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Date: Thu, 3 Nov 1994 15:50:28 +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 #2.

  The mathematical method is (by definition) investigation of fixed models.
What is then mathematics itself? Models can be more or less general (let us
compare, for example, arithmetic of natural numbers, the relativity theory and
some model of the solar system). Very specific models will be investigated
better under management of specialists who are creating and using them. A
combination of special experience with sufficient experience in mathematics (in
one person or in a team) will be here the most efficient strategy. But the
investigation of more general models which can be applied to many different
specific models draws up the contents of a specific branch of science which is
called mathematics.

  For example, the Calculus has many applications in various fields and,
therefore, it is a striking example of a theory which undoubtedly belongs to
mathematics. On the other hand, a model of solar system (used, for example, for
exact prediction of eclipses) is too specific to be encountered as part of
mathematics (although it is surely a mathematical model).

  The fixed character of mathematical models and theories is simultaneously the
force and the weakness of mathematics. The ability of mathematicians to obtain
maximum of information from minimum of premises has shown its efficiency in
science and technique many times. But, the other side of this force is
weakness: no concrete fixed model (theory) can solve all problems arising in
science (or even in mathematics itself). An excellent confirmation of this
thesis was given in the famous incompleteness theorem of K.Goedel.

   And one more weakness. Mathematics, being distracted from real problems of
other fields, controlled only by its "internal needs", is getting more and more
uncontrollable. Theories and whole branches of mathematics are developed, which
do not have and cannot have any applications to real problems. Polish writer
Stanislav Lem joked in his book "Summa Technologiae": a mathematician is like a
mad tailor: he is making "all possible clothes" and hopes to make also
something suitable for dressing... . As we have seen this problem is due to the
very nature of mathematical method. No other branch of science knows such
problems.

  Mathematicians have learned ability "to live" (literally!) in the world of
mathematical concepts and even (while working on some concrete problem) - in a
very specific "world" of a concrete model. Investigation of models is
mathematician's goal for goal's sake, during their work they disregard the
existence of the reality behind the model. Here we have the main source of the
creative power of mathematics: in this way, "living" (sometimes, for many
years) in the "world" of their concepts and models, mathematicians have learned
to draw maximum of conclusions from a minimum of premises.

  After one has formulated some model, it usually appears that in mathematics
some work has already been done on the problem, and some methods or even
algorithms have been created. This allows to draw in real time many important
conclusions about the model. Clearly, if the model looks so specific that no
ready mathematical means can be found to investigate it, the situation becomes
more complicated. Either the model is not good enough to represent a really
interesting fragment of the "reality" (then we must look for another model), or
it is so important that we may initiate investigations to obtain the necessary
new mathematical methods.

  The key to all these possibilities is mathematical platonism - the ability of
mathematicians "to live" in the "worlds" of the models they do investigate, the
ability to forget all things around them during their work. In this way some of
them have got the ill fame of being "rusks", queer customers, etc. Thus we can
say that platonism is in fact the psychology of working mathematicians (and
that it is a philosophy only from their subjective point of view).

  The above stated picture of the nature of mathematics is not yet commonly
acknowledged. Where is the problem, why it is so hard to regard mathematical
theories as fixed models? A personal communication of S.Lavrov from 1988: " ...
Theorems of any theory consist, as a rule, of two parts - the premise and the
conclusion. Therefore, the conclusion of a theorem is derived not only from a
fixed set of axioms, but also from a premise which is specific to this
particular theorem. And this premise - is it not an extension of the fixed
system of principles? ... Mathematical theories are open for new notions. Thus,
in the Calculus after the notion of continuity the following connected notions
were introduced: break points, uniform continuity, Lipschitz's conditions, etc.
... All this does not contradict the thesis about fixed character of principles
(axioms and rules of inference), but it does not allow "working mathematicians"
to regard mathematical theories as fixed ones."

3. Intuition and axiomatics

  The fixed character of mathematical models and theories is not always evident
- because of our platonist habits (we are used to treat mathematical objects as
specific "world"). Only few people will dispute the fixed character of a fully
axiomatized theory. All principles of reasoning, allowed in such theories, are
presented in axioms explicitly. Thus the principal basis is fixed, and any
changes in it yield explicit changes in axioms.

  But can we also fix those theories which are not axiomatized yet? How is it
possible? For example, all mathematicians are unanimous about the ways of
reasoning which allow us to prove theorems about natural numbers (other ways
yield only hypotheses or errors). But most mathematicians do not know anything
about the axioms of arithmetic! And even in the theories which seem to be
axiomatized (as, for example, geometry in "Elements" of Euclid) we can find
aspects of reasoning which are commonly acknowledged as correct, but are not
presented in axioms. For example, the properties of the geometric relation "the
point A is located on a straight line between the points B and C", are used by
Euclid without any foundation. Only in the XIXth century M.Pasch introduced the
"axioms of order", characterising this relation. But it was also until this
time that all mathematicians treated it equally (though they did not realise
how they managed to do it).

  Trying to explain this phenomenon, we are led to the concept of intuition.
Intuition is treated usually as "creative thinking", "direct obtaining of
truth", etc. Now we are interested in much more prosaic aspects of intuition.

  The human brain is a very complicated system of processes. Only a small part
of these electrochemical fireworks can be controlled consciously. Therefore,
similar to the processes going on at the conscious level, there must be a much
greater amount of thinking processes going on at the unconscious level.
Experience shows that when the result of some unconscious thinking process is
very important for the person, it (the result) can be sometimes recognised at
the conscious level. The process itself remains hidden, for this reason the
effect seems like a "direct obtaining of truth" etc., (see Poincare [1908],
Hadamard [1945]).

  Since unconscious processes yield not only arbitrary dreams, but also
(sometimes) reasonable solutions of real problems, there must be some
"reasonable principles" ruling them. In real mathematical theories we have such
unconscious "reasonable principles" ruling (together with the axioms or without
any axioms) our reasoning. Relatively closed sets of unconscious ruling
"principles" are the most elementary type of intuition used in mathematics.

  We can say, therefore, that a theory (or model) can be fixed not only due to
some system of axioms, but also due to a specific intuition. So, we can speak
about intuition of natural numbers which determines our reasoning about these
numbers, and about "Euclidean intuition", which makes the geometry completely
definite, though Euclid's axioms do not contain many essential principles of
geometric reasoning.

  How could we explain the emergence of intuitions, which are ruling the
reasoning of so many people equally? It seems that they can arise because human
beings all are approximately equal, because they deal with approximately the
same external world, and because in the process of education, practical and
scientific work they tend to achieve accordance with each other.

  While investigations are going on, they can achieve the level of complexity,
at which the degree of definiteness of intuitive models is already
insufficient. Then various conflicts between specialists can appear about which
ways of reasoning should be accepted. It happens even that a commonly
acknowledged way of reasoning leads to absurd conclusions.

  In the history of mathematics such situations appeared many times: the crash
of the discrete geometric intuition after the discovery of incommensurable
magnitudes (the end of VI century B.C.), problems with negative and complex
numbers (up to the end of XVIII century), the dispute of L.Euler and
J.d'Alembert on the concept of function (XVIII century), groundless operation
with divergent series (up to the beginning of XIX century), problems with the
acceptance of Cantor's set theory, paradoxes in set theory (the end of XIX
century), the scandal around the axiom of choice (the beginning of XX century).
All that was caused by the inevitably uncontrollable nature of unconscious
processes. It seems, the ruling "principles" of these processes are picked up
and fastened by something like the "natural selection" which is not able to a
far-reaching co-ordination without making errors. Therefore, the appearance of
(real or imagined) paradoxes in intuitive theories is not surprising.

  The defining intuition of a theory does not always remain constant.
Particularly numerous changes happen during the beginning period, when the
intuition (as the theory itself), is not yet stabilised. During this, the most
delicate period of evolution, the greatest conflicts appear. The only reliable
exit from such situations is the following: we must convert (at least partly)
the unconscious ruling "principles" into conscious ones and then investigate
their accordance with each other. If this conversion were meant in a literal
sense, it would be impossible as we cannot know the internal structure of a
concrete intuition. We can speak here only about a reconstruction of a "black
box" in some other - explicit - terms. Two different approaches are usually
applied for such reconstruction: the so-called genetic method and the axiomatic
method.

  The genetic method tries to reconstruct intuition by means of some other
theory (which can also be intuitive). Thus, a "suspicious" intuition is
modelled, using a "more reliable" one. For example, in this way the objections
against the use of complex numbers were removed: complex numbers were presented
as points of a plane and in this way even their strangest properties (as, for
example, the infinite set of values of log x for a negative x) were converted
into simple theorems of geometry. After this, all disputes stopped. In a
similar way the problems with the basic concepts of the Calculus (limit,
convergence, continuity, etc.) were cleared up - through their definition in
terms of epsilon-delta.

  It appeared, however, that some of these concepts, after the reconstruction
in terms of epsilon-delta, possessed unexpected properties missing in the
original intuitive concepts. Thus, for example, it was believed that every
continuous function of a real variable is differentiable almost everywhere
(except of some isolated "break- points"). The concept of continuous function
having been defined in terms of epsilon- delta it appeared that a continuous
function can be constructed, which is nowhere differentiable (the famous
construction of C.Weierstrass).

  The appearance of unexpected properties in reconstructed concepts means, that
here indeed we have a reconstruction - not a direct "copying" of intuitive
concepts, and that we must consider the problem seriously: are our
reconstructions adequate?

  The genetic method clears up one intuition in terms of another one, i.e. it
is working relatively. The axiomatic method, conversely, is working
"absolutely": among commonly acknowledged assertions about objects of a theory
some subset is selected, assertions from this subset are called axioms, i.e.
they are acknowledged as true without any proof. All other assertions of the
theory we must prove using the axioms. These proofs can contain intuitive
moments which must be "more evident" than the ideas presented in axioms. The
most famous applications of the axiomatic method are the following: the axioms
of Euclid, the Hilbert's axioms for the Euclidean geometry, the axioms of
G.Peano for arithmetic of natural numbers, the axioms of E.Zermelo and
A.Fraenkel for set theory.

  The axiomatic method (as well as the genetic method) yields only a
reconstruction of intuitive concepts. The problem of adequacy can be reduced
here to the question, whether all essential properties of intuitive concepts
are presented in axioms? From this point of view the most complicated situation
appears, when axiomatization is used to rescue some theory which had "lost its
way" in paradoxes. The axioms of Zermelo-Fraenkel were developed exactly in
such a situation - paradoxes having appeared in the intuitive set theory. The
problem of adequacy here is very complicated: are all positive contents of the
theory saved?

  What criteria can be set for the adequacy of reconstruction? Let us remember
various definitions of the real number concept in terms of rational numbers,
presented in the 1870s simultaneously by R.Dedekind, G.Cantor and some others.
Why do we regard these reconstructions to be satisfactory? And how can the
adequacy of a reconstruction be founded when the original concept remains
hidden in intuition and every attempt to get it out is a reconstruction itself
with the same problem of adequacy? The only possible realistic answer is: take
into account only those aspects of intuitive concepts which can be recognised
in the practice of mathematical reasoning. It means, first, that all properties
of real numbers, acknowledged before as "evident", must be proved on the basis
of the reconstructed concept. Secondly, all intuitively proved theorems of the
Calculus must be proved by means of the reconstructed concept. If this is done,
it means that those aspects of the intuitive concept of real number which
managed to appear in mathematical practice explicitly all are presented in the
reconstructed concept. But, maybe, some "hidden" aspects of the intuitive real
number concept have not yet appeared in practice. But they will appear in
future? At first glance, it seems hard to dispute such a proposition.

To be continued. #3