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From: Karlis Podnieks <podnieks@mii.lu.lv>
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Subject: Semantics
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University of Latvia
Institute of Mathematics and Computer Science


K.Podnieks, Dr.Math.
podnieks@mii.lu.lv

PLATONISM, INTUITION
 AND
THE NATURE OF MATHEMATICS

Continued from #4


5. Hilbert's program

  At the beginning of the XXth century the honour of mathematics was questioned
seriously. The well known contradictions were found in the set theory. Till
that time set theory was acknowledged widely as the natural foundation and a
very important tool of mathematics. In order to save the honour of mathematics
David Hilbert proposed in 1904 his famous program of "perestroika" in the
foundations of mathematics:

        a) to convert all existing (mainly intuitive) mathematics into a
           formal theory (a new variant of set theory cleared of paradoxes
           included);
        b) to prove consistency of this formal theory (i.e. the proof
           that no proposition can be proved and disproved in it
           simultaneously).

To solve the task (a) - it was meant to complete the axiomatization of
mathematics (this process proceeded successfully in the XIXth century: formal
definition of the notions of function, continuity, real numbers, axiomatization
of arithmetic, geometry etc.).

  The task (b) - contrary to (a) - was a great novelty: an attempt to get an
absolute consistency proof of mathematics. Hilbert was the first to realise
that a complete solution of the task (a) enables one to set the task (b).
Really, if we have not a complete solution of (a), i.e. if we are staying
partly in the intuitive mathematics, then we cannot discuss absolute proofs of
consistency. We may hope to establish a contradiction in an intuitive theory,
i.e. to prove some proposition and its negation simultaneously. But we cannot
hope to prove the consistency of such a theory: consistency is an assertion
about the set of all theorems of the theory, i.e. about the set, explicit
definition of which we do not have in the case of intuitive theory.

  But, if the intuitive theory is replaced by a formal one, the situation is
changed, then the set of all theorems of a formal theory is an explicitly
defined object. Let us remember our examples of formal theories. The set of all
theorems of CHESS is (theoretically) finite, but from a practical point of view
it is rather infinite. Nevertheless, one can prove easily the following
assertion about all theorems of CHESS:

  In a theorem of CHESS one cannot have 10 white queens simultaneously. Really,
in the axiom of CHESS we have 1 white queen and 8 white pawns, and by the rules
of the game only white pawns can be converted into white queens. The rest of
the proof is arithmetical: 1+8<10. Thus we have selected some specific
properties of axioms and inference rules of CHESS which imply our general
assertion about all theorems of CHESS.

  With the theory L we have similar opportunities. One can prove, for example,
the following assertion about all theorems of L:

                if X is a theorem, then aaX is also a theorem.

Really, if X is axiom (X=a), then L |-- aaX by rule2. Further, if for some X: L
|-- aaX, then we have the same for X'=Xb and X"=aXa:

                aaX |--  aa(Xb),        aaX |--  aa(aXa)
                    rule1                   rule2

Thus, by induction, our assertion is proved for any theorem of L.

  Hence, if the set of theorems is defined precisely enough, one can prove
general assertions about all theorems. Hilbert's opinion was that consistency
assertions would not be an exception. Roughly, he hoped to select those
specific properties of the axiom system of the entire mathematics which make
deduction of contradictions impossible.

  Let us remember, however, that the set of all theorems is here infinite, and,
therefore, consistency cannot be verified empirically. We may only hope to
establish it theoretically. For example, our assertion:

                                 L  |-- X --> L |-- aaX

was proved using the induction principle. Then, what kind of theory must be
used to prove the consistency of the entire mathematics? Clearly, the means of
reasoning used to prove consistency of some theory T must be more reliable than
the means used in T itself. How could we rely on the consistency proof when
suspicious means were used in it? But, if a theory T contains the entire
mathematics, then we (mathematicians) cannot know any means of reasoning
outside of T. Hence, proving consistency of such a universal theory T we must
use means from T itself - from the most reliable part of them.

There are two different levels of "reliability" in mathematics:

        1) arithmetical ("discrete") reasoning - only natural numbers and
           similar discrete objects are used;
        2) set-theoretic reasoning - Cantor's concept of arbitrary
           infinite sets is used.

The first level is regarded as reliable (only few people will question it), and
the second one as suspicious (Cantor's set theory was cleared of
contradictions, but...). Hilbert's intention was to prove the consistency of
mathematics by means of the first level.

  As soon as Hilbert announced his project in 1904, Henry Poincare stated
serious doubts about its reality. He pointed out that proving consistency of
mathematics by means of induction principle (the main tool of the first level)
Hilbert would use a circular argument: consistency of mathematics means also
consistency of induction principle ... proved by means of induction principle!
At that time few people could realise the real significance of this hint. But
25 years later Kurt Goedel proved that Poincare was right: an absolute proof of
consistency of essential parts of mathematics is impossible!

6. Some replies to critics

  1. I do not believe that the natural number system is an inborn property of
human mind. I think that it was developed from human practice with sets of
discrete objects. Therefore, the concrete form of our present natural number
system is influenced by both the properties of discrete sets from human
practice and the structure of human mind. If so, how long was the development
process of this system and when it was ended? I think that the process ended in
the VIth century B.C., when first results were obtained about the natural
number system as the whole (theorem about infinity of primes was one of such
results). In human practice only relatively small sets can appear (and
following the modern cosmology we believe that only a finite number of
particles can be found in the Universe). Hence, results about "natural number
infinity" can be obtained in a theoretical model only. If we believe that
general results about natural numbers can be obtained by means of pure
reasoning, without any additional experimental practice, it means that we are
convinced of stability and (sufficient) completeness of our theoretical model.

  2. The development process of mathematical concepts does not yield a
continuous spectrum of concepts but a relatively small number of different
concepts (models, theories). Thus, considering the history of natural number
concept we see two different stages only. Both stages can be described by
corresponding formal theories:

        - stage 1 (the VIth century B.C. - 1870s) can be described by
          first order arithmetic,
        - stage 2 (1870s - today) can be described by arithmetic of ZFC.

I think that the natural number concept of Greeks corresponds to first order
arithmetic and that this concept remained unchanged up to 1870s. I believe that
Greeks would accept any proof from the so called elementary number theory of
today. G.Cantor's invention of "arbitrary infinite sets" (in particular, "the
set of all sets of natural numbers", i.e. P[w])  added new features to the old
("elementary") concept. For example, the well known strong Ramsey's theorem
became provable. Thus the fixed model of stage 1 was replaced by a new model
(stage 2) which also remains principally unchanged up to day.

  Finally, let us consider the history of geometry. The invention of non-
Euclidean geometry could not be treated as "further development" of the old
Euclidean geometry. The Euclidean geometry remains unchanged up to day, and we
can still prove new theorems using Euclid's axioms. The non-Euclidean geometry
appeared as a new theory, different from the Euclidean one, and it also remains
unchanged up to day.

  Therefore, I think, I can retain my definition of mathematics as
investigation of fixed models which can be treated, just because they are
fixed, independently of any experimental data.

  3. I do not criticise platonism as a philosophy (and psychology) of working
mathematicians. On the contrary, platonism as a creative method is extremely
effective in this field. Platonist approach to "objects" of investigation is a
necessary aspect of mathematical method. Indeed, how can one investigate
effectively a fixed model - if not thinking about it in a platonist way (as the
"last reality", without any experimental "world" behind it)?

  4. By which means do we judge theories? My criterion is pragmatic (in the
worst sense of the word). If in a theory contradictions are established, then
any new theory will be good enough, in which main theorems of the old theory
(but not its contradictions) can be proved. In such sense, for example, ZFC is
"better" than Cantor's original set theory.

On the other side, if undecidable problems have appeared in a theory (as
continuum-problem appeared in ZFC), then any extension of the theory will be
good enough, in which some of these problems can be solved in a positive or a
negative way. Of course, simple postulation of the needed positive or negative
solutions leads, as a rule, to uninteresting theories (such as ZFC+GCH). We
must search for more powerful hypotheses, such as, for example, "V=L" or AD
(axiom of determinateness). Theories ZF+"V=L" and ZF+AD contradict each other,
but they both appear very interesting, and many people make beautiful
investigations in each of them.

If some people are satisfied neither with "V=L" nor with AD, they can suggest
any other powerful hypothesis having rich and interesting consequences. I do
not believe that here any convergence to some unique (the "only right") system
of set theory can be expected.

  5. Mathematicians are not in agreement about the ways to prove theorems, but
their opinions do not form a continuous spectrum. The existing few variations
of these views can be classified, each of them can be described by means of a
suitable formal theory. Thus they all can be recognised as "right", and we can
peacefully investigate their consequences.

  6. I think that the genetic and axiomatic methods are used in mathematics not
as heuristics, and not to prove theorems. These methods are used to clarify
intuitive concepts which appear insufficiently precise, and, for this reason,
investigations cannot be continued normally.

  The most striking application of the genetic method is, I think, the
definition of continuous functions in terms of epsilon-delta. The old concept
of continuous function (the one of the XVIIIth century) was purely intuitive
and extremely vague, so that one could not prove theorems about it. For
example, the well known theorem about zeros of a  function f continuous on [a,
b] with f(a)<0 and f(b)>0 was believed to be "obvious". It was believed also
that every continuous function is almost everywhere differentiable (except of
some isolated "break points"). The latter assertion could not be even stated
precisely. To enable further development of the theory a reconstruction of the
intuitive concept in more explicit terms was needed. This was done by Cauchy im
terms of epsilon-delta. Having such a precise definition, the "obvious" theorem
about zeros of f needs already a serious proof. And it was proved. The
Weierstrass's construction of a continuous function (in the sense of the new
definition) which is nowhere differentiable, shows unexpectedly that the
volumes of the old (intuitive) and the new (more explicit) concept are somewhat
different. Nevertheless, it was decided that the new concept is "better", and
for this reason it replaced the old intuitive concept of continuous function.

  In similar way the genetic method was used many times in the past. The so
called "arithmetization of the Calculus" (definition of reals in the terms of
natural numbers) also is an application of the genetic method.

  7. Our usual metatheory used for investigation of formal theories (to prove
Goedel's theorem etc.) is the theory of algorithms (i.e. recursive functions).
It is, of course, only a theoretical model giving us a somewhat deformed
picture of how are real mathematical theories functioning. Perhaps, the new
developing "subrecursive mathematics" will provide more adequate picture of the
real processes. I find especially interesting the paper Parikh [1971].

7. References

Devlin K.J. [1977]
The axiom of constructibility. A guide for the mathematician.
"Lecture notes in mathematics", vol. 617, Springer-Verlag,
Berlin - Heidelberg - New York, 1977, 96 pp.

Hadamard J. [1945]
An essay on the psychology of invention in the mathematical field.
Princeton, 1945,
143 pp.

Jech T.J. [1971]
Lectures in set theory with particular emphasis on the method of forcing.
Springer-
Verlag, Berlin - Heidelberg - New York, 1971

Keldysh L.V. [1974]
The ideas of N.N.Luzin in the descriptive set theory.
"Uspekhi matematicheskih nauk", 1974, vol.29, n5, pp.183-196 (in Russian)

Kleinberg E.M. [1977]
Infinitary combinatorics and the axiom of determinateness.
"Lecture notes in mathematics", vol. 612, Springer-Verlag,
Berlin - Heidelberg - New York, 1977, 150pp.

Mendelson E. [1970]
An introduction to mathematial logic.

Parikh R. [1971]
Existence and Feasibility in Arithmetic.
JSL, 1971, Vol.36, N.3, pp.494-508

Podnieks K.M. [1988a]
Platonism, Intuition and the Nature of Mathematics.
"Heyting'88. Summer School & Conference on Mathematical Logic.
Chaika, Bulgaria, September 1988. Abstracts.";
Sofia, Bulgarian Academy of Sciences, 1988, pp. 50-51.

Podnieks K.M. [1988b]
Platonism, Intuition and the Nature of Mathematics.
Riga, Latvian State University, 1988, 23 pp. (in Russian).

Podnieks K.M. [1981, 1992]
Around the Goedel's theorem.
Latvian State University Press, Riga, 1981, 105 pp. (in Russian).
2nd edition: "Zinatne", Riga, 1992, 191 pp. (in Russian).

Poincare H. [1908]
Science et methode.
Paris, 1908, 311 pp.

Rashevsky P.K. [1973]
On the dogma of the natural number system.
"Uspekhi matematicheskih nauk", 1973,
vol.28, n4, pp.243-246 (in Russian)

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