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, 150
pp.
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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