Dear Colleagues:

        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.

  Studying mathematics Plato came to his surprising philosophy of two worlds:
the "world of ideas" (strong and perfect as the "world" of geometry) and the
world of things. According to Plato, each thing is only an imprecise, imperfect
implementation of its "idea" (which does exist independently of the thing
itself in the world of ideas). Surprising and completely fantastic is Plato's
notion of the nature of mathematical investigation: before a man is born, his
soul lives in the world of ideas and afterwards, doing mathematics he simply
remembers what his soul has learned in the world of ideas. Of course, this is
an upside-down notion of the nature of mathematical method. The end-product of
the evolution of mathematical concepts - a fixed system of idealised objects,
is treated by Plato as an independent beginning point of the evolution of  the
"world of things".

  Nevertheless, being an outstanding philosopher, Plato tried to explain (in
his own manner) those aspects of the human knowledge which remained
inaccessible to other philosophers of his time. To explain the real nature of
idealised mathematical objects, Greeks had insufficient knowledge in physics,
biology, human physiology and psychology, etc.

  Today, any philosophical position in which ideal objects of human thought are
treated as a specific "world" should be called platonism. Particularly, the
philosophy of working mathematicians is a platonist one. Platonist attitude to
objects of investigation is inevitable for a mathematician: during his everyday
work he is used to treat numbers, points, lines etc. as the "last reality", as
a specific "world". This sort of platonism is an essential aspect of
mathematical method, the source of the surprising efficiency of mathematics in
the natural sciences and technology. It explains also the inevitability of
platonism in the philosophical position of mathematicians (having, as a rule,
very little experience in philosophy). Habits, obtained in the everyday work,
have an immense power. Therefore, when a mathematician, not very strong in
philosophy, tries to explain "the nature" of his mathematical results, he
unintentionally brings platonism into his reasoning. The reasoning of
mathematicians about the "objective nature" of their results is, as a rule,
rather an "objective idealism" (platonism) than the materialism.

  A platonist is, of course, in some sense "better" than the philosophers who
consider mathematical objects merely as "arbitrary creatures of human mind".
Nevertheless, we must distinguish between people who simply talk about the
"objective nature" of their constructions, and people who try to understand the
origin of mathematical concepts and ways of their evolution.

  Whether your own philosophy of mathematics is platonism or not, can be easily
determined using the following test. Let us consider the twin prime numbers
sequence:

                (3, 5), (5, 7), (11, 13), (17 ,19), (29, 31), (41, 43), ...

(two prime numbers are called twins, if their difference is 2). In 1742 Chr.
Goldbach conjectured that there are infinitely many twin pairs. The problem
remains unsolved up to day. Suppose that it will be proved undecidable from the
axioms of set theory. Do you believe that, still, Goldbach's conjecture
possesses an "objective truth value"? Imagine you are moving along the natural
number system:

                  0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...

and you meet twin pairs in it from time to time: (3, 5), (5, 7), (11, 13),
(17,19), (29, 31), (41, 43), ... It seems there are only two possibilities:

  a) we achieve the last pair and after that moving forward we do not
     meet any twin pairs (i.e. Goldbach's conjecture is false),

  b) twin pairs appear over and again (i.e. Goldbach's conjecture is
     true).

It seems impossible to imagine a third possibility ... If you think so, you
are, in fact, a platonist. You are used to treat the natural number system as a
specific "world", very like the world of your everyday practice. You are used
to think that any sufficiently definite assertion about things in this world
must be either true or false. And, if you regard the natural number system as a
specific "world", you cannot imagine the third possibility that, maybe,
Goldbach's conjecture is neither true nor false. But such a possibility will
not surprise us if we remember (following Rashevsky [1973]) that natural number
system contains not only some information about the real things of the human
practice, but it also contains many elements of fantasy. Why do you think that
a fantastic "world" (some kind of  Disneyland) will be completely perfect?

  As another striking example of platonist approach to nature of mathematics
let us consider an expression of N.Luzin from 1927 about the continuum-problem
(quoted after Keldish [1974]):

        "The cardinality of continuum, if it is thought to be a set of
         points, is some unique reality, and it must be located on the aleph
         scale there, where it is. It's not essential, whether the
         determination of the exact place is hard or even impossible (as
         might have been added by Hadamard) for us, men".

  The continuum-problem was formulated by Georg Cantor in 1878: does there
exist a set of points with cardinality greater than the cardinality of natural
numbers (the so called countable cardinality) and less than the cardinality of
the continuum (i.e. of the set of all points of a line)? In the set theory
(using the axiom of choice) one can prove that the cardinality of every
infinite set can be measured by means of the so called aleph scale:

                A0 A1 A2 ... An An+1 ... Aw ...
                |___|___|_ ..._|___|__ ... __|__ ...

Here A0 (aleph-0) is the countable cardinality, A1 - the least uncountable
cardinality etc., and Aw is greater than An for every natural number n .

Cantor established that A0<c (c denotes the cardinality of continuum), and then
he conjectured that c=A1. This conjecture is called continuum-hypothesis. Long-
drawn efforts of Cantor itself and of many other outstanding people did not
lead to any solution of the problem. In 1905 D.Koenig proved that c is not
equal to Aw, and that was all ... .

Now we know that the continuum-problem is undecidable if one uses commonly
acknowledged axioms of set theory. Kurt Goedel in 1939 and Paul Cohen in 1963
proved that one can assume without contradiction any of the following "axioms":

                        c=A1, c=A2, c=A3, ...,

and even (as joked N.Luzin): c=A17. Thus, the axioms of set theory do not allow
to determine the exact place of c on the aleph scale, although we can prove
that

                        (Ex)  c=Ax,

i.e. that c "is located" on this scale. The platonist, looking at the picture
of the aleph scale, tries to find the exact place of c ... visually! He cannot
imagine a situation when a point is situated on a line, but it is impossible to
determine the exact place. This is a normal platonism of a working
mathematician. It stimulates investigation even in the most complicated fields
(we never know before whether some problem is solvable or not). But, if we pass
to methodological problems, for example, to the problem of the "meaning" of
Cohen's results, we should keep off our platonist habits. If we think that, in
spite of the undecidability of the continuum-problem "for us, men", some
"objective", "real" place for the cardinality of continuum on aleph scale does
exist, then we assume something like Plato's "world of ideas" - some fantastic
"world of sets", which exists independently of the axioms used in reasoning of
mathematicians. At this moment the mathematical platonism has converted into
the philosophical one. Such people say that the axioms of set theory do not
reflect the "real world of sets" adequately, that we must search for more
adequate axioms, and even - that no fixed axiom system can represent the "world
of sets" precisely. But here they pursue a mirage, of course, no "world of
sets" can exist independently of the principles used in its investigation.

  The real meaning of Cohen's results is very simple. We have established that
(Ex)  c=Ax, but it is impossible to determine the exact value of x. It means
that the traditional set theory is not perfect and, therefore, we may try to
improve it. And it appears that one can choose between several possibilities.

  For example, we can postulate the axiom of constructibility (see Jech [1971],
Devlin [1977]). Then we will be able to prove that c=A1, and to solve some
other problems, which are undecidable in the traditional set theory.

  But we can postulate also a completely different axiom - the axiom of
determinateness (see Kleinberg [1977]). Then we will be forced to reject the
axiom of choice (in its most general form) and as a result we will be able to
prove that every set of points is Lebesgue-mesuarable, and that the cardinality
of continuum is incompatible with alephs (except of A0). In this set theory
continuum-hypothesis can be proved in the following form: every infinite set of
points is either countable or has the cardinality of continuum.

  Both directions (the axiom of constructibility and the axiom of
determinateness) have yielded already a plentiful collection of beautiful and
interesting results. These two set theories are at least as "good" as the
traditional set theory, but they contradict each other, therefore we cannot
speak here about the convergence to some unique "world of sets".

  Our main conclusion is the following: everyday work is permanently moving
mathematicians to platonism (and, as a creative method, this platonism is
extremely efficient), but passing to methodology we must reject such a
philosophy deliberately. Most essays on philosophy of mathematics disregard
this problem.

 2. Investigation of fixed models - the nature of the mathematical method

  The term "model" will be used below in the sense of applied mathematics, not
in the sense of logic (i.e. we will discuss "models intended to model" natural
processes or technical devices, not sets of formulas).

Following the mathematical approach of solving some (physical, technical etc.)
problem, one tries "to escape the reality" as fast as possible, passing to
investigation of a definite (fixed) mathematical model. In the process of
formulating a model one asks frequently: can we assume that this dependency is
linear? can we disregard these deviations? can we assume that this partition of
probabilities is normal? It means that one tends (as fast as possible and using
a minimum of postulates) to formulate a mathematical problem, i.e. to model the
real situation in some well known mathematical structure or to create a new
structure. Solving the mathematical problem one hopes that, in spite of the
simplifications made in the model, he will obtain some solution of the original
(physical, technical etc.) problem.

  After mathematics has appeared, all scientific theories can be divided into
two classes:

        a) theories, based on a developing system of principles,
        b) theories, based on a fixed system of principles.

In the process of development theories of class (a) are enriched with new basic
principles, which do not follow from the principles acknowledged before. Such
principles arise due to fantasy of specialists, supported by more and more
perfect experimental data. The progress of such theories is first of all in
this enrichment process.

  On the other hand, in mathematics, physics and, at times, in other branches
of science one can find theories, whose basic principles (postulates) do not
change in the process of their development. Every change in the set of
principles is regarded here as a passage to new theory. For example, the
special relativity theory of A.Einstein can be regarded as refinement of the
classical mechanics, a further development of I.Newton's theory. But, since
both theories are defined very precisely, the passage "from Newton to Einstein"
can be regarded also as a passage to a new theory. The evolution of both
theories is going on today: new theorems are proved, new methods and algorithms
are developed etc. Nevertheless, both sets of basic principles remain constant
(such as they were during life time of their creators).

  Fixed system of basic principles is the distinguishing property of
mathematical theories. A mathematical model of some natural process or
technical device is essentially a fixed model which can be investigated
independently of its "original" (and, thus, the similarity of the model and the
"original" is only a limited one). Only such models can be investigated by
mathematicians. Any attempt to refine a model (to change its definition in
order to obtain more similarity with the "original") leads to a new model,
which must remain fixed again, to enable a mathematical investigation of it.

  Working with fixed models mathematicians have learned to draw maximum of
conclusions from a minimum of premises. This is why mathematical modelling is
so efficient.

  It is very important to note that a mathematical model (because it is fixed)
is not bound firmly to its "original". It may appear that some model is
constructed badly (in the sense of the correspondence to the "original"), but
its mathematical investigation goes on successfully. Since a mathematical model
is defined very precisely, it "does not need" its "original". One can change
some model (obtaining a new model) not only for the sake of the correspondence
to "original", but also for a mere experiment. In this way we easily obtain
various models (and entire branches of mathematics) which do not have any real
"originals". The fixed character of mathematical models makes such deviations
possible and even inevitable.

  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.

  However, let us suppose that this is the case, and in 2094 somebody will
prove a new theorem of the Calculus using a property of real numbers, never
before used in mathematical reasoning. And then will all other mathematicians
agree immediately that this property was "intended" already in 1994? At least,
it will be impossible to verify this proposition: none of the mathematicians
living today will survive 100 years.

  Presuming that intuitive mathematical concepts can possess some "hidden"
properties which do not appear in practice for a long time we fall into the
usual mathematical platonism (i.e. we assume that the "world" of mathematical
objects exists independently of mathematical reasoning).

  Some of the intuitive concepts admit several different but, nevertheless,
equivalent reconstructions. In this way an additional very important evidence
of adequacy can be given. Thus, let us remember, again, various definitions of
real numbers in terms of rational numbers. Cantor's definition was based upon
convergent sequences of rational numbers. Dedekind defined real numbers as
"cuts" in the set of rational numbers. One definition more could be obtained
using (infinite) decimal fractions. The equivalence of all these definitions
was proved (we cannot prove strongly the equivalence of the intuitive concept
and its reconstruction, but we can prove - or disprove - the equivalence of two
explicit reconstructions).

  Another striking example is the reconstruction of the intuitive notion of
computability (the concept of algorithm). Since the 1930s several very
different reconstructions of this notion were proposed: recursive functions,
"Turing-machines" of A.M.Turing, lambda-calculus of A.Church, canonical systems
of E.Post, normal algorithms of A.Markov, etc. And here, too, the equivalence
of all these reconstructions was proved. The equivalence of different
reconstructions of the same intuitive concept means that the volume of the
reconstructed concepts is not accidental. This is a very important additional
argument for replacing the intuitive concept by an explicit reconstruction.

  The trend to replace intuitive concepts by their more or less explicit
reconstructions appears in the history of mathematics very definitely.
Intuitive theories cannot develop without such reconstructions normally: the
definiteness of the intuitive basic principles gets insufficient when the
complexity of concepts and methods is growing. In most situations the
reconstruction can be performed by the genetic method, but to reconstruct
fundamental mathematical concepts the axiomatic method must be used
(fundamental concepts are called fundamental just because they cannot be
reduced to other concepts).

  Goedel's incompleteness theorem has provoked very much talking about
insufficiency of the axiomatic method for a true reconstruction of the "alive,
informal" mathematical thinking. Such people say that axiomatics are is not
able to cover "all the treasures of the informal mathematics". All that is once
again the usual mathematical platonism, converted into the methodological one
(for a detailed analysis see Podnieks [1981, 1992]).

  Does the "axiomatic reasoning" differ in principle from the informal
mathematical reasoning? Do there exist proofs in mathematics obtained not
following the pattern "premises - conclusion"? If not, and every mathematical
reasoning can be reduced to a chain of conclusions, we may ask: are these
conclusions going on by some definite rules which do not change from one
situation to another? And, if these rules are definite, can they being a
function of the human brain be such that a complete explicit formulation is
impossible? Today, if we cannot formulate some "rules" explicitly, then how can
we demonstrate that they are definite?

  Therefore, it is nonsense to speak about the limited applicability of
axiomatics: the limits of axiomatics coincide with the limits of mathematics
itself! Goedel's incompleteness theorem is an argument against platonism, not
against formalism! Goedel's theorem demonstrates that no fixed fantastic "world
of ideas" can be perfect. Any fixed "world of ideas" leads us either to
contradictions or to undecidable problems.

   In the process of evolution of mathematical theories axiomatization and
intuition interact with each other. The axiomatization "clears up" the
intuition when it has lost its way. But the axiomatization has also some
unpleasant consequences: many steps of intuitive reasoning, expressed by a
specialist very compactly, appear very long and tedious in an axiomatic theory.
Therefore, after replacing intuitive theory by the axiomatic one (this
replacement may be inequivalent because of defects in the intuitive theory)
specialists learn a new intuition, and thus they restore the creative potency
of their theory. Let us remember the history of axiomatization of set theory.
In the 1890s contradictions were discovered in Cantor's intuitive set theory,
they were removed by means of axiomatization. Of course, the axiomatic set
theory of Zermelo-Fraenkel differs from Cantor's intuitive theory not only in
its form, but also in some aspects of contents. For this reason specialists
have developed new, modified intuitions (for example, the intuition of sets and
proper classes), which allow them to work in the new theory successfully.
Today, all serious theorems of set theory are proved intuitively, again.

  What are the main benefits of axiomatization? First, as we have seen,
axiomatization allows to "correct" intuition: to remove inaccuracies,
ambiguities and paradoxes which arise sometimes due to insufficient
controllability of intuitive processes.

  Secondly, axiomatization allows to carry out a detailed analysis of relations
between basic principles of a theory (to establish its dependence or
independence, etc.), and between the principles and theorems (to prove some of
theorems only a part of axioms is needed). Such investigations may lead to
general theories which can be applied to several more specific theories (let us
remember the theory of groups).

  Thirdly, sometimes after the axiomatization we can establish that the theory
considered is not able to solve some of naturally arising problems (let us
remember the continuum-problem in set theory). In such situations we may try to
improve the theory, even  developing several alternative theories.

4. Formal theories

  How far can we proceed in the axiomatization of some theory? Complete
elimination of intuition, i.e. full reduction to a list of axioms and rules of
inference. Is this possible? The work by G.Frege, B.Russell and D.Hilbert (the
end of the XIXth - beginning of XXth century) showed how this can be done even
with the most serious mathematical theories. All these theories were reduced to
axioms and rules of inference without any admixture of intuition. Logical
techniques developed by these men allow us today to axiomatize any theory based
on a fixed system of principles (i.e. any mathematical theory).

  What do they look like - such pure axiomatic theories? They are called formal
theories (formal systems or deductive systems) underlining that no step of
reasoning can be done in them without a reference to an exactly formulated list
of axioms and rules of inference. Even the most "self-evident" logical
principles (like, "if A implies B, and B implies C, then A implies C") must be
either formulated in the list of axioms and rules explicitly or derived from
it.

  The exact definition of the "formal" can be given in terms of the theory of
algorithms (i.e. recursive functions): a theory T is called formal, if an
algorithm (i.e. mechanically applicable computation procedure) is presented for
checking correctness of reasoning via principles of T. It means that when
somebody is going to publish a "mathematical text" calling it "a proof of a
theorem in T", we can check mechanically whether the text in question really is
a proof according to standards of reasoning accepted in T. Thus, the standards
of reasoning in T are defined precisely enough to enable the checking of proofs
by means of a computer program (note that we discuss here checking of ready
proofs, not the problem of provability!).

  As an unpractical example of formal theory let us consider the game of chess,
let us call this "theory" CHESS. All possible positions on a chess-board (plus
a flag: "whites to go" or "blacks to go") we shall call propositions of CHESS.
The only axiom of CHESS will be the initial position, and the rules of
inference - the rules of the game. The rules allow us to pass from one
proposition of CHESS to some other ones. Starting with the axiom we get in this
way theorems of CHESS. Thus, theorems of CHESS are all possible positions which
can be got from the initial position by moving of chess-men according to the
rules of the game.

Exercise 4.1. Can you provide an unprovable proposition of CHESS?

  Why is CHESS called formal theory? When somebody offers a "mathematical text"
P as a proof of a theorem A in CHESS it means that P is the record of some
chess-game stopped in the position A. And we can check its correctness. Here
checking is not a serious problem: the rules of the play are formulated
precisely enough, and we can write a computer program which will execute the
task.

Exercise 4.2. Estimate the size of this program in some programming language.

  Our second example of a formal theory is only a bit more serious. It was
proposed by P.Lorenzen, let us call it theory L. Propositions of L are all
possible words made of letters a, b, for example: a, aa, aba, abaab. The only
axiom of L is the word a, and L has two rules of inference:

                  X               X
                ----- ,         -----
                 Xb              aXa

It means that (in L) from a proposition X we can infer immediately propositions
Xb and aXa. For example, proposition aababb is a theorem of L:

                a |-- ab |-- aaba |-- aabab |-- aababb
                 rule1   rule2  rule1       rule1

This fact is expressed usually as L |-- aababb ( "aababb is provable in L").

Exercise 4.3.
  a) Describe an algorithm which determines whether a proposition
of L is a theorem or not.
  b) Can you imagine such an algorithm for the theory CHESS? Of course, you
can. Thus you see that even having a relatively simple algorithm for proof
correctness the problem of provability can appear a very complicated one.

A very important property of formal theories is given in the following

Exercise 4.4. Show that the set of all theorems of a formal theory is
effectively (i.e. recursively) enumerable.

  It means (theoretically) that for any formal theory a computer program can be
written which will print on an (endless) paper tape all theorems of the theory
(and nothing else). Unfortunately, such a program cannot solve the problem
which the mathematicians are interested in: is a given proposition provable or
not. When, sitting near the computer we see our proposition printed, it means
that it is provable. But until that moment we cannot know whether the
proposition will be printed some time later or it will not be printed at all.

  T is called a solvable theory (or, effectively solvable) if an algorithm
(mechanically applicable computation procedure) is presented for checking
whether some proposition is provable using principles of T or not. In the
exercise 4.3a you have proved that L is a solvable theory. But in the exercise
4.3b you established that it is hard to state whether CHESS is a "normally
solvable" theory or not? Proof correctness checking is always much simpler than
provability checking. It can be proved that most mathematical theories are
unsolvable, elementary (i.e. first order) arithmetic and set theory included
(see, for example, Mendelson [1970]).

  Mathematical theories normally contain the negation symbol not. In such
theories to solve the problem stated in a proposition A means to prove either A
or notA. We can try to solve the problem using the enumeration program of the
exercise 4.4: let us wait sitting near the computer for until A or notA is
printed. If A and notA are printed both, it will mean that T is an inconsistent
theory (i.e. one can prove using principles of T some proposition and its
negation). But totally we have here 4 possibilities:

        a) A will be printed, but notA will not (then the problem A has
           positive solution),
        b) notA will be printed, but A will not (then the problem A has
           negative solution),
        c) A and notA will be printed both (then T is an inconsistent theory),
        d) neither A nor notA will be printed.

In the case d) we will be sitting forever near the computer, but nothing
interesting will happen, using principles of T one can neither prove nor
disprove the proposition A, and for this reason T is called an incomplete
theory. The incompleteness theorem of K.Goedel says that most mathematical
theories are incomplete (see Mendelson [1970]).

Exercise 4.5. Show that  any complete formal theory is solvable.

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, 150pp.

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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