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Date: Thu, 3 Nov 1994 15:44:26 +0200 (EET)
From: Karlis Podnieks
To: QED discussions
Subject: Semantics
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K.Podnieks, Dr.Math.
podnieks@mii.lu.lv
PLATONISM, INTUITION
AND
THE NATURE OF MATHEMATICS
Continued from #1.
Studying mathematics Plato came to his surprising philosophy of
two worlds:
the "world of ideas" (strong and perfect as the "world" of geometry) and
the world of
things. According to Plato, each thing is only an imprecise, imperfect
implementation
of its "idea" (which does exist independently of the thing itself in the
world of ideas).
Surprising and completely fantastic is Plato's notion of the nature of
mathematical
investigation: before a man is born, his soul lives in the world of ideas
and afterwards,
doing mathematics he simply remembers what his soul has learned in the
world of
ideas. Of course, this is an upside-down notion of the nature of
mathematical method.
The end-product of the evolution of mathematical concepts - a fixed
system of
idealised objects, is treated by Plato as an independent beginning point
of the
evolution of the "world of things".
Nevertheless, being an outstanding philosopher, Plato tried to
explain (in his
own manner) those aspects of the human knowledge which remained
inaccessible to
other philosophers of his time. To explain the real nature of idealised
mathematical
objects, Greeks had insufficient knowledge in physics, biology, human
physiology
and psychology, etc.
Today, any philosophical position in which ideal objects of human
thought are
treated as a specific "world" should be called platonism. Particularly,
the philosophy
of working mathematicians is a platonist one. Platonist attitude to
objects of
investigation is inevitable for a mathematician: during his everyday work
he is used
to treat numbers, points, lines etc. as the "last reality", as a specific
"world". This sort
of platonism is an essential aspect of mathematical method, the source of
the
surprising efficiency of mathematics in the natural sciences and
technology. It
explains also the inevitability of platonism in the philosophical
position of
mathematicians (having, as a rule, very little experience in philosophy).
Habits,
obtained in the everyday work, have an immense power. Therefore, when a
mathematician, not very strong in philosophy, tries to explain "the
nature" of his
mathematical results, he unintentionally brings platonism into his
reasoning. The
reasoning of mathematicians about the "objective nature" of their results
is, as a rule,
rather an "objective idealism" (platonism) than the materialism.
A platonist is, of course, in some sense "better" than the
philosophers who
consider mathematical objects merely as "arbitrary creatures of human
mind".
Nevertheless, we must distinguish between people who simply talk about
the
"objective nature" of their constructions, and people who try to
understand the origin
of mathematical concepts and ways of their evolution.
Whether your own philosophy of mathematics is platonism or not,
can be
easily determined using the following test. Let us consider the twin
prime numbers
sequence:
(3, 5), (5, 7), (11, 13), (17 ,19), (29, 31), (41, 43),
...
(two prime numbers are called twins, if their difference is 2). In 1742
Chr.Goldbach
conjectured that there are infinitely many twin pairs. The problem
remains unsolved
up to day. Suppose that it will be proved undecidable from the axioms of
set theory.
Do you believe that, still, Goldbach's conjecture possesses an "objective
truth value"?
Imagine you are moving along the natural number system:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
...
and you meet twin pairs in it from time to time: (3, 5), (5, 7), (11,
13), (17 ,19), (29,
31), (41, 43), ... It seems there are only two possibilities:
a) we achieve the last pair and after that moving forward we do
not meet any
twin pairs (i.e. Goldbach's conjecture is false),
b) twin pairs appear over and again (i.e. Goldbach's conjecture
is true).
It seems impossible to imagine a third possibility ...
If you think so, you are, in fact, a platonist. You are used to
treat the natural
number system as a specific "world", very like the world of your everyday
practice.
You are used to think that any sufficiently definite assertion about
things in this world
must be either true or false. And, if you regard the natural number
system as a specific
"world", you cannot imagine the third possibility that, maybe, Goldbach's
conjecture
is neither true nor false. But such a possibility will not surprise us if
we remember
(following Rashevsky [1973]) that natural number system contains not only
some
information about the real things of the human practice, but it also
contains
many elements of fantasy. Why do you think that a fantastic "world" (some
kind
of Disneyland) will be completely perfect?
As another striking example of platonist approach to nature of
mathematics let
us consider an expression of N.Luzin from 1927 about the
continuum-problem
(quoted after Keldish [1974]):
"The cardinality of continuum, if it is thought to be a set of
points, is some
unique reality, and it must be located on the aleph scale there, where it
is. It's not
essential, whether the determination of the exact place is hard or even
impossible (as
might have been added by Hadamard) for us, men".
The continuum-problem was formulated by Georg Cantor in 1878:
does there
exist a set of points with cardinality greater than the cardinality of
natural numbers
(the so called countable cardinality) and less than the cardinality of
the continuum
(i.e. of the set of all points of a line)? In the set theory (using the
axiom of choice) one
can prove that the cardinality of every infinite set can be measured by
means of the so
called aleph scale:
A0 A1 A2 ... An An+1 ... Aw ...
|___|___|_ ..._|___|__ ... __|__ ...
Here A0 (aleph-0) is the countable cardinality, A1 - the least
uncountable cardinality
etc., and Aw is greater than An for every natural number n .
Cantor established that A0