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Date: Thu, 3 Nov 1994 15:44:26 +0200 (EET)
From: Karlis Podnieks <podnieks@mii.lu.lv>
To: QED discussions <qed@mcs.anl.gov>
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<c (c denotes the cardinality of continuum), and then
he conjectured that c=A1. This conjecture is called continuum-hypothesis. Long-
drawn efforts of Cantor itself and of many other outstanding people did not
lead to any solution of the problem. In 1905 D.Koenig proved that c is not
equal to Aw, and that was all ... .

Now we know that the continuum-problem is undecidable if one uses commonly
acknowledged axioms of set theory. Kurt Goedel in 1939 and Paul Cohen in 1963
proved that one can assume without contradiction any of the following "axioms":

                        c=A1, c=A2, c=A3, ...,

and even (as joked N.Luzin): c=A17. Thus, the axioms of set theory do not allow
to determine the exact place of c on the aleph scale, although we can prove
that

                        (Ex)  c=Ax,

i.e. that c "is located" on this scale. The platonist, looking at the picture
of the aleph scale, tries to find the exact place of c ... visually! He cannot
imagine a situation when a point is situated on a line, but it is impossible to
determine the exact place. This is a normal platonism of a working
mathematician. It stimulates investigation even in the most complicated fields
(we never know before whether some problem is solvable or not). But, if we pass
to methodological problems, for example, to the problem of the "meaning" of
Cohen's results, we should keep off our platonist habits. If we think that, in
spite of the undecidability of the continuum-problem "for us, men", some
"objective", "real" place for the cardinality of continuum on aleph scale does
exist, then we assume something like Plato's "world of ideas" - some fantastic
"world of sets", which exists independently of the axioms used in reasoning of
mathematicians. At this moment the mathematical platonism has converted into
the philosophical one. Such people say that the axioms of set theory do not
reflect the "real world of sets" adequately, that we must search for more
adequate axioms, and even - that no fixed axiom system can represent the "world
of sets" precisely. But here they pursue a mirage, of course, no "world of
sets" can exist independently of the principles used in its investigation.

  The real meaning of Cohen's results is very simple. We have established that
(Ex)  c=Ax, but it is impossible to determine the exact value of x. It means
that the traditional set theory is not perfect and, therefore, we may try to
improve it. And it appears that one can choose between several possibilities.

  For example, we can postulate the axiom of constructibility (see Jech [1971],
Devlin [1977]). Then we will be able to prove that c=A1, and to solve some
other problems, which are undecidable in the traditional set theory.

  But we can postulate also a completely different axiom - the axiom of
determinateness (see Kleinberg [1977]). Then we will be forced to reject the
axiom of choice (in its most general form) and as a result we will be able to
prove that every set of points is Lebesgue-mesuarable, and that the cardinality
of continuum is incompatible with alephs (except of A0). In this set theory
continuum-hypothesis can be proved in the following form: every infinite set of
points is either countable or has the cardinality of continuum.

  Both directions (the axiom of constructibility and the axiom of
determinateness) have yielded already a plentiful collection of beautiful and
interesting results. These two set theories are at least as "good" as the
traditional set theory, but they contradict each other, therefore we cannot
speak here about the convergence to some unique "world of sets".

  Our main conclusion is the following: everyday work is permanently moving
mathematicians to platonism (and, as a creative method, this platonism is
extremely efficient), but passing to methodology we must reject such a
philosophy deliberately. Most essays on philosophy of mathematics disregard
this problem.

 2. Investigation of fixed models - the nature of the mathematical method

  The term "model" will be used below in the sense of applied mathematics, not
in the sense of logic (i.e. we will discuss "models intended to model" natural
processes or technical devices, not sets of formulas).

Following the mathematical approach of solving some (physical, technical etc.)
problem, one tries "to escape the reality" as fast as possible, passing to
investigation of a definite (fixed) mathematical model. In the process of
formulating a model one asks frequently: can we assume that this dependency is
linear? can we disregard these deviations? can we assume that this partition of
probabilities is normal? It means that one tends (as fast as possible and using
a minimum of postulates) to formulate a mathematical problem, i.e. to model the
real situation in some well known mathematical structure or to create a new
structure. Solving the mathematical problem one hopes that, in spite of the
simplifications made in the model, he will obtain some solution of the original
(physical, technical etc.) problem.

  After mathematics has appeared, all scientific theories can be divided into
two classes:

        a) theories, based on a developing system of principles,
        b) theories, based on a fixed system of principles.

In the process of development theories of class (a) are enriched with new basic
principles, which do not follow from the principles acknowledged before. Such
principles arise due to fantasy of specialists, supported by more and more
perfect experimental data. The progress of such theories is first of all in
this enrichment process.

  On the other hand, in mathematics, physics and, at times, in other branches
of science one can find theories, whose basic principles (postulates) do not
change in the process of their development. Every change in the set of
principles is regarded here as a passage to new theory. For example, the
special relativity theory of A.Einstein can be regarded as refinement of the
classical mechanics, a further development of I.Newton's theory. But, since
both theories are defined very precisely, the passage "from Newton to Einstein"
can be regarded also as a passage to a new theory. The evolution of both
theories is going on today: new theorems are proved, new methods and algorithms
are developed etc. Nevertheless, both sets of basic principles remain constant
(such as they were during life time of their creators).

  Fixed system of basic principles is the distinguishing property of
mathematical theories. A mathematical model of some natural process or
technical device is essentially a fixed model which can be investigated
independently of its "original" (and, thus, the similarity of the model and the
"original" is only a limited one). Only such models can be investigated by
mathematicians. Any attempt to refine a model (to change its definition in
order to obtain more similarity with the "original") leads to a new model,
which must remain fixed again, to enable a mathematical investigation of it.

  Working with fixed models mathematicians have learned to draw maximum of
conclusions from a minimum of premises. This is why mathematical modelling is
so efficient.

  It is very important to note that a mathematical model (because it is fixed)
is not bound firmly to its "original". It may appear that some model is
constructed badly (in the sense of the correspondence to the "original"), but
its mathematical investigation goes on successfully. Since a mathematical model
is defined very precisely, it "does not need" its "original". One can change
some model (obtaining a new model) not only for the sake of the correspondence
to "original", but also for a mere experiment. In this way we easily obtain
various models (and entire branches of mathematics) which do not have any real
"originals". The fixed character of mathematical models makes such deviations
possible and even inevitable.

To be continued. #2