explosive force had a higher order of potential than the
passive force emphasized by the ancients. Perhaps inspired by the mission of
the French academy, Leibniz later advised Tsar Peter the Great to establish the
Russian Academy of Sciences. Tsar Peter agreed and invited Leibniz to be its
first president, though Leibniz declined that offer partly due to his existing
responsibilities and commitments, and – possibly – partly because Russia was
regarded as a dangerous place. In any case, Leibniz’s circle of influence
included many nations other than France and Russia.
In the Westphalian era, the idea of Man as magus was
coming into its own, and was being enshrined in forward-thinking policy by
European and Eurasian leaders partly under Leibniz’s influence. The concept of
Man as magus emerged from Neoplatonism. It was joined at the hip with the
optimistic conception of a plentiful and vast physical universe, in which Man
has a decisive and central role. Arguably, it was for promoting such plenitude
and optimism against the established fixedness of prevailing policy that
Giordano Bruno was burned at the stake. Far from being mystical, vague or
otherwise divorced from reality, the Neoplatonic orientation was a living force
which demanded that humanity adopt the scientific stance and take its rightful
place as commander of itself and of the universe. Leibniz’s legacy survives
today in the policies of scientific-industrial nation state. A month before the
closing words of this thesis were written, the NASA rover Curiosity landed on
Mars. Leibniz would be pleased, but much remains to be done.
The entirety of what follows in this Appendix is a
quote from an anonymous circular which we have reason to believe was written by
Leibniz. It is referred to in Chapter 3 of this thesis.
Ad majorem Dei
gloriam [“to the greater
glory of God”][683]
Profound
meditation is urged upon researchers concerning the propulsive power of
gunpowder; and their perceptive intelligence is challenged to realize the possibility
of diverting the vast power of gunpowder to healthier applications than those
hitherto known. Only irreverent thinking can deny such an incontrovertible
truth as that in the sight of God only wholesome applications and uses
appertain to all that is created and can be manufactured therefrom; for
everything exists for the benefit of mankind, if only they had a serious desire
for it. Nevertheless it is only too well known that the corrupted and misled
spirit of man is concerned in countless cases not with salutary application,
but allows itself to become passionately concerned with applying all its acumen
to discovering how to misuse things, which according to the intention of the
Creator should be useful to mankind, so that they should have every cause to
praise him on their account.
Such
misuse had induced some men to describe the discoverer of gunpowder simply as a
sorcerer in monk’s garb instructed by satan;
for owing to its violent power it seems impossible to achieve aught but
explosions, loss of life and destruction. Similarly doubtless in the immemorial
past, people thought of the propulsive power of flowing water and of wind,
before they had been made serviceable for useful purposes for mankind by wise
and industrious mechanical craftsmen who used first simple wheels and later
toothed wheels. The above mentioned view of the satanic origin of gunpowder
should therefore be set aside and replaced by the following:
1.
The inventor of gunpowder, whoever he was, was a capable chemist.
2.
The skilful achievements of chemistry are hated neither by God nor by Nature,
nor are they directed against God’s will; since they
can instantly convert active poisons into blessed and healing potions.
3.
It is possible by means of some form of control to force the aforesaid
propulsive power of gunpowder, however sudden and violent it may now be, into
ordered channels, so that it should be adaptable depending on the arrangements
made for driving an ordinary mill or for performing other work; and this aim
may be obtained if earnest prayer for divine support is combined with
enthusiastic pyro-mechanical labours, and if mind and hand are ceaselessly
busied with this work; and above all if the aforementioned fundamental
demonstration [of a pious use of the gunpowder and the praise of the Almighty
Creator are kept in mind, - rather than its direct tangible use, which divine
Providence will in its wisdom confer on the present century or on future ones.
More
than two and a half years have elapsed since a number of researchers were
publicly charged with the said problem concerning gunpowder, namely that it
could and should be used for other purposes than hitherto. And the researchers
should thereby be urged to the bold attempt gradually to give up their
enthusiastic researches which were placing the misapplied use of the violent
power of gunpowder . . . in a very favourable light. And at the same time they
were to devote a part of their efforts, for the glory of the Creator, to
discovering a new use for gunpowder, which had been latent in it from the
beginning but had gone unheeded, because all who have hitherto occupied
themselves with its application have stood under the spell of that terrible
misconception that gunpowder was useful only for an idle and wasteful display
of its flashes and flying sparks, or else to wound, to kill, to explode, to
burst, to ruin, in short to unhinge the whole world. It is but too easy to
suppose that our researchers, under the influence of this prejudice, have given
either no thought or not sufficiently serious thought to a useful application
of gunpowder, especially as no prospect of an important or obvious application
attracted them in this direction. Only one man has been found who, with a view
to a possible advantageous application, has freed himself from the above
mentioned preconceived opinion … and has honoured
the attempt with his attention, in a French letter of the 24 May 1686.
‘I have received the problem
communicated by you concerning a new application of gunpowder. In my opinion it
may certainly be hoped to attain this end. Seven or eight years ago I showed to
Monsieur Colbert an engine which I had had built for this very purpose, and
which was illustrated in the Proceedings of our Academy.
‘It worked as follows; a tiny
quantity of gunpowder, about a thimbleful, was able to raise some 1,600 lb.
five feet high; and not with such violence as is usual, but with moderate and
steady power. Four or five servants, whom Monsieur Colbert ordered to pull the
rope attached to the engine, were quite easily lifted up into the air.
Nevertheless there was a certain difficulty in constantly reproducing this
Power.’
The
writer of this letter communicated two quite unusual and really incomparable
discoveries:
1.
One or two drachmas of gunpowder, a thimbleful, will raise a weight of 1,600
lb. five feet; and furthermore,
2.
This occurs without the usual violence, but with moderate and steady power.
The
first of these discoveries arouses admiration, but is in conformity with
principles already accepted. The effectiveness of the powder could naturally be
increased either by addition of more powder or by improvement of the piston.
Nevertheless the second discovery would appear to extend beyond these known
principles, and must be the more highly valued since it approaches the
miraculous. Doubtless therefore all those who are plagued by curiosity to see
this simple and really useful experiment could take the trouble to make such a
machine or another which is suitable to impel any weight chosen as desired. They
should allow themselves to be helped therein by men who are familiar not only
with the use and misuse of gunpowder but also with the art of mechanics.
Especially they will need for this, the magnificent work of the late Monsieur
Bondel The Art of Shooting with iron balls filled with powder, a work
that would perhaps be more correctly entitled “The Art of Thoroughly
Understanding the Nature and Characteristics of Natural and Violent Motion”. In
it will be found many demonstrations directed toward the aim here mentioned.
And there can no longer be any doubt that the fact that a tiny quantity of
powder can raise 1,600 lb. so high, can, some day, be put to general use as
soon as an inventor turns his attention to solving the many difficulties,
especially those which obstruct the repetition of regular action. It remains
only to add that at this point besides the description and exhibition, a
drawing of the machine itself could very easily have been shown, by means of
which its manner of working could have been shown; did not the ease with which
this can be manifested and understood seem to make this quite superfluous,
especially since its effectiveness has already been more than sufficiently
demonstrated. Moreover, gifted investigators have had sufficient reason to
believe positively that the making of such experiments, all too rare, each of
which needs special consideration, may ultimately lead to a fruitful
contribution useful to everyone.
Meantime,
the decision rests with God alone. He will according to his merciful judgment
at the right time make it evident that all creation is appointed for the
welfare and service of mankind. It is therefore the duty of man not only to
believe this truth, but to work with all his power that he may use and enjoy
everything with acknowledgement and gratitude. Praised therefore be the most
holy name of Him through whose goodness the first stage of this apparent
impossibility (namely the useful application of gunpowder) has been overcome;
Praised, say I, be his name to all eternity! Amen.
The entirety of what follows in this Appendix is a
quote from a letter by Leibniz.
Tentamen anagogicum
[“an essay proceeding from the
basics to prove something firm”][684]
The inquiry into final causes in physics
is precisely the application of the method which I think ought to be used, and
those who have sought to banish it from their philosophy have not adequately
considered its usefulness. For I do not wish to do them the injury of thinking
that they have evil designs in doing this. Others followed them, however, who
have abused their position, and who, not content with excluding final causes
from physics but restoring them elsewhere, have tried to destroy them entirely
and to show that the Creator of the universe is most powerful, indeed, but
without any intelligence. There have been still others who have not admitted
any universal cause, like the ancients who recognized nothing in the universe
but a concourse of corpuscles. This seems plausible to those minds in whom the
imaginative faculty predominates, because they believe that they need to use
only mathematical principles, without having any need either for metaphysical
principles, which they treat as illusory, or for principles of the good, which
they reduce to human morals; as if perfection and the good were only a
particular result of our thinking and not to be found in universal nature.
I recognize that it is rather easy to
fall into this error, especially when one’s thinking stops at what imagination
alone can supply, namely, at magnitudes and figures and their modifications.
But when one pushes forward his inquiry after reasons, it is found that the
laws of motion cannot be explained through purely geometric principles or by
imagination alone. This is also why some very able philosophers of our day have
held that the laws of motion are purely arbitrary. They are right in this if
they take arbitrary to mean coming from choice and not from geometric
necessity, but it is wrong to extend this concept to mean that laws are
entirely indifferent, since it can be shown that they originate in the wisdom
of their Author or in the principle of greatest perfection, which has led to
their choice.
This consideration gives us the true
middle term that is needed for satisfying truth as well as piety. We know that
while there have been, on the one hand, able philosophers who recognized
nothing except what is material in the universe, there are, on the other hand,
learned and zealous theologians who, shocked at the corpuscular philosophy of
and not content with checking its misuse, have felt obliged to maintain that
there are phenomena in nature which cannot be explained by mechanical
principles; as for example, light, weight, and elastic force. But since they do
not reason with exactness in this matter, and it is easy for the corpuscular
philosophers to reply to them, they injure religion in trying to render it a
service, for they merely confirm those in their error who recognize only
material principles. The true middle term for satisfying both truth and piety
is this: all natural phenomena could be explained mechanically if we understood
them well enough, but the principles of mechanics themselves cannot be
explained geometrically, since they depend on more sublime principles which
show the wisdom of the Author in the order and perfection of his work.
The most beautiful thing about this
view seems to me to be that the principle of perfection is not limited to the
general but descends also to the particulars of things and of phenomena and
that in this respect it closely resembles the method of optimal forms, that is
to say, of forms which provide a maximum or minimum, as the case may be - a
method which I have introduced into geometry in addition to the ancient method of
maximal and minimal quantities. For in these forms or figures the optimum is
found not only in the whole but also in each part, and it would not even
suffice in the whole without this. For example, if in the case of the curve of
shortest descent between two given points, we choose any two points on this
curve at will, the part of the line intercepted between them is also
necessarily the line of shortest descent with regard to them. It is in this way
that the smallest parts of the universe are ruled in accordance with the order
of greatest perfection; otherwise the whole would not be so ruled. It is for
this reason that I usually say that there are, so to speak, two kingdoms even
in corporeal nature, which interpenetrate without confusing or interfering with
each other - the realm of power, according to which everything can be explained
mechanically by efficient causes when we have sufficiently penetrated into its
interior, and the realm of wisdom, according to which everything can be
explained architectonically, so to speak, or by final causes when we understand
its ways sufficiently. In this sense one can say with Lucretius not only that
animals see because they have eyes but also that eyes have been given them in
order to see, though I know that some people, in order the better to pass as
free thinkers, admit only the former. Those who enter into the details of
natural machines, however, must have need of a strong bias to resist the
attractions of their beauty. Even Galen, after learning something about the
function of the parts of animals, was so stirred with admiration that he held
that to explain them was essentially to sing hymns to the honor of divinity. I
have often wished that an able physicist would undertake to prepare a special
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[1] Loemker, p.442. Liberty has been taken to break the passage into two paragraphs.
[2] Petrarca relates in an entertaining way how his differences with an Averroist who came to visit him, with emphasis on the gratuitous disrespect shown by the visitor to the Scriptures and Apostles. Petrarca “An Averroist visits Petrarca” From a letter to Boccaccio from Venice 28 August 1364, pp.140-141 in Cassirer, E., Kristeller, P. O., Randall, J. H. Jr (eds and trans.) The Renaissance Philosophy of Man Phoenix Books, University of Chicago Press 1948
[3] Van Inwagen, P. “Metaphysics” 2007 in Stanford Encyclopedia of Philosophy http://plato.stanford.edu/entries/metaphysics/ accessed 21 Feb 2012
[4] Petrarca denigrates dialectic saying it is appropriate for students but is immature and silly in a philosopher of age and experience. Petrarca “A disapproval of an unreasonable use of the discipline of dialectic” Letter to Tommaso Caloria from Avignon 12 March 1335, pp.134-139 in Cassirer, E., Kristeller, P. O., Randall, J. H. Jr (eds, trans.) The Renaissance Philosophy of Man Phoenix Books, University of Chicago Press 1948, p.139
[5] The historical importance of Nicolaus of Cusa cannot be understated. We quote the closing paragraph of the introduction to a biography of Nicolaus of Cusa, “In the act of coming to the aid of Pius II and his crusade, Nicholas of Cusa, the cardinal and longtime confidant of the pope, died three days before him on the road to Ancona. It was August 11. By then Cusanus had succeeded in participating in, if not actually shaping, nearly every major issue of the more than half-century that life had given him [1401-1464]. In addition to his career as an energetic churchman, he also found time to leave a remarkable intellectual legacy that fascinates an unusually broad range of people to this day, from astronomers and mathematicians to church historians, from political theorists to theologians and philosophers.” Crowner, D., Christianson, G. (trans. and eds) Meuthen, E. Nicholas of Cusa: A Sketch for a Biography The Catholic University of America Press, Washington, D.C. 2010 pp.xxv-xxvi. Accessed at http://cuapress.cua.edu/books/frontmatter/MENC.pdf 26 May 2011
[6] It is
said that Isaac Newton promoted this approach. However, the liveliness of the
Neoplatonic environment in which Newton studied at Cambridge, with people like
Boyle around, and the evident mysticism in some of Newton’s own writings cast
doubt on how “Newtonian” he was himself.
[7] Kielkopf, C.F. Strict Finitism: An examination of Ludwig Wittgenstein’s remarks on the foundations of mathematics Mouton, The Hague and Paris 1970, p.32
[8] Leibniz, G. W. “Body is not a substance” March 1689-March 1690 (?) Sämtliche schriften und briefe series VI volume 4 Deutsche Akademie der Wissenschaften (ed) p 1637 Strickland, L. (trans.) Leibniz Translations 2009 accessed at http://www.leibniz-translations.com/bodysubstance.htm 6 June 2012
[9] Leibniz, G.W. “On the method of distinguishing real from imaginary phenomena” Date unknown in Loemker,, p.364
[10] Leibniz, G. W. New Essays on Human Understanding in Ladd, G. T. (trans.) The Philosophical Works of Leibnitz Tuttle,
Morehouse and Taylor Publishers, New Haven 1890, p.96
[11] Yates 2009, pp.178-181
[12] Petrarca “On his own ignorance and that of many others” c.1368, pp.47-133 in Cassirer, E., Kristeller, P. O., Randall, J. H. Jr (eds and trans.) The Renaissance Philosophy of Man Phoenix Books, University of Chicago Press 1948
[13] Brand, P. Cambridge History of Italian Literature Cambridge University Press 1999, p.134. Cusa continues Petrarca’s programme of demonstrating that intellect as exercised by an intelligent commoner can be more effective than the formal reasoning promoted by the scholastics.
[14] Yates 1964, pp.146-147
[15] This is Edward Gibbon’s date which is conventionally accepted.
[16] Accessed at http://www.goddess-athena.org/Encyclopedia/Friends/Iamblichus/index.htm on 5 April 2012
[17] Hicks, R.D. (ed.) Diogenes Laertius Lives of Eminent Philosophers IV 1 accessed at http://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.01.0258%3Abook%3D4%3Achapter%3D1 20 April 2012
[18] Accessed at http://www.wisdomworld.org/setting/hypatia.html on 5 April 2012
[19] Ibid.
[20] Inge, W.R. Christian Mysticism Plain Label Books, Chumley P. Grumley (series ed.) 1956 pp.153-154 based on The Bampton Lectures 1899 delivered before the University of Oxford
[21] Accessed at http://philosophos.hubpages.com/hub/neoplatonism on 7 April 2012
[22] Wisnovsky, R. Avicenna’s Metaphysics in Context Cornell University Press 2003, p.64
[23] Goodman, L.E. Avicenna Cornell University Press 2005, p.56
[24] Harrison-Barbet, A. “Avicenna” in Philosophical Connections: Islamic Neoplatonism accessed at http://www.philosophos.com/philosophical_connections/profile_036.html 20 April 2012
[25] Zabeeh, F. (ed. and trans.) Avicenna’s Treatise on Logic Martinus Nijhoff, The Hague 1971. For example, the definition of “proposition” and the meaning of “indefinite” pp.20-21 nn.6-10, and p.23 n12
[26] Supra nn.23-25
[27] Bergh, S. van den Averroes’ Tahafut Al-Tahafut (The Incoherence of the Incoherence) Trustees of the E. J. W. Gibb Memorial, London http://www.muslimphilosophy.com/ir/tt/index.html 7 April 2012. See the introduction.
[28] Kilcullen, R.J. Lectures for PHIL252 Medieval Philosophy, Tape 7 Al Ghazali and Averroes Macquarie University, accessed at http://www.humanities.mq.edu.au/Ockham/x52t07.html 7 April 2012. See Kilcullen’s full list of lectures here http://www.humanities.mq.edu.au/Ockham/kilcullen.html accessed 7 April 2012
[29] Yates 1964, p.49
[30] Ibid.
[31] Ibid., p.47
[32] Láng, B. Manuscripts of Learned Magic in the Medieval Libraries of Central Europe Magic in History Series, The Pennsylvnia State University Press, 2008 pp.33, 35
[33] Hourani, G.F. (ed,. trans.) On the harmony of religion and philosophy : a translation, with intro. and notes, of Ibn Rushd's Kitab fasl al-maqal, with its appendix (Damima) and an extract from Kitab al-Kashf 'an manahij al-adilla Luzac London 1961, p.14
[34] Ibid., pp.12-19
[35] Ibid., p.7
[36] Accessed at http://history.hanover.edu/texts/presoc/emp.htm and http://plato.stanford.edu/entries/empedocles/ 7 April 2012
[37] Hourani, G.F. (ed,. trans.) On the harmony of religion and philosophy Luzac London 1961, p.7
[38] Accessed at http://faculty-staff.ou.edu/V/David.R.Vishanoff-1/I-terms/Malikites.htm 7 April 2012
[39] Hourani, G.F. (ed,. trans.) On the harmony of religion and philosophy Luzac London 1961, pp.8-9
[40] A readable and apparently thorough summary of Al Ghazali’s works is at this page http://www.ghazali.org/articles/gz1.htm accessed 7 April 2012
[41] McInerney, R. Aquinas against the Averroists: on there being only one intellect West Lafayette, Purdue University Press 1993, pp.71-99
[42] Ibid., introduction p.ix
[43] Martin, C. 2007 pp.6, 18-19
[44] Barrett, F. The Magus Book III: Biographia Antiqua London 1801 pp.150-151 Accessed at http://www.sacred-texts.com/grim/magus/ma255.htm 9 April 2012. However, as the Sacred Texts website itself notes, “The biographical section has been deprecated by authorities such as Waite, and it's not even certain that it was written by Barrett; it may have been added as filler by the printer.” Barrett’s book was re-published in 2003 by Kessinger Publishing under the title “The Magus a Complete System of Occult Philosophy”.
[45] Yates 1964, p.49
[46] Ibid., pp.51-52
[47] Ibid., p.93
[48] Ibid., p.50 n.1
[49] Paraphrasing the translation in Rowland, I.D. Giordano Bruno: Philosopher and Heretic University of Chicago Press 2009, p.59 of a passage in Giordano Bruno’s Expulsion of the Triumphant Beast 1584
[50] White, M. The Pope and the Heretic: The true story of Giordano Bruno, the Man Who Dared to Defy the Roman Inquisition Harper Collins 2001
[51] For example, http://www.amaluxherbal.com/bnewbooks/hermes%20trismegistus.html accessed 9 April 2012
[52] Yates, F. Art of memory University of Chicago Press, 1966
[53] White, M. The Pope and the Heretic: The true story of Giordano Bruno, the Man Who Dared to Defy the Roman Inquisition Harper Collins 2001, pp.67-68
[54] Giordano Bruno Nolano, Spaccio de la bestia trionfante. Stampato in Parigi MDLXXXIIII, in Dialoghi filosofici italiani, a cura di Michele Ciliberto, Monciodadori, Milano 2000
[55] Paraphrasing the translation from Rowland, I.D. Giordano Bruno: Philosopher and Heretic University of Chicago Press, Chicago 2009, p.59, originally published in 2008 by Farrar, Straus and Giroux
[56] Yates 1964, p.58, Yates 2009, pp.13-14
[57] Jaroszyński, P. Science in culture Rodopi B.V., Amsterdam, New York 2007, p.139
[58] Ibid.
[59] Yates 1964, p.58, Yates 2009, p.63
[60] Ibid., original 1964 edition, pp.84, 87-88
[61] Ibid., pp.86
[62] Ibid., pp.87-88
[63] Láng, B. Unlocked Books: Manuscripts of Learned Magic in the Medieval Libraries of Central Europe Penn State Press 2008, pp.41-42
[64] Ibid., pp.90-91
[65] Yates 2009, p.185 n.25
[66] Ibid., p.60
[67] Ficino “De Vita coelitùs comparanda” in Opera Omnia Basileae 1576 p.493, as analysed and discussed in Walker, D.P. Spiritual and Demonic Magic: From Ficino to Campanella University of Notre Dame Press, Notre Dame and London 1975 (first published by the Warburg Institute, University of London 1958), pp.12-15
[68] Walker, D.P. Spiritual and Demonic Magic: From Ficino to Campanella University of Notre Dame Press, Notre Dame and London 1975 (first published by the Warburg Institute, University of London 1958), p.93
[69] Ibid., p.144
[70] Ibid.
[71] It is on the cover of Fludd’s Tratactus secundus de naturae simia seu technica macrocosm historia “Second Treatise on the Imitation of Nature or the Technical History of the Macrocosm” published in 1618 with the ape sitting in the middle of a circle depicting the sciences and engineering, including music, building construction, surveying and geography. It is also in Utriusque cosmi historia “The history of both worlds” published between 1617 and 1621 in which the ape is sitting on top of the world again in the centre of the arts and sciences, but above the ape a woman is standing atop the world clearly with spiritual power with her wrist chained to heaven above. The image is given the title “Integra naturae” or “Integration of nature”. Both works were first published by Theodore de Bry in Oppenheim.
[72] Yates 1964, pp.58, 145
[73] Ibid.
[74] Ibid., p.148, n.2 with primary sources Smith, C.F. John Dee London 1909 and Calder, I.R.F. John Dee, studied as an English Neoplatonist unpublished Ph.D. thesis London University 1952
[75] Billingsley, H. (trans.) Euclid The Elements of Geometrie London 1570, reprinted by Ann Arbor 1967
[76] Franklin, J. “Diagrammatic reasoning and modelling in the imagination: the secret weapons of the Scientific Revolution” in Freeland, G. and Corones, A. (eds.) 1543 and All That: Image and Word, Change and Continuity in the Proto-Scientific Revolution Dordrecht 1999, p.82
[77] White, M. The Pope and the Heretic: The true story of Giordano Bruno, the Man Who Dared to Defy the Roman Inquisition Harper Collins 2001, p.158
[78] Amerio, R. (ed.) Magia e grazia Fratelli Bocca, Rome 1957, p.180
[79] Yates 1964, pp.155-156
[80] Hunt, H. A. K. A physical interpretation of the universe: the doctrines of Zeno the Stoic Melbourne University Press 1976, p.20
[81] Couliano, I.P. translated by Cook, M. Eros and Magic in the Renaissance University of Chicago Press, Chicago and London 1987, p.105. However, the premise of Couliano’s discussion is the as magician in the sense of controlling education, religion and every other aspect of the lives of citizens to ensure uniformity. Couliano does not consider that such uniformity may merely be a by-product of the organisation needed for operations via grand projects of the kind described in the text above.
[82] Yates 1964, p.145
[83] Ibid., p.146
[84] Eco, U. The search for the perfect language Blackwell 1995, p.272
[85] Gerhardt, C.I. (ed.) Die philosophischen Schriften von Gottfried Wilhelm Leibniz 7 vols., 1875-1890. reprinted by Georg Olms, Hildesheim 1978, volume IV, pp.27-102
[86] Peckhaus, V. “Calculus Ratiocinator Versus Characteristica Universalis? The Two Traditions in Logic, Revisited” History and Philosophy of Logic 2004 Vol.25, No.1, pp.3-14
[87] Eco, U. 1995, p.275. Eco’s source is Leibniz’s Elementa in Couturat, L. Opuscules et Fragments Inedits de Leibniz: Extraits des manuscrits de la Bibliothèque royale de Hanovre F.Alcan, Paris 1903, reprinted George Olms, Hildesheim 1961, pp.42-92
[88] Yates 1964, pp.56-57
[89] Ibid., p.57, second paragraph
[90] Leibniz, G.W. “Studia Felicitatem Dirigenda” winter 1678/9 in Academy of Sciences of Berlin (ed.) Sämtliche Schriften und Briefe Series VI Philosophische Schriften, no. 4 pp.137-138 as quoted and translated in Antognazza, M.R. Leibniz: An intellectual biography Cambridge University Press 2009, p.237
[91] These ideas on the Leibnizian orientation are from Antognazza, M.R. Leibniz: An intellectual biography Cambridge University Press 2009, pp.237-238
[92] Eco, U. The search for the perfect language Blackwell 1995, p.229
[93] Ibid.
[94] Ibid., p.239
[95] Eco, U. The search for the perfect language Blackwell 1995, pp.144-145
[96] For an introduction and historical overview of this domain, see Franklin, J. “Diagrammatic reasoning and modelling in the imagination: the secret weapons of the Scientific Revolution” in Freeland, G. and Corones, A. (eds.) 1543 and All That: Image and Word, Change and Continuity in the Proto-Scientific Revolution Dordrecht 1999, pp.53-115 and Eco, U. The search for the perfect language Blackwell 1995.
[97] Eco, U. The search for the perfect language Blackwell 1995, p.277
[98] Franklin, J. “Diagrammatic reasoning and modelling in the imagination: the secret weapons of the Scientific Revolution” in Freeland, G. and Corones, A. (eds.) 1543 and All That: Image and Word, Change and Continuity in the Proto-Scientific Revolution Dordrecht 1999, p.82
[99] Smith, B. Characteristica Universalis in K. Mulligan, ed., Language, Truth and Ontology (Philosophical Studies Series) Kluwer, Dordrecht/Boston/Lancaster 1990, pp.50–81
[100] White, M. The Pope and the Heretic: The true story of Giordano Bruno, the Man Who Dared to Defy the Roman Inquisition Harper Collins 2001, p.29
[101] Mercer, C. “Leibniz and His Master: The Correspondence with Jakob Thomasiu” Chapter 2 in Lodge, P. ed. Leibniz and his Correspondents CUP 2004
[102] Sturm, J.C. Philosophia eclectica Altdorf, 1686, pp.72-73 Jolley writes, “Sturm’s works were widely read. Leibniz refers to them throughout his life, although he does not refer specifically to Philosophia eclectica.” Jolley, N. The Cambridge Companion to Leibniz CUP 1995, p.116
[103] Mercer, C. Leibniz’s Metaphysics: Its Origins and Development Cambridge University Press: Cambridge, 2001, Chapter 1, pp.23ff, and Mercer, C. “The Platonism at the Core of Leibniz’s Philosophy” Chapter 15 in Studies on Platonism and Early Modern Philosophy in the series Hutton, S. ed. International Archives of the History of Ideas Springer Volume 196, 2007, pp.225ff
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[110] Barker, C. Cultural Studies: Theory and Practice SAGE Publications, London 2008 pp.188ff and pp.211-212
[111] University of Sydney Library http://www.library.usyd.edu.au/libraries/rare/modernity/ accessed on 7 August 2010
[112] Loemker, p.94 first paragraph (Letter to Thomasius, April 20/30, 1669). In this, Leibniz at least approximates Hermes Trismegistus, “…god [sic] is father of the cosmos, but the cosmos is the father of the things in the cosmos; the cosmos is the son of god, and the things in the cosmos are made by the cosmos.” Corpus Hermeticum XI §8 in Copenhaver, B.P. Hermetica Cambridge University Press 1992, p.29
[113] Loemker, p.304
[114] Leibniz, G.W. 1698 in Wiener, pp.138-9
[115]
Leibniz, G.W. “Reflections on the doctrine of a single universal spirit” 1702 in
Loemker, , pp.554-560
[116] Wiener, p.223 §3; also see Leibniz’s response to Dr Clarke’s second letter, pp.222-228
[117] Incidentally, the concept that space is infinite is core to Henry More’s conflating God with space. More is quoted saying this many times in Koyré, A. From the Closed World to the Infinite Universe Harper and Brothers, New York 1957. See Chapter 6 “God and Space, Spirit and Matter” pp.125-154. By contrast, Leibniz does not think that the universe is infinite.
[118] General Scholium in the 1713 (second) edition of Principia II, 311ff quoted in Burtt, E.A. 1932 2nd ed, reprinted 1950, p.257. Burtt suggests that Newton added the General Scholium to counter the disapprobation from the Godlessness of the pure positivism and empiricism of the first edition of the Principia. Was the General Scholium a populist sop to the churchgoing masses? The General Scholium appears to be irreconcilable with the positivism and empiricism of the Principia. In any case, even considering each outlook on its own, neither is supportable.
[119] Burtt, E.A. The Metaphysical Foundations of Modern Physical Science Routledge London 1932 2nd ed. reprinted 1950, p.256
[120] Leibniz, G.W. Letter to Samuel Clark November 1715 §4 Wiener, p.216
[121] Leibniz, G.W. c.1693 in Wiener, , pp. 77-80
[122] Leibniz, G.W. 1677, Ibid., pp.23-24
[123]
Leibniz’s final words in “Response to Bayle’s Dictionary article Rorarius”
1702, Loemker, p.585
[124] Richard Brown notes that Leibniz did not seem to have made much real progress towards the Universal Characteristic. As regards stating more than has actually been delivered, “this is especially the case in his writings on the Universal Characteristic or Lingua Philosophica. It is true that this project motivated his unique approach to calculus and logic, which in itself is a singular achievement. Yet as far as the general project is concerned little had been accomplished except the production of a vast amount of propaganda. Over and over again he describes the wonderful things he expects from the Characteristic, while strongly hinting that it is already is in his possession.” Brown, R. C. Leibniz, unpublished, Chapter 11 “Epilogue” p.167 Judgement may be reserved, however, for there may be much more to find in the vast trove of Leibniz manuscripts.
[125] A term used by Richard Brown in Brown, R. C. Leibniz, unpublished, Chapter 4 “A Young Central European Polymath Between the Scholastics and the Moderns” p.33
[126] Ibid.
[127] Ibid.
[128] Loemker, p.283
[129] While the cube cannot be doubled with straight-edge and compass, Archytas who studied at Plato’s Academy worked this out a solution in the 4th Century BCE. We quote Rivest, F. and Zafirov, S. “Duplication of the cube” accessed at http://www.cs.mcgill.ca/~cs507/projects/1998/zafiroff/Duplication of the Cube.htm 27 April 2011. Archytas used “a bold construction in three dimensions, determining a certain point as the intersection of three surfaces of revolution, (1) a right cone, (2) a cylinder, (3) a tore [sic] or anchor-ring with inner diameter nil. The intersection of the two later surfaces gives (says Archytas) a certain curve (which is in fact a curve of double curvature), and the point required is found as the point in which the cone meets this curve.” (italics in original)
[130] It is not confirmed whether Gauss performed the experiment on the angle sum of a large triangle by placing fires on two distant mountains to check whether the triangle formed by the path travelled by the light had an angle sum of 180 degrees. Mathematics Illuminated Geometries Beyond Euclid published by Annenberg Learner §8.4 Spherical and Hyperbolic Geometry accessed at http://www.learner.org/courses/mathilluminated/units/8/textbook/04.php 4 June 2011
[131]
1686, Wiener p.296
[132] Shaviv, N. J. “Cosmic Ray Diffusion from the Galactic Spiral Arms, Iron Meteorites, and a Possible Climatic Connection” Physical Review Letters 2002 Vol. 89, Iss. 5. The work of Dayton Miller casts doubt on the negative result of the Michelson-Morley ether experiments. For an overview, see Swenson, L. S. “The Michelson-Morley-Miller Experiments before and after 1905” Journal for the History of Astronomy Vol. 1, p.56 accessed at http://articles.adsabs.harvard.edu//full/1970JHA.....1...56S/0000056.000.html 12 May 2011. Also see Miller, D. “The Ether-Drift Experiment and the Determination of the Absolute Motion of the Earth” Reviews of Modern Physics 1933 Vol. 5, Iss. 3, 203-242 in which Miller calculates an absolute cosmic motion of the earth and of the solar system using the results of the Morley-Miller experiments 1902-1906 and subsequent experiment at Mount Wilson 1921-1924.
[133] Leibniz, G.W., White, C. (trans.) An Essay on the Causes of Celestial Motion Fusion Energy Foundation 1986, esp. pp.6-7
[134] Kutateladze, S. “The Mathematical Background of Lomonosov's Contribution” Journal of Applied Industrial Mathematics, 2011, V. 5, No. 2, 155–162. Accessed at http://www.math.nsc.ru/LBRT/g2/english/ssk/mvl_e.html 4 Dec 2011
[135] In John Locke’s Essays on Human Understanding, §18 Chapter III Book IV Volume 2, Locke exhibits a purist positivist stance by writing, “‘Where there is no property there is no injustice,’ is a proposition as certain as any demonstration in Euclid: for the idea of property being a right to anything, and the idea to which the name ‘injustice’ is given being the invasion or violation of that right.” That is, without property rights, it is not possible to do anyone any harm, because all concepts of justice arise from property. This is opposed to confluence of Justice with Goodness in Plato’s The Republic. If we are in doubt as to Locke’s positivist position, Locke then writes, “Again: ‘No government allows absolute liberty.’ The idea of government being the establishment of society upon certain rules or laws which require conformity to them; and the idea of absolute liberty being for any one to do whatever he pleases; I am as capable of being certain of the truth of this proposition as of any in the mathematics. saying that there where is no property there can be no injustice, just as where there is no government there is absolute liberty.” This indicates that liberty is not the freedom to do what is good or right, but “freedom” to do what one pleases or liberty in the sense of libertinism. The very title of Book I Volume 1 “Neither principles nor ideas are innate” is a bold positivist statement with which any concept of happiness distinct from pleasure would find it difficult to coexist. Project Gutenberg edition Volume 1 at http://www.gutenberg.org/cache/epub/10616/pg10616.txt and Volume 2 at http://www.gutenberg.org/cache/epub/10615/pg10615.txt accessed 7 May 2011.
[136] Contrarily, Wikipedia credits John Locke with the “pursuit of happiness” part of the Declaration of Independence. While Wikipedia may not be relevant as an academic or research source, it is a common popular source, and are taking this opportunity to correct in a research context what is being promoted among the public at large. See http://en.wikipedia.org/wiki/Life,_liberty_and_the_pursuit_of_happiness accessed 7 May 2011.
[137] Robertson, A. Fra Paolo Sarpi: The Greatest of the Venetians George Allen & Company, London 1911, pp.84-85
[138] Jalas, J. (ed.) Gilbert, W. Renaissance and Reformation Lawrence Books, Kansas 1997: Carrie E-Books, 1998, Chapter 9: The Northern Renaissance and the background of the Reformation accessed at http://vlib.iue.it/carrie/texts/carrie_books/gilbert/09.html 24 Dec 2011
[139] Broekhuysen, A. Gerard Groote and the Brethren of the Common Life Wisdom’s Goldenrod Center for Philosophic Studies, Hector New York accessed at http://wisdomsgoldenrod.org/publications/misc/gerard_groote.html 24 Dec 2011
[140] Bocken, I. “The Language of the Layman: The Meaning of Imitatio Christi for a Theory of Spirituality” Studies in Spirituality vol. 15, 2005, pp.217-249
[141] Scharpff, F.A. Der Kardinal und Bischof Nikolaus von Cusa als Reformator in Kirche, Reich und Philosophie des fünfzehnten Jahrhunderts (“The Cardinal and Bishop Nicolaus of Cusa as Reformer in the Church, Empire and Philosophy of the Fifteenth Century”), Tubingen 1871
[142] Grosholtz refers to “Leibniz’s infinite-sided polygon” several times Gillies (ed.) “Was Leibniz a Mathematical Revolutionary?” pp.125, 126, 133 However, Cusa deserves credit for the concept.
[143] Available online at http://jasper-hopkins.info/ Accessed 1 Jan 2010
[144] Kepler wrote of Cusa others divinus mihi Cusanus, i.e. “Cusa and others seem to me divine” in drawing the analogy of the circle compared with polygon to God compared with his creatures. Duncan, A.M. (trans.) Kepler, J. Mysterium Cosmographicum Abaris Books, Janus Series Norwalk USA 1981, Chapter II “Outlines of the primary derivation” p.93
[145] Meli, pp.19-21 referring to Cassirer, E. Leibniz’ System in seinin wissenschaftlichen Grundlagen Marburg 1902, pp.362-3 and 503, and other places
[146] Koyré, A. From the Closed World to the Infinite Universe Harper and Brothers, New York 1957 p.6
[147] Mead, G.R.S. Plotinus Theosophical Society London 1895, pp.24, 26, 28
[148] Duncan, A.M. (trans.) Kepler, J. Mysterium Cosmographicum Abaris Books, Janus Series Norwalk USA 1981, p.53
[149] Leibniz, G.W. c.1682-4, Loemker, p.280
[150] Draft of a letter by Leibniz to Thomas Burnett 1699, quoted in Antognazza, M.R. Leibniz: An intellectual biography Cambridge University Press 2009, p.233
[151] “Man is made to be in the visible universe an image and likeness of God himself (Cf. Genesis 1:26.), and he is placed in it in order to subdue the earth (Cf. Genesis 1:26).” (references in original) Opening statement of Laborem Exercens Papal Encyclical of Pope John Paul II, 14 Sep 1981 accessed at http://www.vatican.va/holy_father/john_paul_ii/encyclicals/documents/hf_jp-ii_enc_14091981_laborem-exercens_en.html 8 May 2011
[152] Cassirer, E., Kristeller, P. O., Randall, J. H. Jr (eds and trans.) The Renaissance Philosophy of Man Phoenix Books, University of Chicago Press 1948; for example, see Mirandola pp.223-5 and Pompanazzi pp. 282-3.
[153] Ibid.
[154] See also Cusa Idiota de Mente, p.63 on the precise role of understanding in human thought vis-à-vis God’s thought. Also see the footnote in Merlan, P. From Platonism to Neoplatonism Martinus Nijhoff, The Hague 1968 at p.5, “Understanding (knowledge) is not the only form of significant mental activity of man. We may enjoy something esthetically [sic]; we may be in empathy with an animal or our fellow-man; any mood is some kind of mental engagement. But none of these activities is of the order of understanding – they are attitudes, reactions, modes of being.”
[155] “Through work man must earn his daily bread (Cf. Ps 127(128):2; cf. also Gen 3:17-19; Prov. 10:22; Ex 1:8-14; Jer 22:13) and contribute to the continual advance of science and technology and, above all, to elevating unceasingly the cultural and moral level of the society within which he lives in community with those who belong to the same family.” (references in original) Opening statement of Laborem Exercens Papal Encyclical of Pope John Paul II, 14 Sep 1981 accessed at http://www.vatican.va/holy_father/john_paul_ii/encyclicals/documents/hf_jp-ii_enc_14091981_laborem-exercens_en.html 8 May 2011
[156]
Latta, pp.254-5, §§64-5
[157] Iltis, C. “Leibniz and the Vis viva Controversy” Isis Spring 1971, Vol. 62, No.1, pp.21-35 at p.22
[158] Hankins, T.L. “Eighteenth-Century Attempts to Resolve the Vis viva Controversy” Isis 1965, Vol. 56, No.185, pp.281-297 at p.281
[159] Leibniz, G.W. Specimen Dynamicum 1695 in Loemker, p.442
[160] Ibid., p.439
[161]
Unfortunately, a translation
of De Motu Gravium was not able to be
obtained for this thesis, so we are reliant on Meli’s reading. Meli’s quote is
in reference to Leibniz’s essay De Motu
Gravium lines 95-101. Meli, D.B. Equivalence
and Priority: Newton versus Leibniz Clarendon Press, Oxford 1993, p.154
[162] Ibid., p.442
[163] From Part I of Specimen Dynamicum 1695 in Wiener, p.137
[164] Hankins, T.L. “Eighteenth-Century Attempts to Resolve the Vis viva Controversy” Isis 1965, Vol. 56, No.185, pp.281-297 at p.287
[165] Ibid.
[166] Ibid.
[167] Ibid., pp.283-6, 291ff
[168] Klemm writes, “Near the end of the century Huygens’s assistant Denis Papin who, through working on problems with gunpowder, hit on the steam engine in 1690.” Klemm, F. A History of Western Technology. History of Science and Technology reprint series Iowa State University Press, Ames (1954), 1991 accessed at http://www.cmpense.org/worksinprogress/summary/Klemm.html 17 May 2012
[169] For example in 1672 it was Huygens who suggested Leibniz summing the series 1/1 + 1/3 + 1/6 + … in which the denominators are the triangular numbers. Bos, H. J. M. “Newton, Leibniz and the Leibnizian tradition” in Grattan-Guinness, I. (ed.) From the Calculus to Set Theory, 1630-1920: An Introductory History Duckworth: London 1980, p.61 from Leibniz, G. W. Writings (Mathematische Schriften) Gerhardt, C. I. (ed.) 1849-1864 Berlin and Halle, vol. 5, p.405
[170] Freudenthal, G. “Perpetuum mobile: the Leibniz-Papin controversy” Studies in the History and Philosophy of Science Vol. 33 (2002) 573-637, p.599
[171] Ibid.
[172] According to Klemm, Papin’s contribution to the steam engine which further developed Huygens’ design – and Leibniz’s collaboration with Papin – was an important marker of the end of the Baroque Period and the beginning of the Age of Rationalism early in the 18th Century. Klemm, Chapter 3 “The struggle for a new prime mover” pp.208ff. C.f. Giedion, S. Mechanization takes command: a conttribution to anonymous history Norton & Co. New York 1969 first published by Oxford University Press 1948. The book covers the mid-15th Century to the early 20th Century.
[173] Referring to August 1690 issue of the Acta Eruditorum which explained his steam machine, Papin wrote to Count Philipp Ludwig von Sinzendorff in the early 1690s, “As water has the property that it is converted by fire to steam … and can then be easily condensed again by cold, I thought that it should not be too difficult to build engines in which, by means of moderate heat and the use of only a little water, that complete vacuum could be produced which had been sought in vain by the use of gunpowder. … I have ascertained by experiment that the piston, raised by the heat to the upper part of the cylinder, will immediately return again to the bottom, and this happens several times in succession so that one might suppose that there was no air at all exerting pressure from below or offering any obstacle to the descent of the piston. Now my tube, whose diameter is only 2 ½ inches is nevertheless able to to raise 60 lb. the whole distance through which the piston falls. And the tube does not even weigh 5 oz. … I have also proved that one minute is sufficient time for a moderate fire to force the piston to the top of the tube. … If consideration is given to the magnitude of the force that can be generated by this means, and the small cost of the wood that has to be used, it must certainly be admitted that this method is far preferable to the use of gunpowder. … It would lead us too far afield to discuss how this discovery could be applied to extract water from the mines, to throw bombs, to sail against the wind or other similar applications which might arise.” (Klemm, p.221-222)
[174] Grosholz, E. “Was Leibniz a mathematical revolutionary?” in Gillies, D. Revolutions in Mathematics
Oxford University Press, New York, 1992, pp.117-133
[175]
Ibid., p.126
[176]
Ibid.,
pp.126-127
[177]
Ibid., p.127
[178] In the same vein, an information-theoretic approach might not be helpful when studying 17th century thinkers as Grosholz attempted to do. See Grosholz, E. “Partial Unification of Domains, Hybrids and the Growth of Mathematical Knowledge” in Grosholz, E. and Breger, H. (eds.) The Growth of Mathematical Knowledge Kluwer, Dordrecht 2000
[179] Cohen, A. Music in the French Royal Academy of Sciences: A study in the evolution musical thought, University Press, Princeton NJ 1981, pp.3-6
[180] Bell, A. E. Christian Huygens and the Development of Science in the Seventeenth Century E. Arnold, London 1947, p.47
[181] Mercer, C. Leibniz’s Metaphysics: Its Origins and Development Cambridge University Press, Cambridge 2001, pp. 80-110
[182] White, M. The Pope and the Heretic: The true story of Giordano Bruno, the Man Who Dared to Defy the Roman Inquisition Harper Collins 2001, p.29
[183] Yates 1964, p.176; Yates, F. The French Academies of the Sixteenth Century Warburg Institute, University of London 1947, pp. 236 ff.; and Yates, F. The Valois Tapestries Warburg Institute, University of London 1959, pp. 82 ff.
[184] Leibniz, G.W., Huggard, E.M. (trans.) Theodicy §49 Project Gutenberg edition 2005 accessed at www.gutenberg.com, p.101
[185] ARTFL Project, Project for American and French Research on the Treasury of the French Language, of the Department of Romance Languages and Literatures, University of Chicago. Accessed at http://artfl-project.uchicago.edu/node/60 on 22 May 2012
[186] Lennon, T. M. and Hickson, M. “Pierre Bayle” Stanford Encyclopedia of Philosophy article first published 2003 revised 2008. Accessed at http://plato.stanford.edu/entries/bayle/ on 22 May 2012
[187] Ibid.
[188] Johnson, A.H. “Leibniz’s method and the basis of his metaphysics” Philosophy 1960 vol. 35, p.60
[189] Loemker, L. E. Struggle for Synthesis: The Seventeenth Century Background of Leibniz’s Synthesis of Order and Freedom Harvard University Press: Cambridge USA 1972, p.127
[190] Ibid., p.130
[191] Grosholz, E. “Was Leibniz a mathematical revolutionary?” in Gillies, D. Revolutions in mathematics Oxford University Press: New York, 1992, p.117-133
[192] Ibid., p.133
[193] The literature on the quadrivium is vast. See http://scienceworld.wolfram.com/biography/Plato.html accessed 26 Feb 2012 for a summary. For Plato’s advocacy, see The Republic Book VII in Rouse, W.H.D. (trans.) Great Dialogues of Plato Signet Classic, New American Library a division of Penguin, New York 1999, pp.324-331 culminating at p.332 with the name “dialectic” being given to the progress of thought through the quadrivium, likening it to a turning away from the shadows and progression through a tunnel upward to the sun in the context of the allegory of The Cave. Plato says much more about music in Book III of The Republic.
[194] For example, Lawrenz refers to Kepler’s Mysterium Cosmographicum and Harmonice Mundi as “luxuriant cosmological fantasias” Lawrenz, J. Leibniz: Double-aspect ontology and the labyrinth of the continuum 2007 PhD thesis, University of Sydney, p.23, note 5
[195] Cohen, A. Music in the French Royal Academy of Sciences: A study in the evolution musical thought, University Press: Princeton NJ 1981, pp.3-5. Cohen writes of Mersenne who was influential in the intellectual life of Paris in the early 17th Century, and the parenthetical comments are in the original: “Having derived his thinking from the platonic traditions of the Renaissance (in which music was considered ‘the image of the whole encyclopedia’), Mersenne assigned a special role to music in his own system, where it formed an essential part of mathematics, ‘utile à toutes le sciences.’ Certainly, this view of music reflects its place in the quadrivium of the medieval artes liberales (the four mathematical disciplines that had their common basis in numerical ratio and proportion: “arithmetic - pure number, music - applied number, geometry - stationary number, astronomy - number in motion”), which continued to form the foundation for university studies through the Renaissance and into early modern times.”
[196] Johnson, A. H. “Leibniz’s method and the basis of his metaphysics” Philosophy 1960 vol. 35, 51-61, pp.52-53
[197] Letter to Ferrault, 1676 “All problems in gravitation, magnetism, electricity and light are explicable by the resolution of a few problems of pure geometry” in Wiener, pp.xxi-xxii
[198] Johnson, A. H. “Leibniz’s method and the basis of his metaphysics” Philosophy 1960 vol. 35, 51-61, p.53
[199] Franklin, J. and Newstead, A. “On the Reality of the Continuum; Discussion Note: A reply to Ormell, ‘Russell’s Moment of Candour’, Philosophy” Philosophy (published by the Royal Institute of Philosophy) 83 2008, p.121
[200] Duncan, A.M. trans. Kepler, J. Mysterium Cosmographicum Abaris Books, Janus Series Norwalk USA 1981, p.57
[201] Brown, S. “Leibniz and Berkeley: Platonic Metaphysics and ‘The Mechanical Philosophy’” Chapter 16, pp.239-253 in Hedley, D. and Hutton, S. (eds) Platonism at the Origins of Modernity: Studies on Platonism and Early Modern Philosophy Springer 2008
[202] Lawrenz, J. Leibniz: Double-Aspect Ontology and the Labyrinth of the Continuum of Sydney PhD thesis 2007, p.23, footnote 5
[203] Franklin, J. “The Formal Sciences discover the Philosopher’s Stone” Studies in History and Philosophy of Science Vol. 25 No. 4 1994 pp.513-533, p.18
[204] Klemm, p.208
[205] C.f. Franklin, J. “‘Useful science’, as an idea, is Baconian, but science as an accepted route to profit, military superiority is an eighteenth-century development.” in “Artifice and the Natural World” in Cambridge History of Eighteenth Century Philosophy CUP p.842
[206] Klemm, p.212
[207] Ibid., p.212
[208] Ibid., p.213
[209] Calvör, H. Historisch-Chronologische Nachricht … des Maschinenwesens … auf dem OberharzeBrunswick, 1763 translated by Dorothea Waley Singer in Klemm, pp.208-9
[210] Ibid.
[211] Brown, R. C. Leibniz unpublished, Chapter 11 “Epilogue”, pp.163-4
[212] Klemm writes that Leibniz personally paid for the pump. Richard Brown qualifies this, writing that it was originally agreed with the Leibniz’s employer the Duke Johann Friedrich that Leibniz would personally paid for the prototype mine pump for test purposes, in September 1679. However, in October 1680 Leibniz’s share was reduced to one third with the balance to be provided by the Duke and the Mining Office. Brown, R. C. Leibniz unpublished, Chapter 11 “Epilogue”, p.163
[213] Brown, R. C. Leibniz unpublished, Chapter 11 “Epilogue”, pp.163-4
[214] Klemm, pp. 218-220 from Ad majorem Dei gloriam published in Nouvelles de la Republique des Lettres Amsterdam 1695. The article is extracted in Appendix 1.
[215] Brown, R. C. Leibniz unpublished, Chapter 11 “Epilogue”, pp.175, 179
[216]
“Essay on a new plan of a certain science” in Wiener, p.583
[217]
“On the elements of natural sciences” ca. 1682-4, Loemker, pp.281, 282
[218] E.g. see Franklin, J. “Artifice and the Natural World” in Cambridge History of Eighteenth Century Philosophy Cambridge University Press, p.843
[219] Klemm, pp.226-227
[220] Loemker, p.593 (Letter to Hansch, July 25, 1707)
[221] Thus, Leibniz would agree that the divine mind does “design through us” albeit that we have free will. We disagree with Austin Farrer in his commentary on Theodicy (November 24, 2005 [EBook #17147], pp.32-33):
And perhaps some such element enters into all our choices, since our life is to some extent freely designed by ourselves. If so, our minds are even more akin to the divine mind than Leibniz realized. For the sort of choice we are now referring to seems to be an intuitive turning away from an infinite, or at least indefinite, range of less attractive possibility. And such is the nature of the divine creative choice. The consequence of such a line of speculation would be, that the divine mind designs more through us, and less simply for us, than Leibniz allowed: the ‘harmony’ into which we enter would be no longer simply ‘pre-established’. Leibniz, in fact, could have nothing to do with such a suggestion, and he would have found it easy to be ironical about it if his contemporaries had proposed it.
[222] Letter to Hansch, July 25, 1707 in Loemker, p.592-3
[223] Leibniz, G.W. Theodicy §147. There is also an epigraph to Theodicy which says the same, though this author has been unable to find an edition which includes this epigraph. The epigraph is quoted in Jorgensen, L.M. “Leibniz’s Theodicy” PHIL 375: Advanced History of Philosophy course notes Fall 2008 p.10 accessed at http://faculty.valpo.edu/ljorgens/teachres/Theodicy%20Packet.pdf 12 May 2011. Jorgensen states that the edition being used is an unpublished manuscript of W.H. Warren (trans.), Mulvaney, R.J. (ed.) Leibniz, G.W. Theodicy.
[224] Loemker, p.556
[225] Leibniz, G.W. Beaudry, P. (trans.) “The String Whose Curve Is Described by Bending Under Its Own Weight, and the Remarkable Resources That Can Be Discovered from It by However Many Proportional Means and Logarithms” Acta Eruditorum, Leipzig, June 1691, last sentence accessed at http://www.schillerinstitute.org/fid_97-01/011_catenary.html#1 on 9 June 2012
[226] See Franklin, J. “Mathematical Necessity and Reality” Australasian Journal of Philosophy Vol. 67 No. 3, September 1989
[227] Leibniz, G.W., Beaudry, P. (trans.) “The String Whose Curve Is Described by Bending Under Its Own Weight, and the Remarkable Resources That Can Be Discovered from It by However Many Proportional Means and Logarithms” Acta Eruditorum, Leipzig, June 1691, second-last paragraph accessed at http://www.schillerinstitute.org/fid_97-01/011_catenary.html#1 on 9 June 2012
[228] Ibid., first paragraph
[229] In The Republic (602 d), Plato says, “A stick will look bent if you put it in the water, straight when you take it out, and deceptive differences of shading can make the same surface seem to the eye concave or convex; and our minds are clearly liable to all sorts of confusions of this kind.”
[230] Malcolm, N. “A summary biography of Hobbes” in Sorell, T. (ed.) The Cambridge Companion to Hobbes CUP Cambridge 1996, p.22. On the same page, Malcolm speculates that “Hobbes must have gained a special interest in the writings and political actions of Sarpi, who had defended Venice against the papal interdict of 1606 and developed a strongly anti-papal theory of Church and State”.
[231] Kainulainen, J. “Paolo Sarpi between Jean Bodin and Thomas Hobbes: a study on ‘political animal’ in early modern Europe” Thesis at the European University Institute, Department of History and Civilisation. Accessed at http://www.eui.eu/Personal/VanGelderen/Theses/JaskaKainulainenThesis.shtml 24 April 2012
[232] Ibid.
[233] White, M. The Pope and the Heretic: The true story of Giordano Bruno, the Man Who Dared to Defy the Roman Inquisition Harper Collins 2001, pp.38-39
[234] Franklin, J. and Newstead, A. “On the Reality of the Continuum; Discussion Note: A reply to Ormell, ‘Russell’s Moment of Candour’, Philosophy” Philosophy (published by the Royal Institute of Philosophy) 83 2008 especially p.124. In an earlier article, Franklin discusses the impossibility of the formal sciences were we to reject imperceptible relations between intangible mathematical objections. Franklin, J. “The Formal Sciences discover the Philosopher’s Stone” Studies in History and Philosophy of Science Vol. 25 No. 4 1994 , p.26 of version at http://www.maths.unsw.edu.au/~jim/philosophersstone.pdf accessed 1 Nov 2010
[235] Burtt, E.A. The Metaphysical Foundations of Modern Physical Science Routledge and Kegan Paul, London 2nd ed. 1932, reprinted 1950 pp. 315-6
[236] Latta, p.233
[237] post-1690, Wiener,p.76
[238] This is contrary to the view of Leibniz’s mathematics teacher at the University of Jena, Weigel, who took the view that the NMP was an extension of Aristotle. See Brown, R. C. Leibniz unpublished, Chapter 4 “A Young Central European Polymath Between the Scholastics and the Moderns”, p.33 and Mercer, C. Leibniz’s metaphysics: its origins and development Cambridge University Press 2001
[239] Papal Speech, Apostolic Journey of His Holiness Benedict XVI to München, Altotting and Regensburg 9-14 September 2006, Meeting with the representatives of science, Lecture of the Holy Father, Aula Magna of the University of Regensburg 12 September 2006 “Faith, Reason and the University: Memories and Reflections” accessed at http://www.vatican.va/holy_father/benedict_xvi/speeches/2006/september/documents/hf_ben-xvi_spe_20060912_university-regensburg_en.html 1 May 2009
[240] By adopting the “practical reason” of the Empiricist, Kant declared that he needed to set thinking aside in order to make room for faith. Kant is beyond the scope of this dissertation. However, we are here nudging up against Pope Benedict XVI’s objection to Kant’s outlook on faith. Ibid.
[241] For example, this is precisely the outlook of Ian Bryce who is on the executive committee of the Humanist Society and is the founder of the Australian Secular Party. From conversation with Dr Ian Bryce of the NSW Humanist Society at Eastside Radio 89.7 FM studio in Paddington on Friday 5 Feb 2010
[242] McGuire, J.E. “Newton’s ‘Principles of Philosophy’: An intended preface for the 1704 Opticks and a related draft fragment” The British Journal for the History of Science 1970 vol. 5 no. 2 pp. 178-186, at p.183
[243] Ibid., p.180
[244] 1698, Wiener pp.138-9
[245] “Reflections on the doctrine of a single universal spirit” 1702, Loemker, p.555
[246] c.f. Wiener p.292 that God’s justice is not arbitrary, that God governs Himself by reason
[247] Wiener pp.77-78
[248] Franklin, J. The Science of Conjecture: Evidence and probability before Pascal J. H. Press Baltimore and London 2002, pp.150-1
[249] “In the course of the 14th Century, European primacy in Averroan studies passed from the University of Paris … to the great Italian schools: Bologna and especially Padua.” Carboni, S. Venice and the Islamic world, 828-1797 Yale Univesity Press 2007 p.153. The influence of Padua on English intellectual life is explained in Woolfson, J. Padua and the Tudors: English students in Italy, 1485-1603 University of Toronto Press 1998. A Princeton University webpage for second year undergraduate history says, “The University of Padua was one of the most prominent universities in early modern Europe, known particularly for the rigor of its Aristotelean logic and science.” accessed at http://www.princeton.edu/~his291/Padua.html 10 May 2011 While Thomas Aquinas would disagree with Averroes calling himself an Aristotelean, Averroan studies are regarded as a development of Aristotelean logic, and Averroes was regarded as a Neo-Aristotelean.
[250] Friedrich Schiller, Johnson, S. trans. Of the Aesthetic Estimation of Magnitude (1793) Accessed at http://www.schillerinstitute.org/transl/Schiller_essays/magnitude.html 1 Jan 2011
[251] Schiller was responding to Kant’s The Critique of Pure Reason first published 12 years before Schiller’s essay. Accessed at http://www.gutenberg.org/cache/epub/4280/pg4280.html 1 Jan 2011
[252] Schiller uses this to posit a universal and objective quality of beauty which is equated to absolute truth, in opposition to Kant’s position that beauty is subjective. In Schiller’s Letter XII, he refers to a “unity of idea” according to which “We are no longer individuals but a species”. See next footnote.
[253] Friedrich Schiller, edited by Riikonen, T. and Widger, D. Aesthetical and Philosophical Essays, Letter XII Accessed at http://www.gutenberg.org/files/6798/6798-h/6798-h.htm on 1 Jan 2011
[254] Wiener, p.210
[255] It is this distinction between mathematics and a more general Analysis that Johnson does not recognise. Johnson seems to think that where the usefulness of Mathematics ends for Leibniz, experiment begins. However, for Leibniz, mathematics is only one of many tools for Analysis. In turn, Analysis is one of many modes of human Reason. Johnson, A. H. “Leibniz’s method and the basis of his metaphysics” Philosophy 1960 vol. 35, 51-61. See page 54.
[256] 1715, Wiener, p.201
[257] Boyer, C. The History of the Calculus and its Conceptual Development Dover Publications, New York 1949, p.94
[258] Ibid., p.93
[259] Duncan, A.M. (trans.) Kepler, J. Mysterium Cosmographicum Abaris Books, Janus Series Norwalk USA 1981, Chapter II “Outlines of the primary derivation” p.93
[260] Boyer quotes from Kepler’s Opera omnia II 595, translated from the quoted Latin by this author, “we find that a straight line is an hyperbola obtuse in the extreme. And from Cusanus we learn that a circle is an infinite linear thing. They are several things simultaneously, not discrete alternatives, whose different faces are turned to light by the use of analogy.” Ibid.
[261] post-1690 in Wiener, pp.74-5
[262] This is pretty close to the heart of Leibniz’s critique of Descartes’ subjectivism. Quite simply, “Yes, humans make mistakes, but that does not mean that reality is subjective.” As Raynaud wrote, “Methodological doubt [for Leibniz] does not have the weight which Descartes accords to it.” in McCarthy 1998, p.152 referring to Leibniz’s response to Article 5 of Descartes’ Principles of Philosophy. This response by Leibniz is in Loemker 1969, p.384
[263] Leibniz, G.W. 1685 “Machina arithmetica in qua non additio tantum et subtractio sed et multiplicatio nullo, divisio vero paene nullo animi labore peragantur” Kormes, M. (trans.) “Leibniz on his calculating machine” in Smith, D. E. A Source Book in Mathematics Dover Publications, Mineola N.Y. 1959 (first published by McGraw-Hill 1929), p.181
[264] Leibnitia webpage http://www.gwleibniz.com/calculator/calculator.html accessed 7 August 2010. At the IBM Archives it is described as “a major advance in mechanical calculating. The Leibniz calculator incorporated a new mechanical feature, the stepped drum — a cylinder bearing nine teeth of different lengths which increase in equal amounts around the drum. Although the Leibniz calculator was not developed for commercial production, the stepped drum principle survived for 300 years and was used in many later calculating systems. ” http://www-03.ibm.com/ibm/history/exhibits/attic3/attic3_037.html accessed 7 August 2010
[265] Wiener, last para. p. xxiii and first para. p.xxiv. Also Book IV, Chapter XII, §13 New Essays on Human Understanding in Wiener, p.479
[266] Loemker, p.250 (Letter to Huygens, 1679 around two years after Leibniz’s initiation into Huygens’ experiments with the [French] Royal Academy of Sciences)
[267] The microscope was central to the work of Hooke in his role as demonstrator at the British Royal Society. Arnol’d, V.I. Huygens and Barrow, Newton and Hooke Birkhauser Verlag, Basel 1990, pp.11-14.
[268] Wiener, pp. 73-4
[269] Ibid., p.74
[270] Franklin, J. “The Formal Sciences discover the Philosopher’s Stone” Studies in History and Philosophy of Science Vol. 25 No. 4 1994 pp.513-533, p.25 of online pdf
[271] “nor do I know why you should consider as most absurd the view that everything happens mechanically in nature, that is, according to certain mathematical laws prescribed by God.” Leibniz, G. W. “Letter to Herman Conring” 19 March 1678, in Loemker, L. G. W. Leibniz Philosophical Papers and Letters 2nd ed. Reidel Publishing Company 1969, p.189
[272] Petrarca was an outspoken and influential critic of Averroism as well as of stepwise reasoning as practised under the heading of “dialectic”. Petrarca “On his own ignorance and that of many others” c.1368, pp.47-133 in Cassirer, E., Kristeller, P. O., Randall, J. H. Jr (eds and trans.) The Renaissance Philosophy of Man Phoenix Books, University of Chicago Press 1948 and Brand, P. Cambridge History of Italian Literature Cambridge University Press 1999, p.134. Cusa continues Petrarca’s programme of demonstrating that intellect as exercised by an intelligent commoner can be more effective than the formal reasoning promoted by the scholastics..
[273] Post-1690 “On the horizon of human doctrine”, Wiener pp.76-77
[274] From a preliminary study towards his Theoria Motus Abstracti. Leibniz, G.W. Sämtliche Schriften und Briefe Deutsche Akademie der Wissenschaften zu Berlin (eds) Akademie-Verlag, Berlin, 1923 Series 6, volume 1, p.160, translated in Garber, D. Leibniz: body, substance, monad Oxford University Press 2009, p.15
[275] Brown, R. C. Leibniz unpublished, Chapter 4 “A Young Central European Polymath Between the Scholastics and the Moderns”, pp.41-42
[276] Burtt, Chapter II “Copernicus and Kepler” p.22ff
[277] Stuart Brown addresses this misconception. Brown, S. “Leibniz and Berkeley: Platonic Metaphysics and ‘The Mechanical Philosophy’” Chapter 16 in Platonism at the Origins of Modernity: Studies on Platonism and Early Modern Philosophy Springer 2008, pp.239-253. Edward Rosen’s writing has done something to help perpetuate the misconception. See Rosen, E. “Was Copernicus a Neoplatonist?” Journal of the History of Ideas Vol.44, No.4 (Oct-Dec 1993), pp.667-9. Rosen’s argument that Copernicus was not a Neoplatonist is based on the misconception of Neoplatonism as a mystical doctrine that “all reality has its source in the transcendent One, which produces a series of less unified levels of being, down to the last and lowest, the physical universe, a living creature endowed with a divine soul; at our highest, we humans can join the One in a mystical union.” This sounds more like the doctrine taught in a New Age yoga class. We wonder whether Rosen merely has an innocent misconception, because apparent bias – not only against Plato but against Christianity – is reflected by his opening sentence, “Christian thought was profoundly influenced by the ancient pagan Greek philosopher Plato.”
[278]
Leibniz wrote, “There are some who imagine a world of light in their brains.”
However, “this is not the light but only a heating of their blood.” Loemker p.367. Also see “On the
general characteristic” in Loemker 1969, p.221.
[279] Kuhn, T. S. The Copernican Revolution Harvard University Press 1957
[280] Kessler, J. J. PhD “Giordano Bruno: The Forgotten Philosopher” accessed at http://www.infidels.org/library/historical/john_kessler/giordano_bruno.html 10 May 2011
[281] Rosen, E. “Was Copernicus a Neoplatonist?” Journal of the History of Ideas Vol.44, No.4 (Oct-Dec 1993), pp.667-669
[282]
Burtt, E.A. The Metaphysical Foundations
of Modern Physical Science Routledge London 1932 2nd ed. reprinted 1950,
p.70
[283] “On the general characteristic” in Loemker , L. E. Gottfried Wilhelm Leibniz: Philosophical Papers and Letters 2nd ed. D. Reidel, Dordrecht Holland 1969 1969, p.221
[284] Hence, today’s heavily quantitative society and culture reflect a heavy influence of Neoplatonism in our heritage.
[285] Burtt, E.A. The Metaphysical Foundations of Modern Physical Science Routledge London 1932 2nd ed. reprinted 1950, p.197
[286] Galileo’s debt to Pythagoreanism is explained in Demarco, D. “The dispute between Galileo and the Catholic Church” Homiletic and Pastoral Review 2002 accessed at http://www.catholiceducation.org/articles/science/sc0043.htm 26 May 2011
[287] Burtt, E.A. The Metaphysical Foundations of Modern Physical Science Routledge London 1932 2nd ed. reprinted 1950, Chapter III “Galileo” p.63 and Robertson, A. Fra Paolo Sarpi: The Greatest of the Venetians George Allen & Company, London 1911, pp.73-75
[288] Wootton, D. quoting from Sarpi’s Pensiero no.146 in Paolo Sarpi: Between Renaissance and Enlightenment Cambridge University Press, 2002, p.37
[289]
Quoted in Meli, D.B. Equivalence and Priority: Newton versus
Leibniz Clarendon Press, Oxford 1993, pp.22-23
[290] “Per mia memoriae” (“From my memory”) Schedae Sarpianae (Writings/Manuscripts of Sarpi) in Robertson, A. Fra Paolo Sarpi: The Greatest of the Venetians George Allen & Company, London 1911, pp.73-75
[291] Sarpi himself was excommunicated in 1607 by the Inquisition. Robertson, A. Fra Paolo Sarpi: The Greatest of the Venetians George Allen & Company, London 1911, p.152
[292] Wootton, p.46 quoting from Sarpi’s Pensiero no.146
[293] Robertson, A. Fra Paolo Sarpi: The Greatest of the Venetians George Allen & Company, London 1911 p. 84
[294] Tedeschi, J. (review) “Venetian Phoenix: Paolo Sarpi and Some of His English Friends (1606-1700) by John L. Lievsay” Modern Philology Vol. 75, No. 2 (Nov., 1977), pp. 191-194
[295] Robertson, A. Fra Paolo Sarpi: The Greatest of the Venetians George Allen & Company, London 1911 pp. 73-74
[296] Ibid., p.73
[297] Wootton, D. Paolo Sarpi: Between Renaissance and Enlightenment Cambridge University Press, 2002, p.46
[298] For example, see any English edition of The Ghost Seer by Friedrich Schiller and The Bravo by James Fenimore Cooper. Also refer to Othello: Moor of Venice by William Shakespeare wherein Venetian methods are represented by Iago.
[299] Perfetti, S. “Pietro Pomponazzi” 2004 in Stanford Encyclopedia of Philosophy. Accessed at http://plato.stanford.edu/entries/pomponazzi/ on 28 April 2012
[300] Wootton, p.41
[301] See de Immortalitate Animae by Pompanazzi, chapter 4
[302] Pompanazzi, P. Immortalitate Animae “On the immortality of the soul” in Cassirer, E., Kristeller, P. O., Randall, J. H. Jr (eds and trans.) The Renaissance Philosophy of Man Phoenix Books, University of Chicago Press 1948, pp.280-380
[303] Ibid., pp.380-381
[304] Wootton, p.47
[305] Ibid., p.46
[306] Burtt, E.A. The Metaphysical Foundations of Modern Physical Science Routledge London 1932 2nd ed. reprinted 1950, p.65
[307] Ibid., pp.65-8
[308] Wootton, p.37
[309] Burtt, E.A. The Metaphysical Foundations of Modern Physical Science Routledge London 1932 2nd ed. reprinted 1950, pp.68-9
[310] Leibniz, G.W. New Essays on Human Understanding Book IV Chapter 12 §13 in Wiener, p.479-480
[311] McGuire, J.E. “Newton’s ‘Principles of Philosophy’: An intended preface for the 1704 Opticks and a related draft fragment” The British Journal for the History of Science 1970 vol. 5 no. 2 pp. 178-186, at pp.184-5
[312] Duncan, A.M. trans. Kepler, J. Mysterium Cosmographicum Abaris Books, Janus Series Norwalk USA 1981, pp.77-79
[313]
Selberg, E. “Petrus Ramus” 2006 revised 2011, in Stanford Encyclopedia of Philosophy Zalta, E.N. (ed.). Accessed at http://plato.stanford.edu/entries/ramus/
11 Aug 2012
[314] Mamiani, M. “To twist the meaning: Newton’s regulae philosophandi revisited” in Buchwal, J. and Cohen I. eds. Isaac Newton’s Natural Philosophy MIT Press, Cambridge MA 2001
[315] Burtt, E.A. The Metaphysical Foundations of Modern Physical Science Routledge London 1932 2nd ed. reprinted 1950, pp.211-213
[316] Koyré, A. Newtonian Studies Chapman & Hall, London 1965 p.40
[317] Ibid., pp.214-216
[318] Pompanazzi, P. Immortalitate Animae “On the immortality of the soul” in Cassirer, E., Kristeller, P. O., Randall, J. H. Jr (eds and trans.) The Renaissance Philosophy of Man Phoenix Books, University of Chicago Press 1948, p.293
[319] Wootton, p.41
[320] Ibid.
[321]
Locke, J. Essays on Human Understanding
Vol. 2 Book IV Chapter XI, Project Gutenberg edition accessed at www.gutenberg.org
[322] Leibniz, G.W. New Essays on Human Understanding Vol. 2 Book IV Chapter II §1, Bennett, J. (trans.) 1st ed. Feb 2005, amended April 2008, p.166 Accessed at www.earlymoderntexts.com/jfb/leibne.pdf 10 May 2011
[323] To Hermes Trismegistus, there is for humans a symbiosis between understanding and sensation, but understanding and sensation are distinct because understanding comes to be especially by the agency of the mind. Corpus Hermeticum IX §2 in Copenhaver, B.P. Hermetica Cambridge University Press 1992, p.27
[324] “Hunc intellectus non errans stare docet: Sensus autem fallax suadet moueri.” Giordano Bruno De Umbris Idearum Paris 1582 digital edition Peterson, J.H. (trans.) 1997 accessed at http://www.esotericarchives.com/bruno/umbris.htm 27 April 2012.
[325] Burtt, E.A. The Metaphysical Foundations of Modern Physical Science Routledge London 1932 2nd ed. reprinted 1950, pp.216-8
[326] Ibid., p.219
[327] Koyré, A. Newtonian Studies Chapman & Hall, London 1965, p.40
[328] From Yahuda Manuscript 1.1a in Mamiani, M. “To twist the meaning: Newton’s regulae philosophandi revisited” in Buchwal, J. and Cohen I. eds. Isaac Newton’s Natural Philosophy MIT Press, Cambridge MA 2001, p.6
[329] Wiener, p.296
[330] It may be stretch to draw a connection. However, the seeking after simplicity goes back to Hermes Trismegistus. He writes, “The many make philosophy obscure in the multiplicity of their reasoning.” He says that they (the many) combine philosophy with arithmetic, music and geometry, so making it incomprehensible. Instead, “Pure philosophy that depends only on reverence for god should attend to these other matters [arithmetic, music and geometry]” leaving only problems in astronomy unsolved. “Asclepius” §§12-13 in Copenhaver, B.P. Hermetica Cambridge University Press 1992, p.74
[331] c.1693, Wiener, pp.77-78
[332] 1677, Wiener, p.18
[333] Mamiani, M. “To twist the meaning: Newton’s regulae philosophandi revisited” in Buchwal, J. and Cohen I. eds. Isaac Newton’s Natural Philosophy MIT Press, Cambridge MA 2001, p.7, middle paragraph
[334] Private communication from Prof. Jim Franklin, UNSW, 23 Aug 2009
[335] Cambridge History of 17th Century Philosophy C.U.P. 1987, pp.141-3
[336] Loemker 1969 p.224 and Wiener 1951, pp.23-24. Loemker ascribes this writing to c.1679 and calls it “On the general characteristic”; Wiener dates it 1677 and calls it “Towards a Universal Characteristic”. This writer thinks both translations/interpretations are fair.
[337] Ihmig is similarly sceptical. He writes, “The strict rejection of hypotheses in experimental philosophy, however, hardly seemed to agree with how Newton practiced science. As hypotheses can be found in all editions of the Principia, some authors have tried to determine the meaning of the concept of hypotheses and its various forms more precisely. This has made it possible to separate ‘good’ from ‘bad’ hypotheses, thus confining Newton’s rejection of hypotheses merely to the bad ones.” Ihmig, K.-N. “Newton's Program of Mathematicizing Nature” in Hoffmann, M.H.G., Lenhard, J., Seeger, F. (eds) Activity and sign: grounding mathematics education Springer 2005, p.242
[338] In New Theory about Light and Colours 1672, Newton had already rejected the use of hypotheses. In the Principia 2nd ed. 1713, Newton admits the difficulties that avoiding hypotheses is causing him, but reiterates the validity of avoiding hypotheses: “But hitherto I have not been able to discover the cause of those properties of gravity from phenomena, and I frame no hypotheses; for whatever is not deduced from the phenomena is to be called an hypothesis; and hypotheses, whether metaphysical or physical, whether of occult qualities or mechanical, have no place in experimental philosophy.”
[339] Letter to Herzog Friedrichm, 1679 in Wiener, p.xxiii. Wiener also writes, “It is in his correspondence concerning the experimentalists of the Royal Society of England (whom he had visited in 1673) that Leibniz reveals most clearly what he thought his logical instrument could do, and what the plain empiricists were not doing, namely, to elicit all the knowledge deducible from a given number of presuppositions. Lack of a proper art of demonstration had made it necessary, in Leibniz’s opinion, for Baconian experimental philosophers like Boyle to resort to many observations in order to find out what Galileo and Descartes were able to know by reasoning.” Wiener, p.xxii. Wiener does not say whether the “logical instrument” is the Universal Characteristic.
[340] Letter to Berthey, 1677, in Wiener, pp.xxiii-xxiv
[341] Burtt, E.A. The Metaphysical Foundations of Modern Physical Science Routledge London 1932 2nd ed. reprinted 1950, p.223
[342] Burtt, E.A. The Metaphysical Foundations of Modern Physical Science Routledge London 1932 2nd ed. reprinted 1950, p.226
[343] Koyré, A. Newtonian Studies Chapman & Hall, London 1965, p.52
[344] This author’s MPhil thesis “On object Petri nets” (University of Warwick, Department of Computer Science, 2004) dealt with Petri nets which seek to model complex systems using enhanced graph-like models which carry tokens under deterministic rules, like a pinball machine in which the balls run through tubes. Nested Petri nets use nets themselves as tokens and so can represent greater complexity. Some nets can be reduced to a linear logic proposition. Net-based models are essential for systems such as telecom networks, computer chips and circuit boards. Attempts to use nets to model processes with a human element have seen a resurgence with the modelling of social networks such as those found on Facebook or LinkedIn. As yet, there is no convincing argument that net-based models bear any qualitative resemblance to the mind or to thought processes.
[345] Leibniz, G.W. Theodicy §147
[346] Leibniz, G.W. New Essays on Human Understanding Book IV Chapter II §13 in Wiener, p.479
[347] Boyer, C. The History of the Calculus and its Conceptual Development Dover Publications, New York 1949, p.48
[348] Ibid., p.51
[349] 1685, Wiener, p.52
[350] Manin, Y. I. Mathematics and Physics Trans. by A. and N. Koblitz, Boston-Basel-Stuttgart 1981, p.5 in Nikulin, D. V. Matter, imagination and geometry: ontology, natural philosophy and mathematics in Plotinus, Proclus and Descartes Ashgate, Aldershot and Burlington 2002, p.xii, n.17
[351] Ibid., pp.70-74ff
[352] Ibid., pp.x-xi
[353] Leibniz, G.W. New Essays on Human Understanding Chapter X Book IV §§9-10 in Wiener pp.473-474
[354] Kielkopf, C.F. Strict Finitism: An examination of Ludwig Wittgenstein’s remarks on the foundations of mathematics Mouton, The Hague and Paris 1970, p.32
[355] Stetz, A. Lie Groups in Modern Physics 1996, freely available online p.5 “There can be no question that the modern theory of elementary particles rests heavily on the formalism of Lie groups.” accessed at www.physics.orst.edu/~stetza/Lie.pdf 7 May 2011
[356] This author’s Honours thesis was entitled “A categorical approach to universal algebra” Department of Pure Mathematics, University of Sydney 2000 supervised by Dr Steve Lack. It is a relatively easy-to-read introduction to category theory and universal algebra. Universal algebra was an even earlier form of meta-algebra pioneered by mathematicians such as Whitehead.
[357]
Davies, P. C. W. “The Implications of a Cosmological Information Bound for
Complexity, Quantum Information and the Nature of Physical Law” Arxiv Preprint,
p.2 Accessed at http://arxiv.org/ftp/quant-ph/papers/0703/0703041.pdf on 21 July 2012
[358] Loemker, p.249 (Letter to Huygens, 1679 around two years after Leibniz’s initiation into Huygens’ experiments with the [French] Royal Academy of Sciences)
[359] c.1693, Wiener, pp.78-80
[360] Ibid., pp.77-78
[361] c.1693, Ibid., pp.78-80
[362] Leibniz, G. W. “Letter to Hansch on the Platonic Philosophy or Platonic Enthusiasm” 25 July 1707, in Loemker, L. G. W. Leibniz Philosophical Papers and Letters 2nd ed. Reidel Publishing Company 1969, p.593
[363] “Reply to Bayle’s Dictionary article Rorarius” 1702, Loemker, L. G. W. Leibniz Philosophical Papers and Letters 2nd ed. Reidel Publishing Company 1969, p.582
[364] Leibniz, G. W. “Letter to Hansch on the Platonic Philosophy or Platonic Enthusiasm” 25 July 1707. Ibid., p.593
[365] Leibniz, G. W. “Letter to Herman Conring” 19 March 1678. Ibid., p.187
[366]
cc.1682-4 in Loemker, pp.281-9
[367] Ibid., p.283
[368] Rouse, W.H.D. (trans.) Great Dialogues of Plato Signet Classic, New American Library a division of Penguin, New York 1999, pp.281-311
[369] Ibid., p.309
[370] Child, J.M. (ed. and trans.) The early mathematical manuscripts of
Leibniz; tr. from the Latin texts by Carl Immanuel Gerhardt with critical and
historical notes by J. M. Child The Open Court Publishing Company 1920,
reprinted by the University of Michigan Library, p.145
[371]
Ibid., pp.145-146
[372] Ibid., p.149
[373] Vermij, R.H. “Bernard Nieuwentijt and the Leibnizian Calculus” Studia Leibnitiana 1989 Vol.21 No.1 69-86, p.77
[374]
Ibid., p.149
[375] “It is interesting to note that Gauss did not publish many of his ideas. It is commonly thought that this was because he was a perfectionist and would only make his views known if they were above criticism. To that end, he would not provide the intuitions behind his proofs, preferring instead to give the impression that they came ‘out of thin air’” which, of course, they did not. Mathematics Illuminated Geometries Beyond Euclid published by Annenberg Learner §8.4 Spherical and Hyperbolic Geometry accessed at http://www.learner.org/courses/mathilluminated/units/8/textbook/04.php 4 June 2011
[376] Bonaventura Cavalieri (1598-1647) carried out seminal work with infinitesimals. It was often referenced merely out of a sense of obligation by subsequent writers rather than because Cavalieri’s work bore any relation to the later work. Cavalieri understood the infinitesimal as a fixed but extremely small quantity, quite opposed to that of Leibniz. That is, Cavalieri’s infinitesimal appealed to common sense of the physical rather than the ontological. Beeley, P. “Infinity, Infinitesimals, and the Reform of Cavalieri: John Wallis and his Critics” in Goldenbaum, U. Infinitesimal Differences: Controversies between Leibniz and his Contemporaries Walter de Gruyter, Berlin 2008, p.33
[377]
Child, J. M. (ed. and trans.) The early mathematical manuscripts
of Leibniz; tr. from the Latin texts published by Carl Immanuel Gerhardt with
critical and historical notes Open Court Publishing Company, Chicago and London, 1920 reprinted by
University of Michigan Library, p.149
[378] Grosholz, E. “Was Leibniz a mathematical revolutionary?” in Gillies, D. Revolutions in Mathematics Oxford University Press, New York, 1992, pp.117-133
[379] Duncan, A.M. trans. Kepler, J. Mysterium Cosmographicum Abaris Books, Janus Series Norwalk USA 1981, p.93
[380] Hopkins, J. trans. Nicolaus
of Cusa Apologia De Docta Ignorantia 3rd ed., J. Arthur Banning
Press, Minneapolis, pp.481-2
[381] Hopkins, J. trans. Johannes
Wenck De Ignota Litteratura 3rd ed.,
J. Arthur Banning Press, Minneapolis, p.439
[382] Hopkins, J. trans. Nicolaus
of Cusa Apologia De Docta Ignorantia 3rd ed., J. Arthur Banning
Press, Minneapolis, p.482
[383] Leibniz, G.W. “Letter on a general principle useful in explaining the laws of nature through a consideration of the divine wisdom: to serve as a reply to the response of Rev. Father Malebranche” Nouvelles de la république des lettres July 1687 in Loemker, L.E. Gottfried Wilhelm Leibniz: Philosophical Papers and Letters D. Reidel Publishing Company, Dordrecht 1969 p.352
[384] Hopkins says that Cusa’s likening of human minds to “living mirrors that mirror one another and all of reality” was also adopted by Leibniz though Hopkins does not say where. The concept seems to be encapsulated in Monadology since every monad perceives every other monad. Hopkins notes that in De Docta Ignoranta II, 1 (97), Cusa says that the world is as perfect as it can be, which prefigures Leibniz’s doctrine that this is the best of all possible worlds. Hopkins, J. “Nicholas of Cusa (1401-1464):
first modern philosopher?" in French, P. A., Wettstein, H. K., and Silver, B. (eds) Midwest Studies in Philosophy Volume XXVI (2002): Renaissance and Early Modern Philosophy Blackwell Publishing, Boston and Oxford, pp.16, 20
[385] Child, J. M. (ed. and trans.) The early mathematical manuscripts of Leibniz; tr. from the Latin texts
published by Carl Immanuel Gerhardt with critical and historical notes Open Court Publishing Company,
Chicago and London, 1920 reprinted by University of Michigan Library,
p.97
[386] Ibid., p.98
[387] Ibid., p.149
[388] Ibid., p.150
[389] Ibid.
[390] John Wallis geometrically explained negative numbers and square roots of negative numbers, treating negative as the opposite direction of positive. Wallis geometrically constructs the square root of a negative number by applying Pythagoras’ theorem to a right-triangle in mirror-image of another sharing the same vertical side. Incidentally, Wallis’ argument leads naturally to construction and use of x- and y-axes, which bear Descartes’ name “Cartesian” today. Wallis, J. Algebra 1673, cap. LXVI (Vol. II, p.286) in Smith, D. E. A Source Book in Mathematics McGraw-Hill: New York, 1929 p. 46ff
[391] Boyer, C. The History of the Calculus and its Conceptual Development Dover Publications: New York, 1949 p.214
[392] Cheyne, C. and Pigden, C. “Pythagorean Powers or a Challenge to Platonism” Australasian Journal of Philosophy (1996) 74, 639-645. Leibniz may respond in the same way to Franklin, J. “Mathematical Necessity and Reality” Australasian Journal of Philosophy 67(3), Sept 1989, 11-17.
[393] Smith, D. E, A Source Book in Mathematics
Dover Publications, New York 1959, quotes from Riemann’s 1854 dissertation Über
die Hypothesen welche der Geometrie zu Grunde liegen
[394] Nikulin, D. V. Matter, imagination and geometry: ontology, natural philosophy and mathematics in Plotinus, Proclus and Descartes Ashgate, Aldershot and Burlington 2002, p.xi
[395] Chaitin, G.J. “The Search for the Perfect Language” a talk delivered at The Perimeter Institute for Theoretical Physics, Monday, September 21, 2009. Text accessed at http://www.umcs.maine.edu/~chaitin/pi.html 15 April 2012
[396] Leibniz, G. W., Specimen Dynamicum 1695 in Loemker, p.442
[397]
Loemker 1969, p.221
[398] Cheyne, C. and Pigden, C. “Pythagorean Powers or a Challenge to Platonism” Australasian Journal of Philosophy (1996) 74, 639-645, p.645
[399]
Loemker, p.384
[400] Wolfe, H. E. Introduction to Non-Euclidean Geometry Holt, Rinehart and Winston New York 1945, p.17ff. See esp. pp.20-21 regarding Wolfe’s objections to Playfair’s Axiom.
[401] Ibid., pp. 6, 8
[402]
Loemker, p.384
[403]
1686, Wiener, p.304
[404]
Raynaud, P. “Leibniz, Reason and Evil” in McCarthy, J. C., (ed. and trans.) Modern Enlightenment and the Rule of Reason
CUA Press Washington D.C. 1998, p.152
[405] Franklin, J. The Science of Conjecture: Evidence and probability before Pascal J. H. Press Baltimore and London 2002, pp.150-1
[406]
Loemker, p.478
[407] Wiener (a) efficient and final causes, first half of p.524 from “The Principles of Nature and of Grace, based on Reason” 1714 second half of § 3, and (b) regarding God as architect, or maker of efficient causes, and God as legislator, or maker of final causes first half of p.551-2 from Monadology 1714, §§ 89-90.
[408]
c.1696, Loemker, pp.477-9
[409]
Ibid., Loemker p.588 middle of first
paragraph
[410] A detailed study of the Second Law of Thermodynamics is beyond the scope of this dissertation. However, Hans Reichenbach’s explanation and proof is entirely statistical at the microscopic level. At the macroscopic level it is entirely Empirical except where he is reasoning inductively from the microscopic. See Reichenbach, H. The Direction of Time University of California Press, Berkeley and Los Angeles 1956, pp.49ff and 145ff. Note that Leibniz was of the mind that time does not exist as an independent physical reality; it is merely a creation by humans for ordering events. Similarly, pure space is merely an invention by humans for giving objects position. Leibniz writes, “Time is the order non-contemporaneous things” and, “Extension is the quantity of space” but “It is false to confound extension, as is commonly done, with extended things, and to view it as substance.” 1715 “Metaphysical foundations of mathematics” in Wiener, p.202
[411] The literature on this topic is vast. For a summary, see http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/seclaw.html accessed 26 Feb 2012
[412] It is not only the Leibnizian corpus that casts doubt on the Second Law of Thermodynamics. Prigogine says, “...it must be recognized that the formulation of the second law seems to us today to be more a program than a well-defined statement, because no recipe was formulated by either Thomson or Clausius to express the entropy change in terms of observable quantities. This lack of clarity in its formulation was probably one of the reasons why the application of thermodynamics became rapidly restricted to equilibrium, the end state of thermodynamic evolution. For example, the classic work of Gibbs, which was so influential in the history of thermodynamics, carefully avoids every incursion into the field of nonequilibrium processes (Gibbs 1975).” Moreover, and noting that only irreversible processes contribute to entropy production, recently “a complete change in perspective has arisen, and we begin to understand the constructive role played by reversible processes in the physical world.” (emphasis in original) Prigogine, I. From Being to Becoming W. H. Freeman and Company, San Fransisco 1980, p.78
[413]
“On destiny or mutual dependence” in Wiener 1951, p. 571
[414]
Parkinson, G. H. R. “Science
and metaphysics in Leibniz’s ‘Specimen Inventorum’” Studia Leibnitiana Vol. 6, 1974 p.4
[415]
Ibid., p.15
[416]
Leibniz writes, “Herr Dreier of Königsberg has aptly observed that the true
metaphysics which Aristotle sought, and which he called [Greek: tên
zêtoumenên], his _desideratum_, was theology.” Theodicy PG edition 2005, p.244 §184
[417]
Leibniz, G. W. Theodicy Project
Gutenberg edition 2005, p.28 accessed at www.gutenberg.org
[418] Ibid., §§48-51, pp.101-102
[419]
§5 in Loemker,p.638
[420]
Loemker, p.556 start of 2nd last paragraph
[421] At http://web.ceu.hu/yehuda_einstein_and_god.pdf accessed 30 Oct 2010, §9, pp.4-5. This is quoted as a statement by Albert Einstein in many locations, but we have been unable to find the primary source.
[422]
Leibniz, G.W. “Towards a Universal Characteristic” 1677, Wiener, p.17
[423] Brown, R. C. Leibniz, unpublished, Chapter 11 “Epilogue” p.167
[424] Antognazza, M.R. Leibniz: An intellectual biography Cambridge University Press 2009, pp.241-243
[425]
1702, Loemker, pp.554-560
[426]
Ibid., §14 p.640, and p.641 5th, 4th
and 3rd last lines
[427]
“Precepts for advancing the sciences and arts” 1680, Wiener, p.49
[428] At http://plato.stanford.edu/entries/leibniz-mind/ section 1 second last paragraph, accessed 30 Oct 2010
[429]
Loemker, pp.364-5, i.e. last paragraph of p.364 spilling over to first
paragraph p.365
[430]
Ibid., p.592 last para. 4th line
[431]
Theodicy, p.244 §186, p.428 §21
accessed at www.gutenberg.org
[432]
Monadology §49 in Latta, p.245
[433]
Ibid., §8 p.220
[434]
Ibid., §1, p.217
[435]
Ibid., §16, p.226
[436]
1705, Loemker , L. E. Gottfried Wilhelm Leibniz: Philosophical Papers and
Letters 2nd ed. D. Reidel, Dordrecht Holland 1969 p.586
[437]
Monadology §14, p.224
[438]
Ibid., p.225 mid-footnote 25
[439]
See esp. the 9th and 10th last lines of the 2nd para. p.590 of Loemker, and the
overall 1702 essay on universal harmony Loemker, pp.574-591
[440]
Ibid., p.587
[441]
Monadology §14, p.251 §61
[442]
Latta writes, “the material
action and re-action throughout the universe, such that a change at any one
point affects every other, is a symbol of the Pre-Established Harmony among the
Monads.” Ibid., p.251 footnote
96
[443]
1702, Loemker p.554 2nd paragraph
[444]
Ibid., p.577
[445]
Ibid., §4 Loemker p.638
[446]
Inference from Monadology, p.247 §53;
“On wisdom” c.1693, Wiener p.77
[447]
Loemker p.583
[448]
Loemker p.583
[449]
Loemker p.583 last 20 lines
[450] At http://plato.stanford.edu/entries/leibniz-mind/ section 1 second last paragraph, accessed 30 Oct 2010
[451] “In the course of his analysis he shows himself a Nominalist. To Leibniz nothing exists independently of individuals. Common properties shared by individuals do not actually exist in re; they are purely creations of the mind.” Brown, R. C. Leibniz unpublished, Chapter 4, pp.32-33
[452]
1704, Loemker pp.535-6
[453]
1702, Loemker p.583
[454]
1704, Loemker p.536
[455]
For example, geometry would exist without the universe, but it would have no
object. As Leibniz puts it, “For it is, in my judgement, the divine
understanding which gives reality to the eternal verities, albeit God's will
have no part therein. All reality must be founded on something existent. It is
true that an atheist may be a geometrician: but if there were no God, geometry
would have no object.” Theodicy p.243
§184
[456]
Loemker p.583
[457]
Loemker, p.365 last paragraph
[458]
Loemker, pp.536-7
[459] Brown, R. C. Leibniz, unpublished, Chapter 4 “A Young Central European Polymath Between the Scholastics and the Moderns” p.33
[460] Rodriguez-Pereyra, G. “Nominalism in Metaphysics” 2008 revised 2001, in Stanford Encyclopedia of Philosophy http://plato.stanford.edu/entries/nominalism-metaphysics/ accessed 21 Feb 2012, with which the popular source Wikipedia agrees at http://en.wikipedia.org/wiki/Nominalism accessed 20 April 2011
[461] The “least time” principle of the refraction of light is an example. This “idea” makes sense when we embed the concept of light in the physical world. Theoretically, we may have been able to derive it a priori, but it is unlikely that it ever would have been. Nonetheless, the principle is a consequence of the Best-ness of the universe.
[462] The Corpus Hermeticum implies this, because it posits the concept of “a nature that enables a thing to come to be” and “a nature that prevents a thing from ever existing”. (This also relates to the concept of “the good” as the final cause in Plato’s Timaeus and Phaedo.) This is a unifying concept across all things in the universe and all things that do not exist. If we understand why one thing exists, then we will understand why every thing exists. If there is something missing in our understanding of one thing, then – because the existence of each thing is based on a foundation common to all things – our understanding of everything else must also be incomplete. Corpus Hermeticum II, §§12-13 in Copenhaver, B.P. (trans. and ed.) Hermetica Cambridge University Press 1992, p.11
[463]
“The principles of nature and of grace based upon reason” 1714 Loemker p.636,
§1
[464]
Monadology, p.262 §78
[465]
1705, Loemker pp.586-7
[466]
Also consistent is Hermes Trismegistus. “There is nothing that is not a product
of the cosmic fecundity. In moving, it makes all things live, and it is at once
the location and craftsman of life.” Corpus Hermeticum IX §6 Copenhaver, p.28.
Also see Corpus Hermeticum IX §8 in Copenhaver, p.29. In Corpus Hermeticum V
§5, Trismegistus writes that motion is indispensable to – indeed, part of – the
fecundity of the cosmos and “This is the order of the cosmos, and this is the
cosmos of order” in Copenhaver p.19.
[467]
Monadology, p.245 §49
[468]
“Hence it comes that our soul has so many means of resisting the truth which it
knows, and that the passage from mind to heart is so long. Especially is this
so when the understanding to a great extent proceeds only by faint thoughts, which have only slight power
to affect, as I have explained elsewhere. Thus the connexion between judgement
and will is not so necessary as one might think.” Theodicy, p.311 §314
[469] C.f. Hill, J.C.R. Leibniz’s metaphysics of intentionality PhD 2008, National University of Singapore
[470]
Loemker, p.588 middle of first paragraph
[471] Robinson, H. “Dualism” 2003 revised 2011, in Stanford Encyclopedia of Philosophy. Accessed at http://plato.stanford.edu/entries/dualism/ 21 Feb 2012
[472]
Monadology, p.228 §17
[473]
Loemker, p.588 7th last line
[474]
Monadology, p.245 §49
[475]
Monadology, p.245 footnote 79
[476] Without referring to monads, Friedrich Schiller addressed the impact of the body on the soul and vice-versa in “On the connection between the animal and spiritual nature in Man”, Letter V of Schiller’s Philosophical Letters Tapio Riikonen, T. and David Widger, D. (eds) accessed at http://www.gutenberg.org/files/6799/6799-h/6799-h.htm on 1 Jan 2011
[477]
Monadology, p.263 §79
[478]
Loemker, p.638 §5
[479]
C.f. Hill, J.C.R. Leibniz’s metaphysics
of intentionality PhD 2008, National University of Singapore
[480]
1705, Loemker p.586 and Monadology pp.244-5
§48
[481] To Hermes Trismegistus, death is merely dissolution. Corpus Hermeticum VIII §4 in Copenhaver pp.25-26, Corpus Hermeticum XII §16 in Copenhaver pp.46-47
[482]
Restated in Leibniz’s 1714 essay at Loemker p.636 §1 last two lines, and p.637
§4 first two lines.
[483] For Hermes Trismegistus, there is nothing which is empty. Even hollow containers are “full of air and spirit”. Hermes finds it almost blasphemous to imply that any of the things that are, is empty. Rather, all that is is full of substance. Corpus Hermeticum II §§10-11 in Copenhaver, pp.10-11
[484] Loemker, p.592 (Letter to Hansch,
July 25, 1707)
[485] Corpus Hermeticum IV §4 in Copenhaver, pp.15-16
[486] For, to Cusa, when treating of an individual mind, he says that the mind is the first and the most simple image of the divine enfolding. The mind comprises the power of understanding, reasoning, imaging and sensing as its elements, meaning these various powers can be combined in multifarious ways. When the Philosopher asks the Layman where the mind gets these powers, the Layman answers, “From unity,” which would seem to correspond to Hermes’ mixing bowl which was sent down by God. Miller, C.L. (trans. and intro.) Nicolaus of Cusa Idiota de Mente Abaris Books, New York 1979, pp.51, 83. Credit goes to Miller for tying together p.51 and p.83 and putting them neatly side-by-side in his introduction. The connection we have drawn and not taken from Miller is that between Cusa’s conception of mind as coming “from unity” and the mixing bowl of Hermes.
[487]
Loemker p.589
[488]
Monadology, p.262 §77
[489]
1714, Loemker p.638 §6 last paragraph and p.637 §4
[490]
Monadology, p.262 §77
[491]
Theodicy, §89 p.171
[492]
Langley, A.G. (ed.) New Essays on Human
Understanding, 1949, p.97
[493]
Theodicy, pp.40-41
[494]
Theodicy, §118 p.188
[495]
Theodicy §118, p.189
[496]
Theodicy §70, p.160
[497] Hermes wrote that all creatures should multiply, though of course only intelligent creatures would be able to comprehend the diktat: “[G]od immediately spoke a holy speech: ‘Increase and increasing and multiply in multitude, all you creatures and craftworks.’” Corpus Hermeticum I §18 in Copenhaver, p.4. Of course, in Genesis I:28, God directed God to, “Be fruitful and multiply.”
[498]
Monadology, pp.264-265 §§82-83
[499] Hermes wrote, “Nothing is more godlike than mind, nothing more active nor more capable of uniting humans to the gods and gods to humans. … Blessed is the soul completely full of mind, wretched the soul completely empty of it.” (Corpus Hermeticum X §23 in Copenhaver, p.35) This and the mixing bowl or monadic “Mind” of Hermes (Corpus Hermeticum IV §4 in Copenhaver, pp.15-16) suggest that Hermes regards soul’s embrace of mind as a choice for humans in the degree to which a human mind imbibes of Mind. Hermes does not suggest that animal souls can choose to embrace Mind more, in order to be elevated to the form or state of humans. However, Hermes does say that the greater the degree to which a human soul embraces Mind, the closer they reach to God (though Hermes uses a lower case “g” and sometimes refers to “gods”). Whether a human soul embraces body – and its pleasures – or Mind is a choice that it is within the power of humans to make. This is consistent with Leibniz’s view of a continuum of activity and passivity between human minds. The more active have to a greater degree embraced Mind, in the Hermetic sense.
[500]
Loemker, p.365 14th last to 10th last lines of 2nd-last paragraph
[501]
Loemker p.590 last 3 lines of last paragraph, and Monadology p.248 §56
[502]
“On the method of distinguishing real from imaginary phenomena” Loemker, p. 365
[503]
Loemker, p.640 §13, 2nd sentence, 2nd paragraph
[504] A possible example of galactic events directly impinging on humanity is given by the paper on the effect of our solar systems passage through a spiral arm of the Milky Way on earthly climate. See Shaviv, N.J. “Cosmic Ray Diffusion from the Galactic Spiral Arms, Iron Meteorites, and a possible climatic connection?” Physical Review Letters 2002 Vol. 89 Issue 5
[505] The same apparent contradiction is in Hermes when he says, “sensation and understanding enter from outside … but the cosmos got them once and for all when it came to be, and, having got them, it keeps them by god’s agency.” (Corpus Hermeticum IX §8 in Copenhaver p.29) How can it be said that the cosmos has them once and for all when god’s agency is needed to keep them? Further, as we know from Hermes, once something is, it can never not be. We can only suppose that the contradiction is to be resolved as with Leibniz. The “agency” by which these things continue to exist is only indirectly of God; directly, it is of the cosmos which was fathered by God, to use Hermes’ words.
[506]
Monadology, p.246 §51
[507]
1702, Loemker p.556 final paragraph
[508]
Loemker p.639 §10
[509]
Monadology pp.228-9 §17
[510] Robinet, A. (ed.) Malebranche, N. Oeuvres complètes de Malebranche Vrin Paris 1958–84, Vol. II, p.312. Quote in Lennon, T.M. and Olscamp, P.J. (trans.) Malebranche, N. The Search for Truth and Elucidations of the Search for Truth Cambridge University Press 1997
[511]
Monadology, p.247 §53
[512]
Also, “The dominion of his
will relates only to the exercise of his power, he gives effect outside himself
only to that which he wills, and he leaves all the rest in the state of mere
possibility.” Theodicy, p.243 §183. Also
see Loemker p.640 §12.
[513]
Theodicy, p.313
§310
[514]
Theodicy, p.399
§8
[515]
Loemker p.640 2nd half §15 which is related to last 4 lines of Loemker p.639 §9
[516]
Monadology p.248 §55
[517]
Loemker p.590, end of first paragraph
[518] This indicates that Leibniz has much more fine-grained criteria than Hermes Trismegistus who says, to paraphrase, what is will always be and what is not never will. (Corpus Hermeticum II §13 in Copenhaver p.11) Or has Hermes assumed the best possible world doctrine in referring to that which has it in its nature to be? Even if he has, Leibniz’s view is more refined because Leibniz considers the unfolding of events so that what may be now is different from what may be after, say, the earth has completed its current orbit around the sun. Hermes regards being as a timeless or eternal matter, whereas Leibniz allows for change and trajectories of events.
[519] Leibniz did believe that it is scientifically possible for water to turn into wine, and for the body of Christ to be in several places simultaneously. He wrote metaphysical explanations as to how these phenomena are possible, which we will not address in this thesis.
[520]
Loemker p.575, 16th line from the bottom
[521]
1705, Loemker p.590
[522]
“Hence it comes that our soul has so many means of resisting the truth which it
knows, and that the passage from mind to heart is so long. Especially is this
so when the understanding to a great extent proceeds only by faint thoughts, which have only slight power
to affect, as I have explained elsewhere. Thus the connexion between judgement
and will is not so necessary as one might think.” Theodicy, p.311 §314
[523]
1705, Loemker p.590; Loemker p.637 §4 first paragraph; Loemker p.640 §14;
Loemker p.640 §15 first half
[524] Loemker, p.595 (Letter to Hansch, July 25, 1707)
[525]
Monadology, p.245 §48
[526]
Monadology, §53
[527]
Loemker, p.587, 6th last line of 1st paragraph
[528] Leibniz, G. W. Second letter to Samuel Clarke 1715-16 paragraph 8, “I do not say the material world is a machine or watch that goes without God’s interposition, and I have sufficiently insisted that the creation wants to be continually influenced by its creator. But I maintain it to be a watch that goes without wanting to be mended by him; otherwise we must say that God bethinks himself again. No, God has foreseen everything. He has provided a remedy for everything beforehand. There is in his works a harmony, a beauty, already pre-established.” in Loemker, p.679
[529]
c.1690, Loemker pp.366-8
[530]
c.1690, Loemker pp.367
[531]
p. 368, emphasis added
[532]
p.536 first paragraph
[533]
Ibid., p.368
[534]
Ibid., p.367
[535]
Letter to Electress Sophia in 1691, footnote 4, Loemker p.369
[536]
“Reflections on the doctrine of a single universal spirit” 1702, Loemker, p.555
[537] Dionysius the Areopagite, Mystical Theology http://www.esoteric.msu.edu/VolumeII/MysticalTheology.html accessed 26 October 2010. Another translation is at http://www.ccel.org/ccel/pearse/morefathers/files/areopagite_06_mystic_theology.htm accessed 28 June 2011 which is Parker, J. (trans.) Dionysius the Areopagite, Works London: James Parker and Co. 1897, transcribed by Roger Pearse, Ipswich, UK, 2004.
[538] “For by the resistless and absolute ecstasy in all purity, from thyself and all, thou wilt be carried on high, to the superessential ray of the Divine darkness, when thou hast cast away all, and become free from all.” Section I, Caput I “What is the divine gloom?” of Mystic Theology accessed at http://www.ccel.org/ccel/pearse/morefathers/files/areopagite_06_mystic_theology.htm 28 June 2011
[539] Ibid.
[540] Loemker, p.368
[541] Ibid.
[542] Ibid.
[543]
Loemker, p.283
[544]
Wiener, pp.78-80
[545]
Loemker, p.537, 4th paragraph
[546]
Loemker, p.536, end of 1st paragraph
[547]
Letter to Burcher de Volder professor at the University of Leyden, 30 June
1704, Loemker, p.536
[548]
Loemker, p.536, end of 1st paragraph
[549]
Loemker, p.583
[550]
God is not like “the blind nature of the mass of material things, which acts
according to mathematical laws, following an absolute necessity, as the atoms
do in the system of Epicurus.” Theodicy,
p.399 §8. Yet the fact that he acts for the best has a certain comprehensible
and reasoned harmony to it, if not “absolute necessity”.
[551] On the other hand, the example in (b), for Leibniz, might lead to something better, just as Westphalian sovereignty followed the 1648 Treaty of Westphalia which concluded the Thirty Years War. The example in (c) is humans acting like the divine in improving the universe and enhancing its plenitude.
[552]
Monadology, pp.228-9 §17
[553]
Loemker, pp.639-640 §11
[554]
Monadology, p.228 §17
[555]
Monadology, p.229 §18
[556]
Loemker, pp.639-640 §11
[557]
Loemker, p.640 §13 first paragraph, and Loemker p.641 §16 first sentence
[558]
Loemker, p.640 §13, 1st paragraph
[559]
Milton, J. Paradise Lost Book III,
line 108 in Ricks, C. (ed.) Paradise Lost
and Paradise Regained The Signet Classic Poetry Series, The New American
Library, New York 1968
[560]
See Thomas Aquinas’ Summa Theologica accesssed at http://www.newadvent.org/summa/ 26 May 2011
[561] For Cusa, see De Docta Ignorantia and Idiota de Mente, and for Einstein see http://web.ceu.hu/yehuda_einstein_and_god.pdf accessed 30 Oct 2010
[562]
Loemker p.638 §5
[563] Loemker, p.592 (Letter to Hansch, July 25, 1707)
[564] Recall the above footnote regarding Plato’s view on reason versus sense perception. In The Republic (602 d), Plato says, “A stick will look bent if you put it in the water, straight when you take it out, and deceptive differences of shading can make the same surface seem to the eye concave or convex; and our minds are clearly liable to all sorts of confusions of this kind.”
[565]
Theodicy §310, p.314
[566]
Bennett, J. (trans.), Leibniz, G.W. New
Essays on Human Understanding , pp.97-99 and Chapter III, 1st
ed. Feb 2005, amended April 2008. Accessed at www.earlymoderntexts.com/jfb/leibne.pdf 10 May 2011
[567]
Theodicy §65, p.109
[568] Bennett, J. (trans.), Leibniz, G.W. New Essays on Human Understanding Book IV Chapter XII §13 “Philalethes: Although I recommend experimentation, I don’t lack respect for probable hypotheses; they can lead us to new discoveries and are at least great helps to the memory. But our mind is very apt to go too fast, and to be content with flimsy conjectures rather than taking the time and trouble needed to test them against a multitude of phenomena.” 1st ed. Feb 2005, amended April 2008, pp.212-213 Accessed at www.earlymoderntexts.com/jfb/leibne.pdf 10 May 2011
[569]
Ibid., p.19
[570] Hermes writes, “Both sensation and understanding flow together into humans, intertwined with one another, as it were. For without sensation it is impossible to to understand, and without understanding it is impossible to have sensation.” Corpus Hermeticum IX §2 in Copenhaver p.27 Even Plato’s Meno, written centuries after the Corpus Hermeticum, bears this out because the uneducated servant boy is given prompts in the form of appropriate questions and a diagram to aid him in “remembering” geometrical truths.
[571]
Langley, G.A. (trans.), Leibniz, G.W. New
Essays on Human Understanding Open Court, La Salle, Illinois 1949, p.105
[572]
Loemker, p.640 §13 1st paragraph
[573]
Loemker, p.640 §13 2nd paragraph
[574] There is evidence that the way that the “plastic brain” is structured is partly a result of culture, causing physically different wiring of synapses between cultures. Thus the question goes beyond sense perception but to what the term objective means and what it means to think; these concepts might have different meanings to radically different brains. This does not mean that ideals differ between minds, but perceptions of ideals could differ with the resultant conclusion that what a particular mind perceives as idea I is in fact idea J or is K which is not an idea at all. Refer to Doidge, N. The Brain That Changes Itself: Stories of Personal Triumph from the Frontiers of Brain Science Penguin 2007
[575] Gose, M. Right Reason: Milton’s Ethical Standard http://globalvillage.pepperdine.edu/GoseWriter/rreason.html accessed 4 October 2010
[576] Milton, J. Paradise Lost Book III, line 108 in Ricks, C. (ed.) Paradise Lost and Paradise Regained The Signet Classic Poetry Series, The New American Library, New York 1968
[577] Nicolaus of Cusa wrote that he experienced an epiphany during a sea voyage, in which God opened up his understanding.
[578] Echoed by Alexander Hamilton http://www.catholiceducation.org/articles/history/us/ah0015.html accessed 4 Oct 2010 “Freedom was a great treasure, Hamilton agreed with the ideologues, propagandists, social engineers, and manipulators of all ages, but it must be organized against abuse. Unrestrained man’s freedom degenerated into license and anarchy.”
[579] ca.1690, in Loemker pp.367-370 at p.368
[580] Davies, P. The Mind of God Penguin Books Camberwell Victoria 1992, p.173
[581] Ibid.
[582] Kepler refers to plenitude in passing when he says that “mere space without body is a contradiction” and “Being has precedence over not being”. M.C., Chapter XI, p.129 Duncan, trans., Abaris Books
[583] Davies, P. The Mind of God Penguin Books Camberwell Victoria 1992, p.175
[584] We return to this below under the heading “Interdependence of ideas”.
[585] From http://www.siue.edu/~evailat/psr-Lut-Er.html accessed 4 Oct 2010
[586]
Milton, J. Paradise Lost Book III,
lines 100 to 130 in Ricks, C. (ed.) Paradise
Lost and Paradise Regained The Signet Classic Poetry Series, The New
American Library, New York 1968
[587] “We must rather act in accordance with the presumptive will of God, so far as we are able to know it, trying with all our might to contribute to the general welfare…” with emphasis in the original. (Loemker, p.305 §4)
[588]
“Discourse on metaphysics” 1686 in Loemker, p.307
[589]
Leibniz writes that it is, “true that God co-operates in evil in the actual
performance of introducing these forms into matter” Theodicy, p.353 §381. Separately, Leibniz explains, “how it is to
be understood that God's will takes effect, and concurs with sin, without
compromising his wisdom and his goodness.” p.399 §8
[590]
“On the true theologia mystica”
c.1690, Loemker p.368
[591]
“Discourse on metaphysics”, Loemker, p.304
[592]
“Discourse on metaphysics” §30, 1686, Loemker 1969 p.322
[593]
“Reply to Bayle’s Dictionary article Rorarius”, 1702, Loemker 1969 p.582
[594]
Leibniz’s Preface to his New Essays Concerning
Human Understanding, p.2
[595] Hill, J. C. R. Leibniz’s metaphysics of intentionality PhD Thesis, National University of Singapore 2008, p.3, viewed 27 August 2010 <https://scholarbank.nus.edu.sg/bitstream/handle/10635/16595/HillJCR.pdf?sequence=1>
[596] Franklin, J. The Science of Conjecture: Evidence and probability before Pascal J. H. Press Baltimore and London 2002, pp.150-1
[597] Loemker, p.592 (Letter to Hansch,
July 25, 1707)
[598]
Hill, J.C.R. Leibniz’s metaphysics of
intentionality PhD National University of Singapore 2008, p.iii
[599]
Hill, J.C.R. Leibniz’s metaphysics of
intentionality PhD National University of Singapore 2008, pp.8-9
[600]
Loemker, p.309
[601]
Farrer, A. (commentator), Huggard, E.M. (trans.) Theodicy, p.12
[602] “God is the sun and light of souls … and this opinion has not been invented only today. In addition to the Holy Scriptures and the Fathers, who were always more Platonists than Aristotelians, I recall having observed long ago that at the time of the Scholastics, several believed that God is the light of the soul … ‘the active intellect of the rational soul’. The Averroists gave this a bad turn of meaning, but others … understood it in a way worthy of God and capable of elevating the soul to a knowledge of its true good.” (“Discourse on metaphysics” 1686 §28, Loemker, p.321)
[603]
Leibniz, G. W. “Discourse on
metaphysics” 1686 §28, Loemker, p.321
[604]
Leibniz, G. W. “Discourse
on metaphysics” 1686, Loemker p.319
[605] Rodriguez-Pereyra, G., “Nominalism in Metaphysics” Stanford Encylopedia of Philosophy Zalta, E.N. (ed.) Article first published 2008 and substantially revised 2011 accessed at http://plato.stanford.edu/entries/nominalism-metaphysics/ 28 July 2012
[606]
Hill, J.C.R. Leibniz’s metaphysics of
intentionality PhD National University of Singapore 2008, p.10
[607]
Loemker p.320 §26, lines 6-9
[608]
Hill, J.C.R. Leibniz’s metaphysics of
intentionality PhD National University of Singapore 2008, p.11
[609]
Leibniz, G. W. “Discourse on
metaphysics” 1686, §23, Loemker p.318
[610]
Ibid., §25 p.319
[611]
Hill, J.C.R. Leibniz’s metaphysics of
intentionality PhD National University of Singapore 2008, pp.32-33
[612] Friedrich Schiller reiterated this concept of a single act of thought: “Reason insists, in accordance with its necessary laws, upon absolute totality of perception, and without letting itself be rebuffed by the necessary limitation of the power of imagination, the mind requires from it a complete summation of all the parts of a given quantum in one simultaneous mental image.” Johnson, S. trans. Of the Aesthetic Estimation of Magnitude (1793) Accessed at http://www.schillerinstitute.org/transl/Schiller_essays/magnitude.html 1 Jan 2011
[613] Hermes Trismegistus identifies God with and as good, whereas Leibniz identifies God with that than which there is no greater. Nicolaus of Cusa identifies God with “Absolute Maximality” or that than which there is no greater, meaning the infinite. De Docta Ignorantia Book I Chapter 2 §5 in Hopkins, J. (trans.) Nicolaus of Cusa De Docta Ignorantia The Arthur J. Banning Press, Minneapolis 3rd ed. 1988, p.6
[614]
Riemann, B. Gesammelte mathematische Werke,
wissenschaftlicher Nachlass und Nachträge “Collected mathematical works,
scientific deductions and supplements” , Springer-Verlag, Berlin Heidelberg
New York 1990. In particular, see I.
Zur Psychologie und Metaphysik “I. On psychology and metaphysics” pp.541-2 of Fragmente
philosophischen Inhalts “Fragments philosophical content” for a concise
introduction.
[615]
Langley, A.G. (ed.) Leibniz, G.W. New
Essays Concerning Human Understanding Open Court Publishing, Illinois 1949,
p.95ff
[616] As does Hermes Trismegistus. Corpus Hermeticum II §13 in Copenhaver, B.P. Hermetica Cambridge University Press 1992, p.11
[617] Wiener, p.xxvi
[618] Letter to Berthey 1677, Ibid., p.xxiv
[619] This is to borrow the words of Meli but to use them in a different context. Meli, D.B. Equivalence and Priority: Newton versus Leibniz Clarendon Press, Oxford 1993, p.19
[620] Franklin, J. “Two caricatures II: Leibniz’s best world” International Journal for the Philosophy of Religion 52 (2002), 45-56, accessed at http://web.maths.unsw.edu.au/~jim/caric2.pdf
[621] Bagni, G.T. 94–103 “Exhaustion argument and limit concept in the History of Mathematics: educational reflections” In Furinghetti, F. Kaiser, S. & Vretblad, A. (Eds.), Proceedings of HPM–2004, History and Pedagogy of Mathematics, Uppsala July 12–17, 2004, at p.4
[622]
Leibniz, G. W. “Analytical quadrature by means of centers of gravity” in Child,
J. M. (ed. and trans.) The early
mathematical manuscripts of Leibniz; tr. from the Latin texts by Carl Immanuel
Gerhardt with critical and historical notes by J. M. Child The Open Court
Publishing Company 1920, republished by the University of Michigan Library, p.66
[623] “In 1629, Cavalieri, a Jesuati – an adherent to the Rule of St. Augustine – was appointed to the chair in mathematics at the University of Bologna, a post he occupied until his death, largely through the recommendation of Galileo, who proclaimed him the foremost Italian mathematician of the day. His Geometria indivisibilibus contains the first systematic exposition, as it pertains to the principles of summation, of what we now know as the calculus. He accomplished this by employing the concept of ‘indivisibles,’ or ‘infinitesimals,’ which served the same purpose as ‘the method of exhaustions’ employed by Archimedes and other Greek mathematicians. In principle these approaches were the same but the system of notation for indivisibles was much more concise and convenient.” Accessed at http://www.brown.edu/Facilities/University_Library/exhibits/math/textfr.html#bon.html 11 May 2011. The original text referred to is Geometria indivisibilibus continuorum nova quadam ratione promota ... Typographia de Duciis, Bologna 1635
[624] Child, J.M. (ed. and trans.) The early mathematical manuscripts of Leibniz; tr. from the Latin texts published by Carl Immanuel Gerhardt with critical and historical notes Open Court Publishing Company, Chicago and London, 1920 reprinted by University of Michigan Library, pp.208-9 n.25
[625] Ibid., p.215. Also see further quotes at p.215 n.36.
[626] Ibid., p.119
[627] Miller, C.L. (trans. and intro.) Nicolaus of Cusa Idiota de Mente Abaris Books, New York 1979, pp.71-73
[628] Katz, M. and Sherry, D. “Leibniz’s Infinitesimals: Their Fictionality, Their Modern Implementations, and Their Foes from Berkeley to Russell and Beyond” Erkenntnis Springer 2012 Vol. 77
[629] Grosholtz, E. “Was Leibniz a Mathematical Revolutionary?” pp.126, 133
[630]
Child, J.M. (ed. and trans.) The early mathematical
manuscripts of Leibniz; tr. from the Latin texts published by Carl Immanuel
Gerhardt with critical and historical notes Open
Court Publishing Company, Chicago and London, 1920 reprinted by University of
Michigan Library, p.38
[631]
Ibid., p.39
[632] Brown, R. C. Leibniz unpublished, Chapter 11 “Epilogue”, p.177. For Leibniz, dy/dx “is simply a device to calculate the subtangent t from the equation dy/dx = y/t.” However, “Had Leibniz been influenced by Newton” we would expect Leibniz to have treated dy/dx as an independent object, function or rate of change.” We have avoided the priority debate and it remains beyond the scope of this thesis. However, Leibniz knew that curves can be used to represent all manner of quantity such as distance, velocity, rate of change itself, or any quantity one seeks to measure. So dy/dx might have represented many quantities and Leibniz was certainly aware that it was a rate of change. By finding the subtangent, Leibniz knew that he was opening up the facility for calculation with any quantity one happened to be analyzing in the form of the curve to which one sought the subtangent. He was aware of the generality of mathematics and he knew that the tangent represented an instantaneous rate of change. However, it is true that Leibniz’s focus on dy/dx was due to its role with respect to the Characteristic Triangle which was so important to the development of his method.
[633]
Child, J.M. (ed. and trans.) The early mathematical manuscripts of
Leibniz; tr. from the Latin texts published by Carl Immanuel Gerhardt with
critical and historical notes Open Court Publishing Company, Chicago and
London, 1920 reprinted by University of Michigan Library, p.77
[634]
Ibid., p.75
[635]
Ibid., p.80
[636]
Ibid., p.80
[637]
Ibid., pp.97-103
[638]
Ibid., pp.93-103
[639]
Ibid., p.98 and in footnote 34 on the same page,
Child notes the advance in Leibniz’s thinking exhibited by Leibniz’s use of the
idea that one quantity is “infinitely small compared
with” another allowing the former to be discarded.
[640] This is referred to by Boyer many times uncritically. See Boyer, C. The History of the Calculus and its Conceptual Development Dover Publications: New York 1949
[641] Spivak, M. A Comprehensive Introduction to Differential Geometry Vol. 2 Publish or Perish, Inc., Houston, 1999 p.160
[642] Torretti, R. “Nineteenth Century Geometry” in Stanford Encyclopedia of Philosophy 1999 revised 2010. Accessed at http://plato.stanford.edu/entries/geometry-19th/ on 18 March 2011. Indeed, the word Euclid uses is “Αίτήματα” (pronounced aiteemata) from Mourmouras, D.E. (ed.) Euclid The Elements accessed at http://www.physics.ntua.gr/~mourmouras/euclid/book1/elements1.html 19 March 2011 which Google translates as “requests”. The Greek Lexicon translates “aίτημα” as a request or demand. (An Intermediate Greek-English Lexicon: founded upon the Seventh Edition of Liddell and Scott’s Greek-English Lexicon, Oxford University Press)
[643] Keyser, C. J. Mathematical philosophy : a study of fate and freedom E. P. Dutton & Co., New York 1922
[644] Boyer, C.B., Merzbach, U.C. A History of Mathematics John Wiley and Sons 2011, Chapter 11 “The Islamic Hegemony”
[645] Mathematics Illuminated Geometries Beyond Euclid published by Annenberg Learner §8.3 Non-Euclidean Geometry accessed at http://www.learner.org/courses/mathilluminated/units/8/textbook/03.php 4 June 2011
[646] Ibid.
[647] Lobachevsky, N.I., Papadopoulos, A. (trans.) Pangeometry, European Mathematical Society Zurich, 2010 p.3
[648] Ibid., p.4 Papadopoulos notes that the definition of plane is akin to Leibniz’s: that surface which divides space into two congruent parts.
[649] Ibid., p.75
[650] Riemann, B., Clifford, W.K. (trans.) “On the Hypotheses which lie at the Bases of Geometry” Nature, 8 (1873), pp.14-17, 36-37. Accessed at http://www.mat.ub.es/EMIS/classics/Riemann/index.html on 15 March 2011 and in Spivak, M. A Comrehensive Introduction to Differential Geometry Vol. 2 Publish or Perish, Inc., Houston, 1999 p.160
[651]
Wolfe, H.E. Non-Euclidean Geometry,
Holt, Rinehart and Winston 1945,
p.8
[652] Ibid., p.6
[653] Brown, R. C. “The Deconstruction of Mathematics” in Brown, R. C. Are Science and Mathematics Socially Constructed? A Mathematician Encounters Postmodern Interpretations of Science World Scientific Publishing Co. Singapore 2009, p.165
[654]
Joyce, D.E. Euclid’s Elements accessed at http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI16.html
on 27 Apr 2011
[655] Riemann, B., Clifford, W.K. (trans.) “On the Hypotheses which lie at the Bases of Geometry” Nature, 8 (1873), pp.14-17, 36-37. Accessed at http://www.mat.ub.es/EMIS/classics/Riemann/index.html on 15 March 2011 and in Spivak, M. A Comrehensive Introduction to Differential Geometry Vol. 2 Publish or Perish, Inc., Houston, 1999 p.160
[656] Riemann, B., Clifford, W.K. (trans.) “On the Hypotheses which lie at the Bases of Geometry” Nature, 8 (1873) pp.14-17, 36-37. Accessed at http://www.mat.ub.es/EMIS/classics/Riemann/index.html on 15 March 2011 and in Spivak, M. A Comrehensive Introduction to Differential Geometry Vol. 2 Publish or Perish, Inc., Houston, 1999 p.160
[657]
Huffman, C. “Archytas” first
published 26 Jun 2003 with substantial revision 25 Jul 2007, accessed at http://plato.stanford.edu/entries/archytas/
on 27 Apr 2011
[658]
Rivest, F. and Zafirov, S. “Duplication of the cube” undated, accessed at http://www.cs.mcgill.ca/~cs507/projects/1998/zafiroff/
on 27 Apr 2011 where it is said of Archytas’ solution, “it is not a
construction in a plane but a bold construction in three dimensions,
determining a certain point as the intersection of three surfaces of
revolution, (1) a right cone, (2) a cylinder, (3) a tore or anchor-ring with inner diameter nil. The intersection of
the two later surfaces gives (says Archytas) a certain curve (which is in fact
a curve of double curvature), and the point required is found as the point in
which the cone meets this curve.”
[659]
Wolfe, H.E. Non-Euclidean Geometry,
Holt, Rinehart and Winston 1945,
p.9
[660]
Ibid., p.13
[661]
Ibid., p.10
[662] Ibid.
[663] Ibid., p.12
[664] Kepler, J. Duncan, A.M. (trans.) Mysterium Cosmographicum Abaris Books, Janus Series, Opal Publishing 1981 pp.123-125
[665] But since ideas do not exist, then to Leibniz the archetypes do not exist, but they do subsist.
[666]
Huggard, E.M. (trans.), Leibniz, G. W. Theodicy:
Essays on the Goodness of God, the Freedom of Man and the Origin of Evil
Released 24 Nov 2005 [EBook #17147] from C.J. Gerhardt's Edition of the Collected Philosophical Works, 1875-90
La Salle, Illinois 61301
[667] Leibniz, G.W. Tentamen anagogicum c.1696, in Loemker, pp.477-9
[668] Duncan, A.M. (trans.), Kepler, J., Mysterium Cosmographicum Abaris Books, Janus Series, Opal Publishing 1981 p.125, footnote 2, fifth-last line
[669] Lipton, B. “Embracing The Immaterial Universe Toward a New Noetic Science” Shift: At the Frontiers of Consciousness No. 9, Dec 2005-Feb 2006, pp. 8-12 the quarterly publication of the Institute of Noetic Sciences (IONS); website: www.noetic.org accessed at http://www.brucelipton.com/biology-of-belief/embracing-the-immaterial-universe
[670] Henry, R.C. “The Mental Universe” Nature 436: 29, 2005
[671] Also see Penrose, R. The Emperor’s New Mind Penguin 1991
[672] Loemker, L. E., trans. and ed., Gottfriend Wilhelm Leibniz: Philosophical Papers and Letters 2nd ed. D. Reidel Publishing Company, Dordrecht Hollard 1969, p.387; first ed. 1956 University of Chicago Press, Chicago
[673] Loemker, L. E. Struggle for Synthesis: The Seventeenth Century Background of Leibniz’s Synthesis of Order and Freedom Harvard University Press: Cambridge USA 1972, p.127
[674] Ibid., pp.127-8
[675] Mercer, C. Leibniz’s Metaphysics: Its Origins and Development Cambridge University Press: Cambridge, 2001, pp. 80-110
[676] Raynaud, P. “Leibniz, Reason and Evil” in McCarthy, J. C., ed. and trans. Modern Enlightenment and the Rule of Reason CUA Press Washington D.C. 1998, p.150
[677] Ibid., p.152
[678]
Ibid.
[679]
Ibid.
[680] Plauché, G. A. writes in his essay Ancient vs. Modern Political Thought 9 April 2011, “The premodern political philosophers whose thought achieved dominance – Plato, Aristotle, Cicero, Augustine, Aquinas, and others – were primarily concerned with the search for right order. They generally accepted essentialism, teleology, eudaimonism, and natural law-type virtue or deontic ethics. Modern political philosophers tend to be more concerned with the search for peace and order, consequentialist or deontic ethical systems concerned primarily with social order, and are more likely to be rationalists or empiricists and base their theories on reductionist foundations.” In this sense, Kepler’s philosophy was more pre-modern while Leibniz’s was modern. Accessed at http://gaplauche.com/blog/2011/04/09/ancient-vs-modern-political-thought/ 12 May 2011
[681]
Leibniz, G. W. c1696 “Tentamen Anagogicum:
An anagogical essay in the investigation of causes” Loemker 1969, p.477
[682] These are precisely the words of Cicero in De natura rerum. Also see Boyle V, 515ff. quoted in Burtt, E.A. The Metaphysical Foundations of Modern Physical Science Routledge London 1932 2nd ed. reprinted 1950 p.189, “the consideration of the vastness, beauty, and regular motion of the heavenly bodies; the excellent structure of animals and plants; besides a multitude of other phenomena of nature, and the subserving of most of these to man; may justly induce him as a rational creature, to conclude, that this vast, beautiful, orderly, and (in a word) many ways admirable system of things, that we call the world, was framed by an Author supremely powerful, wise and good, can scarce be denied by an intelligent and unprejudiced considerer.”
[683] Klemm from Ad majorem Dei gloriam published in Nouvelles de la Republique des Lettres Amsterdam 1695, pp. 218-220
[684] c.1696, Loemker 1969, pp.477-9