Karlis Podnieks (
Thu, 3 Nov 1994 15:50:28 +0200 (EET)

K.Podnieks, Dr.Math.


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
of specialists who are creating and using them. A combination of special
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
(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
maximum of information from minimum of premises has shown its efficiency
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
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
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
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
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
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
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
complicated. Either the model is not good enough to represent a really
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
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
acknowledged. Where is the problem, why it is so hard to regard
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
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
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.
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 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
principles" ruling them. In real mathematical theories we have such
"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
and about "Euclidean intuition", which makes the geometry completely
though Euclid's axioms do not contain many essential principles of
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
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
should be accepted. It happens even that a commonly acknowledged way of
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
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
uncontrollable nature of unconscious processes. It seems, the ruling
"principles" of
these processes are picked up and fastened by something like the "natural
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
The defining intuition of a theory does not always remain
Particularly numerous changes happen during the beginning period, when
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
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
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
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"
intuitive concepts, and that we must consider the problem seriously: are
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
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
than the ideas presented in axioms. The most famous applications of the
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
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,
axiomatization is used to rescue some theory which had "lost its way" in
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
aspects of intuitive concepts which can be recognised in the practice of
mathematical reasoning. It means, first, that all properties of real
acknowledged before as "evident", must be proved on the basis of the
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
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