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Date: Thu, 3 Nov 1994 15:39:44 +0200 (EET)
From: Karlis Podnieks <podnieks@mii.lu.lv>
To: QED discussions <qed@mcs.anl.gov>
Subject: Semantics
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Dear Colleagues:

        The current text presents the first chapter of my book "Around
the Goedel's theorem"
published in Russian (see Podnieks [1981, 1992] in the reference list).
The main ideas were published also in Podnieks [1988a].
        The contents of the book is the following:
1.  The  nature of mathematics
        1.1. Platonism - the philosophy of working mathematicians
        1.2. Investigation of fixed models - the nature of the
mathematical method
        1.3. Intuition and axiomatics
        1.4. Formal theories
        1.5. Logics
        1.6. Hilbert's program
2. The axiomatic set theory
        2.1. The origin of the intuitive set theory
        2.2. Formalization of the inconsistent set theory
        2.3. Zermelo-Fraenkel axioms
        2.4. Around the continuum problem
3. First order arithmetics
        3.1. From Peano axioms to first order axioms
        3.2. How to find arithmetics in other formal theories
        3.3. Representation theorem
4. Hilbert's Tenth problem
        4.1. - 4.7.
...............................................................................
5. Incompleteness theorems
        5.1. The Liar's paradox
        5.2. Self reference lemma
        5.3. Goedel's incompleteness theorem
        5.4. Goedel's second theorem
6. Around the Goedel's theorem
        6.1. Methodological consequences
        6.2. The double incompleteness theorem
        6.3. The "creativity problem" in mathematics
        6.4. On the size of proofs
        6.5. The "diophantine" incompleteness theorem
        6.6. The Loeb's theorem
Appendix 1. About the model theory
Appendix 2. Around the Ramsey's theorem

______________________________________________________________________


University of Latvia
Institute of Mathematics and Computer Science


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

PLATONISM, INTUITION
 AND
THE NATURE OF MATHEMATICS


CONTENTS
1. Platonism - the philosophy of working mathematicians
2. Investigation of fixed models - the nature of the mathematical method
3. Intuition and axiomatics
4. Formal theories
5. Hilbert's program
6. Some replies to critics
7. References
8. Postscript


1. Platonism - the philosophy of working mathematicians

  Charles Hermite has said once he is convinced that numbers and functions are
not mere inventions of mathematicians, that they do exist independently of us,
as do exist things in our everyday practice. Some time ago in the former USSR
this proposition was quoted as the evidence for "the naive materialism of
outstanding scientists".

  But such propositions stated by mathematicians are evidences not for their
naive materialism, but for their naive platonism. Platonist attitude of
mathematicians to objects of their investigations, as will be shown below, is
determined by the very nature of the mathematical method.

  First let us consider the "platonism" of Plato itself. Plato, a well known
Greek philosopher lived in 427-347 B.C., at the end of the Golden Age of
Ancient Greece. In 431-404 B.C. Greece was destroyed in the Peloponnesus war,
and in 337 B.C. it was conquered by Macedonia. The concrete form of the Plato's
system  of philosophy was determined by Greek mathematics.

  In the VI-Vth centuries B.C. the evolution of Greek mathematics led to
mathematical objects in the modern meaning of the word: the ideas of numbers,
points, straight lines etc. stabilised, and thus they got distracted from their
real source - properties and relations of things in the human practice. In
geometry straight lines have zero width, and points have no size at all. Such
things actually do not exist in our everyday practice. Instead of straight
lines here we have more or less smooth stripes, instead of points - spots of
various forms and sizes. Nevertheless, without this passage to an ideal (partly
fantastic, but simpler, stable and fixed) "world" of points, lines etc., the
mathematical knowledge would have stopped at the level of art and never would
become a science. Idealisation allowed to create an extremely effective
instrument - the well known Euclidean geometry.

  The concept of natural numbers (0, 1, 2, 3, 4, ...) rose from human
operations with collections of discrete objects. This development ended already
in the VIth century B.C., when somebody asked how many prime numbers do there
exist? And the answer was found by means of reasoning - there are infinitely
many prime numbers. Clearly, it is impossible to verify such an assertion
empirically. But by that time the concept of natural number was already
stabilised and distracted from its real source - the quantitative relations of
discrete collections in the human practice, and it began to work as a fixed
model. The system of natural numbers is an idealisation of these quantitative
relations. People abstracted it from their experience with small collections
(1, 2, 3, 10, 100, 1000 things). Then they extrapolated their rules to much
greater collections (millions of things) and thus idealised the real situation
(and even deformed it - see Rashevsky [1973]).

  For example, let us consider "the number of atoms in this sheet of paper".
From the point of common arithmetic this number "must" be either even or odd at
any moment of time. In fact, however, the sheet of paper does not possess any
precise "number of atoms" (because of, for example, nuclear reactions). And,
finally, the modern cosmology claims that the "total number" of particles in
the Universe is less than 10**200. What should be then the real meaning of the
statement "10**200+1 is an odd number"? Thus, in arithmetic not only
practically useful algorithms are discussed, but also a kind of pure fantastic
matter without any direct real meaning. Of course, Greek mathematicians could
not see all that so clearly. Discussing the amount of prime numbers they
believed that they are discussing objects as real as collections of things in
their everyday practice.

  Thus, the process of idealisation ended in stable concepts of numbers,
points, lines etc. These concepts ceased to change and were commonly
acknowledged in the community of mathematicians. And all that was achieved
already in the Vth century B.C. Since that time our concepts of natural
numbers, points, lines etc. have changed very little. The stabilisation of
concepts testifies their distraction from real objects which have led people to
these concepts and which continue their independent life and contain an immense
variety of changing details. When working in geometry, a mathematician does not
investigate the relations of things of the human practice (the "real world" of
materialists) directly, he investigates some fixed notion of these relations -
an idealised, fantastic "world" of points, lines etc. And during the
investigation this notion is treated (subjectively) as the "last reality",
without any "more fundamental" reality behind it. If during the process of
reasoning mathematicians had to remember permanently the peculiarities of real
things (their degree of smoothness etc.), then instead of a science (effective
geometrical methods) we would have art, simple, specific algorithms obtained by
means of trial and error or on behalf of some elementary intuition. Mathematics
of Ancient Orient stopped at this level. But Greeks went further... .

To be continued. #1