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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