MATHEMATICS AS A
SCIENCE OF REALITY:
PYTHAGORAS, PLATO,
DESCARTES
© 2007
Aleksandar Kandić
ABSTRACT:
Albert Einstein
proclaimed that "as far as the laws of mathematics refer to reality, they are
not certain; and as far as they are certain, they do not refer to reality". How
much modern conceptions of mathematics differ from the genuine? In this paper it
will be demonstrated not only that mathematics was primarily conceived as a science which purports to discover the
underlying laws and principles of reality, but also that all theoretical or
practical disciplines aiming to acquire systematic, organized knowledge, are to
be regarded as branches of mathematics. Herein, firstly we consider
the nature of axiomatic systems; in the sequel, it will be explained the way in
which both deduction and induction are to be regarded as the forms of intuition; and finally, we clarify the
distinction between the ordinary (i.e. formal) mathematics, and the universal mathematics, as Descartes is
prone to call it. To provide plausible argumentation for the claims proposed in
this paper, Descartes' Regulae,
Plato's Timaeus and Republic VI-VII, as well as some
relevant passages in Aristotle, will be used beside the other published
sources.
KEY
WORDS:
universal
mathematics,
reality, intuition, axiom
1.
Introduction
The first
disciplines to study the unchanging, timeless aspects of nature and reality, the
first ones to look into the language of
nature, were philosophy and mathematics. Aristotle thought that
mathematical method should be applied only to investigation of
immaterial things.
Material, movable things belong to the sensible world, and therefore should be
the subject of physics[1].
But, Aristotle was also aware that
...the so-called
Pythagoreans, who were the first to take up mathematics, not only advanced this
study, but also having been brought up in it they thought its principles were
the principles of all
things.[2]
It is widely
accepted that Pythagoras was the first to coin the word philosophy, which literally translates
to the lover, or friend of wisdom, and mathematics, which is derived from Greek
word mathema, and primarily means learning, study, science. Mathematical knowledge is
nothing like sensorial, unreliable and without justification. It is systematic,
well-organized knowledge, in the sense that conclusions must follow necessarily from premises, or axioms,
postulates. Philosophy and mathematics go hand by hand, because they both aim to
acquire systematic knowledge, as well as to found such knowledge on certain constants, certain unchanging, universal
laws and prinicples abstracted from experience.
It is one of
my deepest beliefs that if one wants to learn about specific concept, the best
way to begin is to look for the earliest evidence, particularly before the subsequent and contemporary
developments take effect on one's understanding of given concept. Among the
first thinkers who begun to value proof and the powers of reasoning, who disregarded mythical
thinking, were Thales, Heraclitus, Zeno of Elea, and the most influential of
all, Pythagoras. Therefore, if one wants to learn about philosophy and mathematics - these two being
inseparable - one should try to dig out as much evidence as possible on the
thinkers mentioned: that includes their own writings, if available, and the
writings of their criticists, commentators, biographers, etc. Mathematics is
what Pythagoras thought it to be, not an empty word which may be given any
meaning that pleases us mostly, in particular moment.
How, then,
should we interpret Albert Einstein's claim that
...as far as the
laws of mathematics refer to reality, they are not certain; and as far as they
are certain, they do not refer to reality?[3]
And what does it mean to say that
mathematics is a science of reality,
or, if you wish, a science of what is
real? Einstein here, obviously, confuses mathematics not only with physics,
but also with applied, formal mathematics (perhaps he wouldn’t make that mistake
if he was acquainted with Aristotle’s thoughts on this topic). This is just as
bad as confusing things necessary in
themselves with things necessary in
our opinion. The other kind of necessity is sometimes called by Aristotle hypothetical. Let me explain. Physics
deals with laws which are not necessary in the most rigorous sense of the word,
but are only considered necessary
because we cannot find any exceptions to them. For example, it is not necessary
that a pencil, or any other object, falls to the ground when thrown, or that
gravitational constant on our planet has the same value as it has now. But,
since certain events repeat without exception, we proclaim these repetitions to
be necessary laws, and since certain
values change very slowly over time, we proclaim them to be constants. It is only in our opinion, it is only in our
interpretation of reality, that these things are necessary. Also, formal
mathematics allows the symbols and relations to be defined in such a manner that
conclusions necessarily follow from
the premises, but, at the same time, not to be in any way connected to reality,
to the real state of things[4].
Therefore, its laws are necessary only in
our opinion, and not in
themselves.
Genuine, ancient mathematics, as envisioned by
the Pythagoreans, was primarily conceived as a science which purports to discover the
underlying laws and principles of reality, thus providing rational explanations
of experiential phenomena. Mathematical entities are, therefore, abstracted from
sensible things. It is only in our minds that such entities are taken to be
imaginary, transcendent, separated from particulars. They represent the principles which are
necessary in the way that if they ceased to exist, everything else would cease
to exist. They are uncaused, but their relationship to sensible things is
that of cause to effect. It is rather naive to think of sensible, composite
things as causes, since they are also caused and thus require
explanation.
I will try to clarify the
distinction between the ordinary (i.e. formal) mathematics, primarily understood
as a problem solving apparatus, and,
on the other hand, the universal
mathematics - as Descartes is prone to call it - which is genuine ancient mathematics, existing solely to
the purpose of demystifying, depicting and formulating those principles which
constitute the fundamentals of reality. After we consider the notion of axiom, the nature of axiomatic systems,
as well as how the axioms are to be understood if they represent the basic
constituents of reality, we shall explain the way in which both deduction and
induction are to be regarded as the forms of intuition, often naively confused with
perception. It will be demonstrated
that all theoretical or practical disciplines aiming to acquire systematic,
organized knowledge are to be regarded as branches of mathematics. To provide plausible
argumentation for the claims proposed, the Descartes' Regulae, Plato's Timaeus and Republic VI-VII, as well as some
relevant passages in Aristotle’s Metaphysics, will be used beside the other
published sources. Descartes’ earliest philosophical writing has to be one of
the most ambitious attempts to revive the ancient conception of mathematics. The
reader will realize what do the three figures from the title of this paper have
in common, as well as why Descartes suspected that
...they (the
ancients, A.K.) had knowledge of a species of mathematics very different from that which passes
current in our time.[5]
2.
The
Structure of Human Knowledge
In everyday life, we employ the word
"knowledge" in a much weaker sense than epistemologists, or, generally,
philosophers and scientists. To the ordinary men - and by ordinary men I think
of those who uncritically accept numerous beliefs about the world as if those
were facts - knowledge appears to be
equated with mere perception. Truly, in the world of information, in the world
where struggle to survive has priority over any practice of rational,
philosophical thinking, mainly because one is not given enough time to reflect
upon phenomena and events which happen almost automatically, it is very
difficult to realize distinctions such is the one between true knowledge and
sensory knowledge. In such a world, everything is turned upside down: that which
is ephemeral, doubtful, which demands explanation and which is, at least in
metaphysical sense, completely irrelevant, is considered to be factual, known,
and of greatest importance. If I perceive it through my senses, and if it has
certain use, certain function within the system of beliefs I have uncritically
accepted, then I know what it is.
But, what happens if we ask
ourselves about the reasons? What
happens if we bring up the questions such is how do you know? why do you consider certain beliefs to be
undoubtfuly true? and how do you
differ true beliefs from false ones? Those who are primarily oriented
towards practical, productive way of living will try to avoid such questions by
all means, since they could easily put them to despair, confusion and
hopelessness. However, even such people are bound to employ mathematical
thinking, although they might be unaware of it or just won’t admit. Even in
everyday situations, or practical work, a certain level of justification has to
be given to our beliefs. It is up to us how deep we want to go. We realize that
justification has to be founded on some other beliefs, which might be unfounded,
too. It seems that justification entails an infinite chain of beliefs, all of
which might be liable to doubt. But explanation, or proof, has to be finite, and its premises undoubtfuly true. In order to avoid
infinite regression, we have to look for a belief, or several beliefs which are
undoubtfuly true, and thus may
constitute a foundation of our knowledge. Such beliefs may be utilized as
starting points in explanation. Then, we could deduce all facts about the world
with mathematical rigorosity.
Although we never call for it in everyday reasoning, we do incline towards mathematical rigorosity.
Because only then, the sensory knowledge will be justified in the proper sense
of the word.
For Plato, knowledge of particular,
sensory objects is possible, but ephemeral, momentarily and dependant of
perspective. At the end of the The
Republic VI, Plato felt the need to differ among the four cognitive states
which correspond to the levels of justification a person has acquired[6],
respectively. The lowest cognitive state, called eikasia, is unreflected, almost passive
reception of phenomena during which we never raise questions about the origin of
things perceived, used and possessed. We simply act as if there are no deeper
reasons, we imitate surroundings. Next comes pistis, or faith, which together with eikasia constitutes a lower kind of
knowledge, non-systematic and based only on the testimonies of our senses. The
difference between pistis and eikasia lies in the realization that
there exist reasons for which things
are what they are, but a person is not even close to discovering what precisely
these reasons are. The lower among two higher kinds of knowledge is dianoia, or mathematical reasoning.
Mathematicians investigate ex
hypothesi: they postulate certain principles and take them to be starting
points in explanation. Then, they try to deduce as much facts as possible about
the world, in such a manner that conclusions necessarily follow from postulated
principles. But, ordinary mathematicians and physicists admit that their
principles are nothing but hypotheses, as well as that such
hypotheses cannot explain all things.
So, there must be an even higher kind of knowledge which elevates the mind to
the first principles of reality, the highest cognitive state. Plato calls it noesis, or proper understanding. Only in this
state, the mind is able to know things without any help of the physical senses,
as he now possesses the knowledge of the Form of the Good. This “mystical”
object is the proper cause of all things, and thus a starting point in any kind
of explanation. However, it cannot be equated with any of the things perceived
through physical senses.
And this brings us to the discussion
of one of the most important concepts in philosophy, mathematics, and generally,
science, as well as in all disciplines aiming to acquire systematic, justified
knowledge. It is the concept of axiom. The word axiom comes from the Greek word axioma, which means that which is deemed worthy, or that which is
considered self-evident. In Republic VI, Plato used the term archen anupotheton to denote such
unhypothetical first principle. Axiom is a proposition which is accepted without
any proof. In contemporary mathematics, the concept of axiom is very loose. Any
proposition might be considered an axiom, as long as it’s properly defined in
terms of the formal language we employ and integrated into the formal system
through the rules of derivation. On the other hand, in ancient, or universal mathematics, axiomatic
propositions represent certain universal principles, things which are abstracted
from experience and aren’t a mere product of human imagination. They are real.
But, what things fulfill such conditions? What things might be accepted as
self-evident and undoubtfuly true? Is there any such thing at all? Sensory
objects are not proper candidates for the status of axioms, since they are in
constant flux, in constant change. Our senses are unreliable, they show us
things from a certain perspective. If we considered some of our perceptions to
be an axiom, it would be a subjective opinion, and no one else would have to
accept it. Axioms are to be things upon whose existence all observers inhabiting all possible worlds agree, independently
of perspective, interpretation, or circumstances.
In Regulae, Descartes clarifies the
relationship between axioms, or simple things which might be utilized as
starting points in explanation, and composite, contingent things, through the
relationship between absolute and relative:
I call that absolute which contains within itself
the pure and simple essence of which we are in quest. Thus the term will be
applicable to whatever is considered as being independent, or a cause, or simple, universal, one,
equal, like, straight, and so forth; and the absolute I call the simplest and
the easiest of all, so that we can make use of it in the solution of questions.
... But the relative is that which,
while participating in the same nature, or at least sharing in it to some degree
which enables us to relate it to the absolute and to deduce it from that by a
chain of operations, involves in addition something else in its concept which I
call relativity. Examples of this are found in whatever is said to be dependant,
or an effect, composite, particular,
many, unequal, unlike, oblique, etc. These relatives are further removed from
the absolute, in proportion as they contain more elements of relativity
subordinate the one to the other.[7]
Descartes here speaks of axioms as
simple things, building blocks of reality. These things are uncaused, unchanging
and immovable, but they combine into composite things. They literally become objects. If there exists a common
element of all things, it should be considered an axiom. We could employ any word or
symbol to signify such thing, but first we must discover what it is. Axioms are
not linguistic propositions only, or symbols which may be given any meaning:
they represent the basic constituents of reality. Axioms represent the proper
causes of all things, the proper explanations of all phenomena. It is the relationship between absolute and
relative, between cause and effect, which is problematic and which has led
thinkers like Einstein to make claims about mathematics such as the one quoted
in the first section. The knowledge of the laws and principles of reality
cannot help us to predict events: this is one of the greatest illusions of
Western science, the illusion that knowledge of the cause entails knowledge of
the effect. Effect is ephemeral, chaotic, and thus cannot be known in the same
manner as the cause, the principle, the absolute.
There are two more things I want to
discuss. First, is it really appropriate to speak of axioms, of plurality of principles?
Whenever we deal with a plurality of things which are of the same kind, we feel
the need to postulate one thing by which they become what they are. A plurality
of principles would require an explanation of a higher order, a principle by
which these principles are principles. So, instead of inferring
from particular towards universal, we should rather infer from universal towards
particular. We should descend from
the first principle. And second, if axioms (and propositions deduced from them)
refer to things, if they’re not
symbolic constructions entirely deprived of meaning, then the structure of our knowledge should
reflect the structure of reality. All of our words are nothing but the names
of things, and names are countless in number since one thing may be called
countless ways. What is important here is that the order of symbols, words and
propositions reflects the natural
order: in scholastics, this was known as the distinction between ordo cognessendi (the order of
knowledge) and ordo essendi (the
order of reality). If this condition isn’t fulfilled, then what we’re dealing
with surely isn’t the case of knowledge, but perhaps, someone’s
fantasy.
Having all this in mind, we conclude
that true knowledge is essentially mathematical. In all philosophical
disciplines, such as metaphysics, ontology, epistemology, or even ethics and
aesthetics, as well as all theoretical or practical disciplines aiming to
acquire systematic, organized knowledge, we sort facts in such a manner that
principles and axioms, which represent the universal, absolute and self-evident,
constitute a foundation of our epistemological structure, while relative facts
are to be deducible from the first
principles. Aristotle thought that each science has its own principles, or archai[8].
Only theology deals with principles which are necessary in the way that if they ceased to exist, everything else
would cease to exist. So, one of the principal differences between universal mathematics and the other
disciplines which we regard as branches of mathematics is in the understanding
of necessity: the first deals with things which are necessary in themselves, the other with things
which are necessary only in our
opinion, or let’s say, in particular circumstances.
3.
Intuition
= {Deduction,
Induction}
In order to increase amount of our
knowledge and gather new facts about the world, we have to ensure that our
conclusions are true. And they will be true if - and only if - both our premises
and implications are true, in the sense that the structure of our reasoning
reflects the structure of reality. Furthermore, the conclusions should follow necessarily from the premises. But, as
we have seen, the notion of necessity has two senses. Our minds are capable of
relating any two things, or facts, in
such a manner that their relationship is like that of cause to effect, and
although it may not be necessary in
itself, it surely becomes necessary in our opinion. Descartes thought
that
...none of the
mistakes which men can make (men I say, not beasts) are due to faulty inference; they are caused merely
by the fact that we found upon a basis of poorly comprehended experiences, or
that propositions are posited which are hasty and groundless.[9]
This means
that we never err in employing the rules of derivation, for these rules are
inherent faculties of human mind. We actually err in differing the true premises, or beliefs,
from false ones. Technically, any proposition can be deduced from any other
proposition, but the proper question is whether these propostions and the causal
relations esstablished between them correspond to the real state of things.
There can be no faulty inference, but
there could be faulty, or unjustified knowledge.
Descartes was also aware that by inference
...we
can get only things from words, cause from effect, or effect from cause, like
from like, or parts or the whole itself from the parts...[10]
Unlike
formalists, Descartes points out that things are the objects of our inquiry.
It is quite pointless to make inferences from words to other words, as much as
it is pointless to give a piece of paper with the word “water“ written on
it to a thirsty person who asked for some water. Mathematical and logical
formalism completely distort the primary idea of inference, because they take
words instead of things to be their objects. When I say that “A implies B“ I mean that a thing
called by the name “A“, or described by the proposition “A“ implies, or causes,
a thing called by the name “B“, or described by the proposition “B“. However,
formalists tend to ignore this fact. They pretend these words and symbols have
no meaning at all, so that they could inspect their relations only by looking
into the logical form. This seems ironical, because the very same logical form
they want to inspect is grounded on the meaning of sentences expressed in
natural / scientific language, and again, these sentences employ the words which
refer to certain things, objects, certain relations, etc. So, we emphasize,
words do not refer to other words, but to things.
How, then, do we acquire new knowledge? In the second section, we have
agreed upon the following facts: 1) true knowledge cannot be equated with
perception, 2) it has to be founded on certain beliefs which are undoubtfuly,
universally true and which are called axioms, 3) axioms cannot be propositions
about sensory, unstable objects, but some universal principles. And if these
axioms are self-evident truths, the starting points in explanation, then they
cannot be deduced from other truths. They have to be known by immediate comprehension, or intuition. Descartes
writes:
By intuition I
understand, not the fluctuating testimony of the senses, nor the misleading
judgment that proceeds from the blundering constructions of imagination, but the
conception which an unclouded and attentive mind gives us so readily and
distinctly that we are wholly freed from doubt about that which we understand.
Or, what comes to the same thing, intuition is the undoubting conception
of an unclouded and active mind, and springs from the light of reason alone; it
is more certain than deduction itself, in that is simpler, though deduction, as
we have noted above cannot by us be erroneously conducted.[11]
Clearly,
Descartes' understanding of intuition is very much different from the popular
one. Ordinary people, as well as some philosophers, usually employ the term
“intuition“ as
synonymous with the term “perception“. By intuition they understand immediate
comprehension of sensory objects.
However, if that would be the case, then we wouldn’t need three different terms
such as perception, intuition and knowledge, but only one. Because all of them
would signify a single faculty of mind. Descartes, however, thinks of intuition
as an ability to abstract certain
necessary properties of sensory objects and to investigate them independently of
things to which they belong, to investigate them in reason alone. These
properties, or simple, absolute things, although perceived through the senses,
come in a mixture with other, contingent things and properties. One has to
employ his reasoning abilities in order to abstract that which is simple, stable
and undoubtful from that which is composite, unstable and uncertain. Both simple
and composite things are shown to us immediately, but the first have to be
discerned from the others by reasoning. Therefore, let it be said that intuition is rationalized perception,
and, on the contrary, perception is
non-rationalized intuition. Knowledge is awareness of the relationship of
these two terms.
Being that many things we know are not self-evident, but still are
considered absolutely certain because they are deduced from a principle or
premise intuitively known, through
series of implications which are also intuitively known, it is reasonable to
speak of deduction as a form of
intuition. According to Descartes, deduction is nothing but intuition
realized in time, with the help of memory and imagination: it is a step by step
examination of the process of inference, during which the mind passes over all
facts, or premises, involved[12].
We often infer immediately, without thinking about the process of inference
itself. When we discern the steps we undertook to reach the conclusion, they
appear to us as seperate and chained. Thus, we think of intuition as deduction.
This could be visualized in the following manner:
A -> A’ -> A’’ -> A’’’
-> A’’’’ -> A’’’’’ -> ...
“A“ stands for axiom, the self-evident truth or
unhypothetical first principle which constitutes the foundation of our
knowledge. If the causal relations esstablished between the first principle and
all subsequent conclusions are considered necessary, then it is natural to assume
that absolute certainty may be attributed to all of the premises in the chain,
and not only to the first one. If absolute certainty is transferable through
implication, then any of the subsequent premises might be considered a starting
point in explanation, an axiom. But, as the number of premises involved inclines
towards unlimited, we are not able to
hold all of them in memory anymore. The
further they are removed from the first principle, or absolute truth, the more
uncertain they appear to be.
Now, for a moment, stop thinking
about these premises as linguistic propositions, and try to think of them as
elements, the building blocks of reality. Notice how the structure represented
by our formula reflects the structure of reality, the structure of perception.
Each event has a succeeding event, and
they keep changing unpredictably. All that is undoubtful and self-evident is
the moment. All that is necessary is
the change itself. We can never predict what kind of change will take place (at
least not with 100% certainty). This process continues indefinitely, just like
the working of perpetuum mobile. The
goal of mathematician is to see the unfolding of time and worldly events as a
logico-deductive structure and to increase the certainty of his predictions.
Interestingly enough, Descartes thought of induction as a reliable source of
knowledge, too. As the main, and possibly single condition for validity of
inductive inference he states adequate or methodical enumeration[13].
This means that we have to provide relevant examples of particular things by
which we infer about certain universal principle. His treatment of induction is
not as thoughtful as that of intuition and deduction, and although he admits
that induction is much more often in danger to be defective and erroneous than
intuition is, he looses several important facts out of sight. Instead of
criticizing our understanding of inductive inference in somewhat Humean way, I
would propose here that induction is a form of intuition as well. For, just as
deductive inference begins from certain principle which is intuitively known, so
too inductive inference begins from certain observations which are immediately
comprehended (in the sense that they are abstracted from sensory objects), and
then progresses towards the universal principle. Induction is a reversed process of
deduction, during which a mind
infers from particular towards universal. By constantly reexamining, or
redefining the hypothesis which
aims to express the first principle, the common property of all things, one
eventually reaches the first, unhypothetical principle and then, just as in
Plato’s vision[14]
and with metaphysical necessity
involved, he descends towards particular, contingent things. Let us visualize
the inductive inference in the following manner:
... -> A’’’’’ -> A’’’’ ->
A’’’ -> A’’ -> A’ -> A
Again, “A“ stands for axiom, or unhypothetical first
principle. The knowledge of such principle is induced by potentially countless number of
observations. It is important to see that implications in this formula represent
causal relations which are necessary only in our opinion, and not in themselves. Modern science, in its
unreasonable attempt to predict and control worldly events, errs precisely in
not giving adequate enumeration of examples by which it inductively infers about
the laws and principles of reality: such laws are merely probable, but not necessary. To pretend
that there are no exceptions to these laws is nothing but wishful thinking.
I will remind the reader that we
have discussed deduction and induction primarily as faculties of mind by which
we acquire knowledge of things belonging to the world of experience. We have
discredited formalism as being similar to fantasy. Both deduction and induction
appear to be the forms of intuition,
since immediate comprehension is
involved in the process of inference. But, this immediate comprehension is not
to be equated with perception, or unreflected, passive reception of sensory
objects. Also, it seems that deduction and induction are nothing but the two
directions of the same path of inference. “The way up and the way down are one
and the same“, Heraclitus used to say. As Descartes often points out, it is
highly preferable to found our knowledge on certain undoubtful and absolute truths (such as those of
arithmetics and geometry), but there yet remains a problem of relating them to
the unstable, contingent world of experience. Now that we have clarified what
true knowledge is, and by which faculties we learn new facts about the world, it
is appropriate to discuss universal
mathematics in a more detailed manner.
4.
Universal
Mathematics
A common error is to associate the
concept of universal mathematics with
the name of G.W. Leibniz, as well as with foundation of algebra and artificial
languages[15].
However, the idea of universal
mathematics is old as philosophy, and what happened in the Renaissance and
the centuries to follow is nothing but an attempt to revive this ancient
conception of mathematics. The most notabable figures were Galileo, Giordano
Bruno, and subsequently, Descartes. Descartes’ earliest philosophical writing Regulae, though unfinished, has to be
one of the most ambitious, and most systematic attempts to put the ancient
science back to practice. Unfortunately, it failed. Algebra and analytical
geometry developed, and this opened the door for development of formalism. Mathematical entities and
symbols lost their reality and meaning. Descartes probably anticipated this when
he wrote that
...science known by
the barbarous name algebra should be
extricated from that vast array of numbers and inexplicable figures by which it
is overwhelmed, so that it might display the clearness and simplicity which, we
imagine, ought to exist in a genuine mathematics.[16]
What is, then, universal, or genuine mathematics, and what might be
the subject matter of its study? Let’s hear what else Descartes has to
say:
It was these
reflections that recalled me from particular studies of arithmetic and geometry
to a general investigation of mathematics, and thereupon I sought to determine
what precisely was universally meant by that term, and why not only the above
mentioned sciences, but also astronomy, music, optics, mechanics and several
others are styled parts of mathematics. Here indeed it is not enough to look at
the origin of the word; for since the name “mathematics” means exactly the same
thing as “scientific study”, these other branches could, with as much right as
geometry itself, be called mathematics. ... But as I considered the matter
carefully it gradually came to light that all those matters only were referred
to mathematics in which order and measurement are investigated, and that
it makes no difference whether it be in numbers, figures, stars, sounds or any
other object that the question of measurement arises. I saw consequently that
there must be some general science to explain that element as a whole which
gives rise to problems about order and measurement, restricted as these are to
no special subject matter.[17]
In Discourse on Method, a work which mostly
recapitulates Regulae, he
continues:
But for all that, I
had no intention of trying to master all those particular sciences that receive
in common the name of mathematics; but observing that, although their objects
are different, they do not fail to agree in this, that they take nothing under
consideration but the various relationships and proportions which are present in these
objects, I thought that it would be better if I only examined these proportions
in their general aspect, and without viewing them otherwise than in the objects
which would serve most to facilitate a knowledge of them. Not that I should in
any way restrict them to these objects, for I might later on all the more easily
apply them to all other objects to which
they were applicable.[18]
According to Descartes, the proper candidates for the object of
scientific inquiry are certain proportions. A proportion is an equality among two
ratios, while ratio is a sort of relation in respect
of size between magnitudes of the same kind[19].
A multitude of proportions constitutes something I’d like to call a system of proportions. Why is this so
important? In the second section, we have agreed that composite, sensory objects
cannot be considered axioms, since they also require explanation. Descartes’
idea is, basically, to abstract certain properties of a thing, properties which
might be expressed in terms of ratios
and proportions, and which are
necessary for that thing to be what it is. Later, he could apply these
proportions to all things of the same kind. A ratio is not a material thing: it
is only abstracted from material things, and represented by drawings, formulas,
or pictures in our imagination. And some ratios, being immaterial, immovable and
timeless, might turn out to be things necessary in themselves, while this could never be
said for any equation, or a formula which merely aims to describe chaotic
movements and changes in sensory objects. Ratios are proper candidates for axioms, because they are simple,
absolute, universal, they are applicable to many particulars. Furthermore, they
are known by intuition, since the
abilities of reasoning have to be employed in order to abstract them from
composite, sensory objects shown to us immediately. All of reality is real, but some properties of things
appear to be contingent and some necessary, or stable, the second being proper objects
of scientific, mathematical knowledge. It is the knowledge of timeless and
unchanging aspects of reality.
But, unlike Descartes, Plato not only proclaimed the highest and most valuable
kind of knowledge to be of certain universal measures and proportions. He
attempted to discover what precisely these proportions were. It is highly
probable that Descartes was inspired by Plato’s Timaues, the late-period dialogue, which
is - in opinion of many commentators - an exposition of genuine Pythagorean
cosmology, metaphysics, or simply mathematics[20].
Plato’s Timaeus, without providing any plausible argumentation, goes on to say
that the so-called world soul, an
entity which encompasses all individual souls and living organisms, is
structured according to musical
ratios[21].
Even if he’s completely wrong, even if his theory is nothing more than wishful
thinking, it still makes a perfect example to explain what Descartes meant by
saying that the subject matter of universal mathematics are measures and proportions in
themselves.
It is well-known that Pythagoras and
his followers conducted experiments by which they have demonstrated that all musical ratios can be expressed in terms
of the ratios of magnitudes[22]:
an octave is produced by the ratio of 2:1, a fifth by the ratio of 3:2, a fourth
by the ratio of 4:3, and so on. Therefore, a musician should study musical
ratios independently of the things and instruments which utilize them (such as
strings, jars filled with water, or as in modern times, electronic instruments
and computers). Knowing these ratios, a musician would not only comprehend the
nature of his skill, but also the universal which appears through particular. Also, he could create music
without any help of the auditive senses, only by utilizing mathematical,
theoretical knowledge, by shaping certain material in such a manner that it
produces harmony. This is very
similar to one of Plato’s ideas expressed in The Republic, the idea that anyone who
reaches unhypothetical, universal knowledge will be able to perceive without any
help of the physical senses[23].
On the other hand, Descartes is very clear about the role and purpose of
symbolic representation in mathematical studies, a part of which is called algebra:
...Having carefully
noted that in order to comprehend the proportions I should sometimes require to
consider each one in particular, and sometimes merely keep them in mind, or take
them in groups, I thought that, in order the better to consider them in detail,
I should picture them in the form of
lines, because I could find no method more simple nor more capable of being
distinctly represented to my imagination and senses. I considered, however, that
in order to keep them in my memory or to embrace several at once, it would be
essential that I should explain them by
means of certain formulas, the shorter the better. And for this purpose it
was requisite that I should borrow all that is best in geometrical analysis and
algebra, and correct the errors of one by the other.[24]
So, algebra, just as any other
language, is nothing but an auxiliary
device. One has to keep in mind that things come first, and not the words.
It is pointless to perform operations on symbols that have no meaning, or do not
refer to real things. For example, the musical ratios mentioned in the previous
paragraph can be represented by lines
(by ratios of magnitudes), or by numbers, ciphers. What is important here is
that our drawings and formulas refer to certain ratios abstracted from
experience. We are not supposed to invent symbols, words and formulas first, and
look for the structures which might correspond to them after. Rather, we should
rationalize our experiences and express their necessary, universal components in
terms of formulas. This is crucial difference. Only if we know the meaning of
the symbols, if we know what things they refer to in the real world, we can do
proper mathematics. Everything else is “mathematical
poetry“.
Surprisingly or not, Plato already
had to distinguish between ordinary and universal mathematics. He called the
later dialectics, and made it the
first science whose name is, actually, unimportant[25],
but which studies the universal Forms and their relationship to sensible,
particular things. Dialectics is, in many ways, similar to universal
mathematics. However, Plato’s teaching generated a great deal of confusion, and
opened the door for Aristotle’s criticism[26].
In the dialogues, Plato did not explain very well what the Forms are and how
they relate to particular things, leaving his claims poorly argumented and
mostly written in poetic metaphors, or allegories. One has to read all the
dialogues and put enormous efforts into exegesis of Timeaus, in order to decipher his
”alchemical language”, and, possibly, to reconstruct the theory of Forms[27].
Certain testimonies made by Aristotle gave rise to the myth of the “unwritten
doctrine” according to which Plato, during his esoteric lectures in Academy,
gave quite a different teaching on the Forms and philosophy of mathematics from
that which is written in the dialogues: a teaching more detailed, and better
argumented. But it sure is difficult to make any judgments about this from our
perspective.
The distinction between ordinary and
universal mathematics appears in
Aristotle, too. Whether it is clearer than in Plato, I’d leave it to reader to
decide. In Metaphysics E, Aristotle
explains the task of theology, or the
first philosophy. And he does that through analogy with universal mathematics. Although these
two are not equated, they are found to be similar in certain important aspects.
The following passage, I believe, contains the first appearance of the term
“universal mathematics” in philosophical literature:
...That physics,
then, is a theoretical science, is plain from these considerations. Mathematics
also, however, is theoretical; but whether its objects are immovable and
separable from matter, is not at present clear; still, it is clear that some
mathematical theorems consider them qua immovable and qua separable from matter.
But if there is something which is eternal and immovable and separable, clearly
the knowledge of it belongs to a theoretical science - not, however, to physics
(for physics deals with certain movable things), nor to mathematics, but to a
science prior to both. For physics deals with things which exist separately but
are not immovable, and some parts of mathematics deal with things which are
immovable but presumably do not exist separately, but as embodied in matter;
while the first science deals with things which both exist separately and are
immovable. Now all causes must be
eternal, but especially these; for they are the causes that operate on so
much of the divine as appears to us. There must, then, be three theoretical
philosophies, mathematics, physics, and what we may call theology, since it is
obvious that if the divine is present anywhere, it is present in things of this
sort. And the highest science must deal with the highest genus. Thus, while the
theoretical sciences are more to be desired than the other sciences, this is
more to be desired than the other theoretical sciences. For one might raise the
question whether first philosophy is universal, or deals with one genus, i.e.
some one kind of being; for not even the mathematical sciences are all alike in
this respect: geometry and astronomy deal with a certain particular kind of thing, while universal mathematics applies alike to all. We answer that if
there is no substance other than those which are formed by nature, natural
science will be the first science; but if there is an immovable substance, the
science of this must be prior and must be first philosophy, and universal in
this way, because it is first. And it will belong to this to consider being qua
being - both what it is, and the attributes which belong to it qua
being.[28]
5.
Conclusions
Perhaps our
youngsters make a good point when they claim to dislike algebra. Mathematics
became much like poetry, or fiction: formalists create their own formal
languages and systems, they define symbols, words and rules of derivation
without paying any respect to reality. Some day, their systems may find
application in technology, particularly computing technology, but they may
remain a fantasy as well. It doesn't really matter. All it matters is that the
coherence of formal system is achieved, and that internally everything operates very
well. But, before one gets acquainted with the work of David Hilbert and, for
example, his concept of meta-mathematics, or the work of Kantor, Reimann,
Gödel, as
well as other modern and contemporary mathematicians, one should study the work
of those thinkers who are considered to be the fathers of the mathematical
discipline, for their conception of mathematics is genuine. There is no point in building a
tower without a good foundation.
We have
exposed the principal differences between ordinary, formal mathematics and universal mathematics, as envisioned by
Pythagoras and his followers. The laws of mathematics are to be abstracted from
experience, they have to correspond to real things. They cannot tell us much
about future events, lesser to provide us with 100% accurate predictions. But
they can make a good foundation for our scientific explanations. Unlike formal
mathematics, which takes away the meaning from its symbols and reality from its
entities, true mathematics always keeps a connection to reality. Its goal is to
demystify, depict and
formulate those principles which constitute the fundamentals of experience, and
to express them in the form of propositions organized into the logico-deductive
structure. All sciences incline towards mathematical rigorosity.
If there
exists a unique principle, or at least several universal laws necessary and
sufficient to explain the governing of our world, then these laws seem natural
to be rediscovered over and over again by thinkers and great minds of all times.
True knowledge is of that which is unchangeable, timeless, constant, or - if you
wish - real in the most rigorous
sense of the word. True knowledge is of things which exist objectively and
independantly of our limited perspectives. Without getting himself into the
error of proclaiming that perceptual world is unreal, one should realize that
composite things, being in a state of eternal flux, cannot be proper objects of
knowledge. No matter how stable they might seem, they are still liable to doubt
as much as testimonies of our senses are. Rather, one should value the things
which underlie the changes, which are
the proper cause of changes in
sensible things. Thus, the principle(s) transcend temporal and spatial
existence. Reality is thought of as a mixture of simple, static, unchangeable
elements, and composite, contingent, perishable things, only the first being
proper objects of scientific inquiry. And these objects, I claim, are standing
right in front of our eyes, waiting to be grasped by the mind, waiting to be known.
Yet, even if
we come to know the laws and principles of reality, we are still bounded by the
possibilites of our natural / scientific language. No matter if we do, or do not
intend to communicate the facts acquired, we still have to represent them in our
own minds, by using symbols, or images. And this representation is entirely
subjective. It is the expression of objective reality. Each individual, being
"trapped" in its own,
unique world perspective, has its own, unique interpretation of reality. This is
why we need hermeneutics, and this is why we are mistaken when we proclaim
certain discovery to be entirely
original, to be achieved exclusively by us. Not knowing the language of nature, one fails to see
the common forms reappearing through the words of great minds and
scientists.
And this language of nature is, I believe, the
subject matter of mathematics.
BIBLIOGRAPHY
Printed
sources
Annas, Julia:
An Introduction to Plato's Republic;
New York, Oxford University Press,
1981.
Annas, Julia:
Aristotle’s Metaphysics, Books M and
N; New York, Oxford University
Press,
1976.
Annas, Julia:
Predmeti matematike kod Aristotela in
Gregorić, P. and Grgić, F. (ed.):
Aristotelova
Metafizika, zbirka rasprava; Zagreb, Kružok, 2003.
Aristotel: Metafizika; Beograd, Kultura,
1971.
Aristotle: Metaphysics in Barnes, Jonathan (ed.):
The Complete Works of Aristotle;
Princeton,
New Jersey, Princeton Univeristy Press, 1984.
Aristotle: Nicomachean Ethics in Barnes, Jonathan
(ed.): The Complete Works of
Aristotle;
Princeton, New Jersey, Princeton Univeristy Press, 1984.
Aristotle: Posterior Analytics; in Barnes,
Jonathan (ed.): The Complete Works of
Aristotle;
Princeton, New Jersey, Princeton Univeristy Press, 1984.
Barker,
Stefan: Filozofija matematike;
Beograd, Nolit, 1973.
Burnyeat,
Myles: Platonizam i matematika, preludij
za raspravu in Gregorić, P. and
Grgić, F.
(ed.): Aristotelova Metafizika, zbirka rasprava; Zagreb, Kružok,
2003.
Cornford, F.
M: Mathematics and Dialectics in The
Republic in Allen, Reginald E. (ed.):
Studies in
Plato's Metaphysics; London, Routledge & Kegan Paul,
1965.
Curley, E. M:
Descartes Against the Skeptics;
Oxford, Basil Blackwell, 1978.
Dekart, Rene:
Praktična i jasna pravila rukovođenja
duhom u istraživanju istine in
Dekart, Rene:
Rasprava o metodi; Valjevo, PŽM, 1999.
Dekart, Rene:
Reč o metodi dobrog vođenja svoga uma i
istraživanja istine u naukama in
Dekart, Rene:
Rasprava o metodi, Valjevo, PŽM, 1999.
Descartes,
René: Discourse on the Method of Rightly
Conducting the Reason in Great
Books of
the Western World, Vol. 31;
Descartes,
René: Rules for the Direction of the Mind in
Great Books of the Western
World,
Vol. 31;
Diels,
Hermann: Predsokratovci, fragmenti,
Sv. 1-2;
Ghyka,
Matila: The Geometry of Art and Life;
New York, Dover Publications, Inc, 1977.
Guthrie,
K. S. and Fideler, D. R: The Pythagorean
Sourcebook and Library;
Guthrie,
W. K. C: A History of Greek
Philosophy, Vols. 1-6;
University
Press, 1962-1981.
Hale,
Bob: Is Platonism Epistemologically
Bankrupt? in Schirn, Matthias (ed.): The
Philosophy
of Mathematics Today;
Heath,
sir Thomas: A History of Greek
Mathematics; Vols. 1-2;
Publications,
Inc, 1981.
Heidel,
W. A: The Pythagoreans and Greek
Mathematics in Furley, David J. (ed.):
Studies
in Presocratic Philosophy, Vol. 1;
Huntley,
H. E: The Divine Proportion; New
York, Dover Publications, Inc, 1970.
Lawlor,
Robert: Sacred Geometry; London,
Thames & Hudson Ltd, 1982.
Lemmon, E. J:
Beginning Logic; Indianapolis,
Hackett Publishing Company, 1965.
Livio, Mario:
The Golden Ratio; New York, Broadway
Books, 2002.
Pavlović,
Branko:
Tajne dijaloga Timaj in Platon:
Timaj;
Plato: The Republic in Hamilton, Edith and
Cairns, Huntington (ed.): The Collected
Dialogues of
Plato; Princeton, New Jersey, Princeton University Press,
1989.
Plato: Timaeus in Hamilton, Edith and Cairns,
Huntington (ed.): The Collected Dialogues
of Plato;
Princeton, New Jersey, Princeton University Press, 1989.
Platon: Država; Beograd, BIGZ,
1993.
Platon: Timaj; Beograd, Mladost,
1981.
Popper, sir
Karl: Back to the Presocratics
in
Furley, David J. (ed.): Studies in Presocratic
Philosophy,
Vol. 1;
Popper,
sir Karl: Objective Knowledge: An
Evolutionary Approach;
Press,
1974.
Russell,
Walter: The Universal One, Vol. 1;
Swannanoa, University of Science and
Philosophy,
1974.
Vilijams,
Bernard: Dekart, projekat čistog
istraživanja in THEORIA br. 4, Beograd,
Filozofsko
društvo Srbije, 1996.
Electronic
sources
Einstein,
Albert: Sidelights on
Relativity;
http://www.gutenberg.org/dirs/etext05/slrtv10.txt
(14.07.2007.)
Descartes,
René: The Principles of
Philosophy;
http://www.philosophy.leeds.ac.uk/GMR/hmp/texts/modern/descartes/principles/princindex.html
(14.07.2007.)
Descartes,
René: The
World;
http://www.princeton.edu/~hos/mike/texts/descartes/world/world.htm
(14.07.2007.)
Kandić,
Aleksandar: Dekartov projekat univerzalne
matematike;
http://frag005.com/dekartov-projekat.htm
(30.05.2007.)
Stanford
Encyclopedia of Philosohy: Descartes’
Epistemology;
http://plato.stanford.edu/entries/descartes-epistemology/
(14.07.2007.)
Wikipedia:
Pythagorean
Tuning;
http://en.wikipedia.org/wiki/Pythagorean_tuning
(14.07.2007.)
Wikipedia: Mathesis
Universalis;
http://en.wikipedia.org/wiki/Mathesis_universalis
(14.07.2007.)
[1] Aristotle: Metaphysics,
995a
[2] Ibid,
985b
[3] Address to the
[4] Barker, S: Filozofija Matematike, p. 166-170 and p.
176-180
[5] Descartes, R: Rules
for the Direction of the Mind, p. 6
[6] Plato: The
Republic, 509d-511e
[7] Descartes, op.
cit, p. 8
[8] Aristotle: Posterior Analytics, 76a-b and Metaphysics,
1005a
[9] Descartes, op. cit, p. 3
[10] Ibid, p.
24
[11] Ibid, p. 4
[12] Ibid, p.
4
[13] Ibid, p.
10-11
[14] Plato, op. cit, 511b-c
[16] Descartes, op.
cit, p. 7
[17] Ibid, p.
7
[18] Descartes, R: Discourse on the Method, p.
47
[19] See
[20] Pavlović, B: Tajne dijaloga
Timaj, p. 11-14
[21] Plato: Timaeus,
34c-36e
[22] Guthrie, W. K. C: A
History of Greek Philosophy, p. 212-226
[23] Plato: The
Republic, 511c
[24] Descartes, op.
cit, p. 47
[25] Plato, op. cit, 533d-e
[26] Aristotle, op.
cit, 990b-993a and Nicomachean Ethics
1096b-1097a
[27] It ought to be said that Plato himself never used the expression “theory of Forms”.
[28] Aristotle, op. cit,
1026a