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PLATONISM, INTUITION
AND
THE NATURE OF MATHEMATICS

by Karlis Podnieks
Associate Professor
Karlis.Podnieks@mii.lu.lv
University of Latvia
Institute of Mathematics and Computer Science

This paper presents the translation of the first chapter of my book "Around Goedel's theorem" published in 1992 in Russian. View copy of the Russian original.

Note. The 1st edition of the book was published in 1981 in Russian. In English, its concepts were first presented in 1988 at the Heyting'88 Summer School & Conference on Mathematical Logic (view copy of Abstract), after that - in 1994, on the QED mailing list (in 5 parts: #1, #2, #3, #4, #5).

CONTENTS

1. Platonism - the philosophy of working mathematicians
2. Investigation of stable self-contained models - the true nature of the mathematical method
3. Intuition and axioms
4. Formal theories
5. Hilbert's program
6. Some replies to critics
7. References

1. Platonism - the Philosophy of Working Mathematicians

Charles Hermite once wrote that numbers and functions are not mere inventions of mathematicians; that they do exist independently of us, as do things in our everyday practice... Some time ago, in the former USSR this proposition was quoted as the evidence for "the naive materialism of distinguished scientists".

Still, such propositions stated by mathematicians are evidences not for their naive materialism, but for their naive Platonism. As will be shown below, Platonist attitude of mathematicians to objects of their investigations is determined by the very nature of the mathematical method.

For more exact Hermite's quotes see Objectivism Without Platonism: Hao Wang on Kurt Godel by Peter Saint-André, and Goedel, Cantor and Plato by Denis Paul.

First let us consider the "Platonism" of Plato (427-347 BC) himself (click here for other Plato web-sites, and here - to read about Plato's lifetime in the history of Greece). The particular form of Plato's system of philosophy was determined by Greek mathematics.

In VI-V centuries BC the evolution of Greek mathematics led to mathematical objects in the modern meaning of the word: the ideas of numbers, points, straight lines etc. stabilized, and thus they were detached 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. Actually, such things do not exist in our everyday practice. Here, instead of straight lines 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, yet simpler, stable, fixed, self-contained) "world" of points, lines etc., the mathematical knowledge would have stopped at the level of art and never would become a science. Idealization allowed creating of an extremely efficient instrument - Euclidean geometry.

The concept of natural numbers (0, 1, 2, 3, 4, ...) developed from human operations with collections of discrete objects. This development process was completed already in the VI century BC, 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. Still, by that time the concept of natural number was already stabilized and detached from its real source - the quantitative relations of discrete collections in the human practice, and it began to work as a stable self-contained model. The system of natural numbers is an idealization of these quantitative relations. People abstracted it from their experience with small collections (1, 2, 3, 10, 100, 1000 things). They extrapolated their rules onto much greater collections (millions of things) and thus idealized the real situation (and even deformed it - see Rashevsky [1973] and van Dantzig [1955]).

Pyotr Konstantinovich Rashevsky (translated also as Rashevskii)

P.K.Rashevsky on the FOM list:
http://www.cs.nyu.edu/pipermail/fom/1998-April/001874.html and http://www.cs.nyu.edu/pipermail/fom/1998-October/002396.html

Note. The idea of the Infinite deviates significantly from the situation in the physical Universe - this fact was clearly stated already in the famous June 4, 1925 lecture by David Hilbert "On the Infinite":

D.Hilbert. Ueber das Unendliche. "Math. Annalen", 1925, Vol.95, pp.161-190 (see also van Heijenoort [1967])

My 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, spontaneous nuclear reactions). Moreover, the modern cosmology claims that the "total number" of particles in the Universe is far less than 10²⁰⁰. What should be then the real meaning of the statement "10²⁰⁰+1 is an odd number"? Hence, in arithmetic not only practically useful algorithms are discussed, yet also a kind of pure fantastic matter without any direct real meaning. (Rashevsky proposed to develop a new kind of arithmetic allowing a more adequate - ''realistic"- treatment of large natural numbers.) Of course, Greek mathematicians could not see all that so clearly. Discussing the total 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 idealization ended in stable, fixed, self-contained 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 V century BC. Since that time our concepts of natural numbers, points, lines etc. have changed very little. The stabilization of concepts is an evidence of their detachment from real objects that have led people to these concepts and that are continuing 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 stable notion of these relations - an idealized, 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 (efficient geometrical methods) we would have an 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. Greeks went further.

(See online paper "Babylonian and Egyptian mathematics" in the MacTutor History of Mathematics archive).

Studying mathematics Plato came to his surprising philosophy of two worlds: the "world of ideas" (as 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.

For a materialist, this may seem an upside-down notion of the real situation. Plato treats the end product of the evolution of mathematical concepts - a stable, self-contained system of idealized objects - as an independent beginning point of the evolution of the "world of things"? Still, it was the first attempt (and, for the IV century BC - a brilliant one!) to explain the specific position of the mathematical knowledge among other branches of human knowledge. Materialists tend to think of mathematics as "one of" the branches of science. It's true that the process of genesis of the first mathematical concepts (arithmetic and Euclidean geometry) was very similar to the process of genesis of the first concepts of "other" branches of science. But Plato realized that mathematical concepts may become stable, fixed, self-contained, and hence, "independent forever". Of course, for us, Plato's radical simple explanation of this independence seems completely fantastic. Today, we do not need the hypothesis of "platonic worlds" to explain the nature of mathematical concepts.

An today, any philosophical position treating ideal objects of human thought as a specific independent "world", should be called Platonism. Particularly, the everyday 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 independent "world". This sort of Platonism is an essential aspect of mathematical method, the true 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 a kind of "objective idealism", not "materialism".

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

(3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61), (71,73), (101, 103), (107,109),
(137, 139), (149, 151), (179, 181), (191, 193), ..., (1787, 1789), ..., (1871, 1873), ...,
(1931, 1933), (1949, 1951), (1997, 1999), (2027, 2029), ...

(two prime numbers are called twins, if their difference is 2). It is believed (at least since 1849) that there are infinitely many twin pairs (the famous twin prime conjecture), yet the problem remains unsolved up to day. Suppose, someone has proved that the twin prime conjecture is unprovable in set theory. Do you believe that, still, the twin prime 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, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, ...

And you meet twin pairs in it from time to time: (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61), (71,73), ... It seems there are only two possibilities:

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

b) Twin pairs appear over and again (i.e. the twin prime 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 independent "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 natural number system as a specific "world", you cannot imagine a third possibility: maybe, the twin prime conjecture is neither true nor false. Still, such a possibility would not surprise you if you would realize (following Rashevsky [1973]) that the natural number system contains not only some information about real things of the human practice, it also contains many elements of fantasy. Why do you think that such a fantastic "world" (a kind of Disneyland) should be completely perfect?

"Does the Bernay's number 67**275**729 [** stands for exponentiation - K.P.] actually belong to every set which contains 0 and is closed under the successor function? The conventional answer is yes yet we have seen that there is a very large element of fantasy in conventional mathematics which one may accept if one finds it pleasant, but which one could equally sensibly (perhaps more sensibly) reject." (Parikh [1971], p.507).

As another striking example of a Platonist approach to the nature of mathematics let us consider an expression by N.N.Luzin from 1927 about the continuum problem (quoted after Keldysh [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 had been added 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:

aleph₀|----| aleph₁|----| aleph₂|----| ... |----| aleph_n|----| aleph_n+1 |----| ... |----| aleph_w|----| ...

Here aleph₀ is the countable cardinality, aleph₁ - the least uncountable cardinality etc., and aleph_w follows after all aleph_n with natural number n.

Cantor established that aleph₀< c (c denotes the cardinality of the continuum), and after this he conjectured that c = aleph₁. This conjecture is called continuum hypothesis. Long-drawn efforts by Cantor himself and by many other outstanding people did not lead to any solution of the problem. In 1905 J.Koenig proved that c is not equal to aleph_w, and that is almost all we have today...

Now we know that the continuum problem is undecidable, if we are using commonly acknowledged axioms of set theory. Kurt Goedel in 1938 (the first half of the proof) and Paul Cohen in 1963 proved that one could assume safely (i.e. without contradiction) any of the following "axioms":

c = aleph₁, c = aleph₂, c = aleph₃, ...

And even (a joke by N.N.Luzin, see Keldysh [1974]): c = aleph₁₇. Thus, the axioms of set theory do not allow determining the exact place of c on the aleph scale, although we can prove that (Ex) c = aleph_x, i.e. that c "is located" on this scale.

A 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, yet 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 (usually, we do not know before whether some problem is solvable or not). Still, 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. Do we think that, in spite of the undecidability of the continuum problem "for us, men", some "objective", "real" place for the cardinality of the continuum on the aleph scale does exist? If yes, 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 (a pure psychological phenomenon) has converted into a kind of philosophy. 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 particular axiom system can represent the "world of sets" precisely. 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 = aleph_x, yet 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 also Jech [1971], Devlin [1977]). Then we will be able to prove that c = aleph₁, and to solve some other problems that are undecidable in the traditional set theory.

We can postulate also a completely different axiom - the axiom of determinateness. 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-measurable, and that the cardinality of continuum is incompatible with alephs (except of aleph₀). In this set theory continuum hypothesis can be proved in its initial form: every infinite set of points is either countable or has the cardinality of the continuum.

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

My main conclusion: everyday work is permanently moving mathematicians to Platonism (and, as a creative method, this Platonism is extremely efficient), still, passing to methodology we must reject such a philosophy deliberately. Many essays on philosophy of mathematics disregard this problem.

Thus, a correct philosophical position of a true mathematician should be: a) Platonism - on weekdays - when I'm doing mathematics (otherwise, my "doing" will be inefficient), b) Advanced Formalism (see below) - on weekends - when I'm thinking "about" mathematics (otherwise, I will end up in mysticism). (For the initial version of this aphorism, see Hersh [1979]).

2. Investigation of Stable Self-contained Models - the True Nature of the Mathematical Method

"The human mind has first to construct forms, independently, before we can find them in things." (Albert Einstein, see Quotations by Albert Einstein).

The term "model" will be used below in the sense of applied mathematics, not in the upside-down sense used in mathematical 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 (stable, self-contained) 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 appeared as a science, all scientific theories can be divided into two classes:

a) Theories based on a developing system of principles,

b) Theories based on a stable system of principles.

In the process of development theories of class (a) are enriched with new basic principles that 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 a new theory. For example, A.Einsteins's special relativity theory can be regarded as a refinement of the classical mechanics, as a further development of I.Newton's theory. Still, 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: new theorems are being proved, new methods and algorithms are developed etc. Nevertheless, both sets of basic principles remain constant (such as they were at the lifetime of their creators).

For me, a stable self-contained system of basic principles is the distinctive feature of mathematical theories. A mathematical model of some natural process or a technical device is essentially a stable model that 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 the model (to change its definition in order to obtain more similarity with the "original") leads to a new model that must remain stable again, to enable a mathematical investigation of it.

Hence, mathematical theories are "the part of science you could continue to do if you woke up tomorrow and discovered the universe was gone" (I do not know the author of this elegant definition put on the web by Dave Rusin).

Working with stable self-contained models mathematicians have learned to draw a maximum of conclusions from a minimum of premises. This is why mathematical modeling is so efficient.

"In mathematics you don't understand things. You just get used to them." (John von Neumann, see Quotations by John von Neumann).

It is very important to note that a mathematical model (because it is stable) is not bound firmly to its "original" source. It may appear that some model is constructed badly (in the sense of the correspondence to their "original" source), yet its mathematical investigation goes on successfully. Since a mathematical model is defined very precisely, it "does not need" its "original" source any more. One can change some model (obtaining a new model) not only for the sake of the correspondence to the "original" source, but also for a mere experiment. In this way people may obtain easily various models that do not have any real "originals". In this way, even entire branches of mathematics have been developed that do not have and cannot have any applications to real problems.The stable self-contained character of mathematical models makes such "mutants" possible and even inevitable. No other branch of science knows such problems.

Polish writer Stanislaw Lem joked in his book "Summa Technologiae": mathematicians are like mad tailors: they are making "all the possible clothes" hoping to make also something suitable for dressing... (sorry - my own English translation, the initial version of this aphorism may be due to David van Dantzig, see Quotations by David van Dantzig).

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

Mathematicians have learned the ability "to live" (literally!) in the world of mathematical concepts and even (while working on some particular problem) - in a very specific "world" of a particular model. Investigation of models is mathematician's goal for goal's sake, during their work they disregard the existence of "realities" 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 a maximum of conclusions from a minimum of premises.

After one has formulated some model, it usually appears that in mathematics some work already has been done on the problem, and some methods and algorithms have been already created. This allows drawing, in real time, of many important conclusions about the model. In this way we usually obtain - relatively quickly - a new useful knowledge about the "original". 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.

Have all useful mathematical structures been created in this way, i.e. as a response to real problems external to mathematics? Of course, not. "The human mind has first to construct forms, independently, before we can find them in things." (Albert Einstein, see Quotations by Albert Einstein). Thus, at least in the fundamental physics, a more efficient source of the necessary mathematical structures is ... making "all the possible clothes" (i.e. internal mathematical "mutations")!

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

2a. What is Mathematics?

The above-stated picture of the nature of mathematics (I would like to call it Advanced Formalism) is not yet commonly acknowledged. Where is the problem, why it is so hard to look at mathematics as the investigation of stable self-contained models?

A personal communication by Svyatoslav Sergeevich 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 that 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 the fixed character of principles (axioms and rules of inference), yet it does not allow "working mathematicians" to regard mathematical theories as fixed ones."

About S.S.Lavrov - see also Kosmos. A Portrait of the Russian Space Age, R-3 page, and in Russian - http://journal.spbu.ru/2001/10/4.html,

"When the working mathematician speaks of axioms, he or she usually means those for some particular part of mathematics such as groups, rings, vector spaces, topological spaces, Hilbert spaces and so on... They are simply definitions of kinds of structures which have been recognized to recur in various mathematical situations. I take it that the value of these kinds of structural axioms for the organization of mathematical work is now indisputable." (Feferman [2000]). Of course, the fixed character of the fundamental axioms (discussed above) does not restrict the diversity and complexity of mathematical structures that can be introduced by various structural axioms.

The mathematical method is (by definition) investigation of stable self-contained models. What is, then, mathematics itself?

The first (trivial) idea: models can be more or less general. Let us compare, for example, the arithmetic of natural numbers, the relativity theory and some specific model of the Solar system. Very specific models can be investigated more successfully under the management of specialists who are creating and using them. A combination of specific experience with sufficient skills in mathematics (in one person or in a team) will be here the most efficient strategy. Investigation of general models that can be applied to many different specific models draws up contents of a specific branch of science that is called mathematics?

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

The second idea could be inspired by the following concept by Sergei Yu. Maslov on two kinds of modeling activities:

a) "left-hemispherical" activities - working in a fixed formal theory (on a fixed mathematical structure),

b) "right-hemispherical" activities - changing a theory/structure (or, inventing a new one).

See

S.Yu.Maslov. Theory of deductive systems and its applications. With a foreword by Nina B. Maslova. Translated from the Russian by Michael Gelfond and Vladimir Lifschitz. MIT Press Series in the Foundations of Computing. MIT Press, Cambridge, MA-London, 1987, xii+151 pp.

See also "G.Mints on S.Maslov" at http://www.mathsoc.spb.ru/pers/maslov/mints.html

Thus, the above conclusion that stable self-contained system of basic principles is the distinctive feature of mathematical theories, can be regarded only as the first step in discovering the nature of mathematics. Without the next step, we would end up by representing mathematics as an unordered heap of mathematical theories!

But, in fact, mathematics is a complicated system of interrelated theories each representing some significant mathematical structure (for example, natural numbers, real numbers, sets, groups, fields, algebras, all kinds of spaces, graphs, categories, computability, all kinds of logic, etc.).

Thus, in a sense, we should think of mathematics as a "two-dimensional" activity. Most of a mathematician's working time is spent along the first dimension (working in a fixed mathematical theory, on a fixed mathematical structure), but, sometimes, he/she needs also moving along the second dimension (changing his/her theories/structures or, inventing new ones).

And thus, Bourbaki's "Elements de Mathematique" can be regarded as an attempt of a systematic treatment of the second dimension of mathematics.

Do we need more than this, to understand the nature of mathematics?

3. Intuition and Axioms

4. Formal Theories

5. Hilbert's Program

6. Some Replies to Critics

7. References

Barwise J. (1942-2000) [1977]

Handbook of Mathematical Logic, Elsevier Science Ltd., 1977, 1166 pp. (Russian translation available)

Brouwer L.E.J. [1912]

Intuitionism and formalism. Inaugural address at the University of Amsterdam (October, 14, 1912). Published in Bull. Amer. Math. Soc., 1913, vol. 20, pp.81-96 (online copy).

Burris S.N. [1998]

Logic for Mathematics and Computer Science, Prentice Hall, 1998, 425 pp. (see also online Supplementary Text to this book)

Dantzig van, D. [1955]

Is 10**10**10 a finite number? "Dialectica", 1955, vol.9, N 3/4, pp. 273-278

Davis M. [1965]

The Undecidable. Basic papers on undecidable propositions, unsolvable problems and computable functions. - Raven Press, New York, 1965, 440 pp.

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.

Feferman S., Friedman H.M., Maddy P., Steel J.R. [2000]

Does mathematics need new axioms? "The Bulletin of Symbolic Logic", 2000, vol.6, N 4, pp. 401-446 (see online copy at www.math.ucla.edu/~asl/bsl/0604/0604-001.ps).

Hadamard J. [1945]

An essay on the psychology of invention in the mathematical field. Princeton, 1945, 143 pp. (Russian translation available)

Hajek P., Pudlak P. [1993]

Metamathematics of First-order Arithmetic. Springer Verlag Perspectives in Mathematical Logic, 1993, 460 pp.

Heijenoort van J. [1967]

From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931. Harvard University Press, 1967 (see online An extremely brief guide to the contents by Chad E. Brown)

Hersh R. [1979]

Some Proposals for Reviving the Philosophy of Mathematics. "Advances in Mathematics", 1979, vol. 31, pp. 31-50.

Hilbert D., Bernays P. [1934, 1939]

Grundlagen der Mathematik. Vol. I, 1934, 471 pp. Vol. II, 1939, 498 pp., Berlin (Springer) (Russian translation available)

Keldysh L.V. [1974]

The ideas of N.N.Luzin in descriptive set theory. "Russian Mathematical Surveys", 1974, vol. 29, N 5, pp. 179-193 (Russian original: "Uspekhi matematicheskih nauk", 1974, vol. 29, N 5, pp. 183-196).

Kleene S.C. [1952]

Introduction to Metamathematics. Van Nostrand, 1952 (Russian translation available)

Mendelson E. [1997]

Introduction to Mathematical Logic. Fourth Edition. International Thomson Publishing, 1997, 440 pp. (Russian translation available)

Parikh R. [1971]

Existence and Feasibility in Arithmetic. JSL, 1971, Vol.36, N 3, pp.494-508

Podnieks K.M. [1975]

The double incompleteness theorem. Proc. Latvian State University, 1975, Vol. 233, pp.191-200 (in Russian).

Podnieks K.M. [1976]

The double incompleteness theorem. Abstracts IV All-union conference on mathematical logic, Kishinev, 1976, p.118 (in Russian)

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 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. (Russian translation available)

Rashevsky P.K. [1973]

On the dogma of the natural number system. "Uspekhi matematicheskih nauk", 1973, vol.28, N 4, pp.243-246 (in Russian). See also "Russian Mathematical Surveys", 1975, vol.28, N4,

Reid C. [1996]

Hilbert. Springer-Verlag, 1996 (Russian translation available)