Mathematics: Modern and Ancient
© 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, and would the fathers of the discipline be
willing to approve of them? In this paper, I will expose the principal
differences between the ordinary (i.e. formal) mathematics, primarily
understood as a problem solving apparatus, and, on the other hand, Pythagorean,
or universal mathematics - as Descartes is prone to call it - which is genuine
ancient mathematics, existing solely to the purpose of discovering, depicting
and formulating those principles which underlie experience and physical
reality. The differences are so vast that they entirely distort the genuine
meaning of the terms such as mathematics, axiom, and related.
A careful reader of philosophical literature will
certainly notice that the kind of mathematics taught in today’s schools is in
many ways different than the one elaborated in famous ancient writings. It is
generally accepted that Pythagoras and his followers invented the word
“mathematics”, which is derived from the Greek word mathema and primarily means learning, study, science. The
Pythagoreans also conceived methods, goals and procedures of the mathematical
discipline. They were divided into mathematicians
(in ancient Greek, mathematikoi), or
those who possess true, systematic knowledge, and listeners - as we could translate the Greek word akusmatikoi - who only transmit
information without proper understanding. Since then, mathematics suffered
through many changes. From being a science which aims to discover, depict and
formulate those principles which underlie experience and reality, the
principles of all things, it turned to being a mere play of words and symbols,
much more similar to poetry than science. Its laws and principles are
considered to be completely abstract and unrelated to physical reality.
First,
I will discuss the modern the view on the nature of mathematics, adequately
expressed by the following words of Albert Einstein:
...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. (Sidelights on Relativity, Geometry and Experience)[1]
Einstein was one of the scientists who questioned
the consistency of physical laws, the universality of physical constants.
Therefore, it was unacceptable to him that laws and principles which underlie
physical reality should be investigated in the same manner as the laws of
mathematics. The laws of mathematics are necessary in the most rigorous sense
of the word, however, in reality,
nothing appears to be absolutely necessary, unchangeable and timeless. I believe
Einstein was right to consider physical laws inconsistent, but I also believe
he made a crucial mistake when he rejected the possibility of an unchangeable law, or principle,
responsible for any kind of change.
Such a principle would certainly be mathematical, and physical entity at the
same time.
In a
few points, what do the upholders of this view assume? And to which paradoxes
they have to commit themselves?
(1)
There are no unchangeable, necessary aspects of physical reality, and,
therefore, the concept of physical law is not rigorous at all. Since the laws
of mathematics express certain necessary, universal truths, there is no way
they could relate to physical reality.
(2)
Mathematical entities cannot be experienced through physical senses. Instead,
they are considered to be mere constructions of the human mind. As a
consequence, we are forced to believe that mathematical entities reside in a
completely different realm than the one displayed to our senses.
(3)
Points (1) and (2) inevitably lead to the so-called two worlds paradox. The relation between mathematical, unchangeable
entities and sensible things remains unexplained and completely obscure. These
two classes of objects inhabit two distinct worlds. Some commentators have
unreasonably accused Plato of committing himself to this paradox, but his
“theory of Forms”[2] was
actually an ambitious (and most probably, successful) attempt to explain this
relation between mathematical and sensible entities.
Being
that modern mathematicians do not even try to solve this paradox, they have to
represent it as nonsensical in certain way. So, they escape to formalism. Mathematical entities become
nothing but mere symbols, completely deprived of their reality and meaning. For
example, David Hilbert gave much support to formalistic conception of
mathematics. In such mathematics, we are allowed to define symbols, operations,
rules of derivation, etc. in such a manner that they need not refer to any
real, existing objects. All that matters is that the internal coherence of
formal system is achieved. It’s a closed system. And it really is difficult not to compare such mathematics to
science-fiction, or poetry, simply because it doesn’t even try to have some
explanatory, scientific value. It finds its application in technology, arts and
empirical sciences only as some kind of modeling device, a problem solving
apparatus.
Now,
let’s turn to the second view which represents the genuine conception of
mathematics, usually ascribed to Pythagorean mathematicians. Aristotle writes:
...the
Pythagoreans, as they are called, devoted themselves to mathematics; they were
the first to advance this study, and having been brought up in it they thought
its principles were the principles of all things. (Metaphysics, 985b)
And these words of Plato add up nicely:
Geometry
is the knowledge of that which always is, and not of a something which at some
time comes into being and passes away. (The
Republic, 527b)
As we all know, for ancient Greeks mathematics was
primarily geometry. They didn’t use
algebra, or symbolic representation. Instead, they literally draw their proofs, they represented them
visually.
What
beliefs could be attributed to the upholders of this view?
(1)
Contrary to the modern conception of mathematics we have just discussed, there
should be certain unchanging, immovable and timeless aspects of physical
reality which constitute the proper object of mathematical studies, and thus,
receive the name of “mathematical entities”. If such entities didn’t exist in
reality, we wouldn’t be able to conceive them in our minds. Furthermore, their
relationship to sensible things is that of cause
to effect.
(2)
Mathematical entities are not mere constructions of the human mind. Instead,
they are abstracted from experience.
For example, mathematical entities could be some elemental, simple things which
combine into sensible, physical objects.
(3) By
abstraction, by observation and constant reexamination of sensory data, the
mind dicovers mathematical entities and the first principles of reality. Then,
it is able to deduce all the facts
about world beginning from these first principles, which have to be some kind
of patterns encompassing all possibilities. That makes them universal laws.
Clearly,
the modern and the ancient view on the nature of mathematics seem to be
opposite in many ways. While the concept of axiom
is rather loose in contemporary mathematics, which means that certain
propositions are considered to be axiomatic only by convention, by agreement
among mathematicians, in ancient, Pythagorean mathematics axioms are supposed
to symbolically express objective
realities, things which are necessary not in our opinion but by themselves and
thus constitute the fundamental principles of reality. Axioms might be
understood as the building blocks of reality, the elements which cannot be
decomposed into smaller elements but combine in order to produce any imaginable
shape, or possibility.
What
things might be proper candidates for the status of axioms? According to
Descartes, who expressed fascination with ancient, or universal mathematics – as he tends to call it – in his early
epistemological treatises, the subject of mathematics should be certain
relationships and proportions abstracted from sensible objects. Descartes
writes:
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. (Rules, p. 7)
Then, in Discourse on Method, he continues:
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. (Discourse on Method, p.
47)
This philosophical language is familiar to any
reader of Plato’s Timaeus, in which
the structure of the so-called “world-soul” is explained according to musical
ratios[3].
In
conclusion, the word “mathematics”
has received a very different meaning than it used to have in ancient times. Of
particular interest would be to observe and affect the future developments in
mathematics. I believe there still exists a chance to bring back the glory
mathematics used to have, to return to those concepts which made it so special
and powerful after all. One of the primary goals should be the reconciliation
of formalism and realism, as well as the clarification of the role of symbolic
representation. We should not take sides, as I did in this paper. Instead, we
should follow the middle path, incline towards some kind of fusion between
ancient science and modern mathematics, modern information technologies, etc.
And of course, we mustn’t forget the psychological, cathartic dimensions of
mathematics which were so important to Pythagoreans and Platonists. The
understanding of the fundamental principles of reality leads to
self-improvement, bliss, and liberates us from many fears which might arise out
of ignorance of such principles. So, in my opinion, the goal of mathematician
is to see the unfolding of time and worldly events as some kind of
logico-deductive structure and to increase the certainty of his predictions, being aware that they could never become absolutely certain.
References
Aristotle: Metaphysics in Barnes, Jonathan (ed.): The Complete Works of Aristotle, Vol. 2;
(Princeton
University Press, Princeton, New Jersey 1984), p.1559
Barker, Stefan: Filozofija matematike; (Nolit, Beograd
1973), pp.166-170 and pp.176-180
Descartes, René: Discourse on the Method of
Rightly Conducting the Reason in Great
Books of the Western World, Vol. 31; (The University of Chicago, Chicago 1952), p.47
Descartes, René: Rules for the Direction of the
Mind in Great Books of the Western
World, Vol. 31; (The
University of Chicago, Chicago 1952), p.7
Einstein, Albert: Sidelights on Relativity;
http://www.gutenberg.org/dirs/etext05/slrtv10.txt
(20.10.2007.)
Kandić, Aleksandar: Dekartov projekat univerzalne matematike;
http://frag005.com/dekartov-projekat.htm
(30.05.2007.)
Kandić, Aleksandar: Mathematics as a Science of Reality: Pyhtagoras, Plato, Descartes;
http://frag005.com/mathematics-athens.htm
(22.07.2007.)
Plato: The Republic in Hamilton, Edith and
Cairns, Huntington (ed.): The Collected
Dialogues of
Plato;
(Princeton University Press, Princeton, New Jersey 1989), p.759
Plato: Timaeus in Hamilton, Edith and Cairns,
Huntington (ed.): The Collected Dialogues
of Plato;
(Princeton
University Press, Princeton, New Jersey 1989), pp.1165-1166