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