Plato, Kant, Hilbert, Engels, Kolmogorov, formalism, platonism, Wigner, unreasonable effectiveness of mathematics, Bernays, Lloyd, Lobachevsky, Gauss, Bolyai
Part 2. Mathematics  an "ordinary" branch of science? Let us try approaching to mathematics from another direction. In many ways, mathematics is similar to physics, chemistry, biology, economics and other branches of science. And similarly, in any of these branches, one can find some ritual aspect. However, in the natural and social sciences, the ritual is steadily confronted with an independent "regulator"  the object of investigation located outside the investigator. Most of extreme fantasies are stopped by this confrontation. For example, in zoology, can one imagine a longterm investigation of fantastic nonexisting creatures? Or, in biology  investigation of "live structures" using fluorine instead of oxigen (an idea taken from science fiction)? [Added August 16, 2008. However, shortterm investigation of fantastic things is used in chemistry and in biology as well. For example, the threesex model  trying to understand, why that is impossible in the nature.] Is mathematics one of such "externalobjectoriented" branches of science, or, its position among other branches of science is, in a sense, "orthogonal", i.e. absolutely specific? Before the nonEuclidean geometries were invented in 18201850s, people could, indeed, think of mathematics as "one of"... For example, they could think of the Euclidean geometry as the "true" theory of the physical space. Materialist and marxist philosophers tried to maintain this view of mathematics as an "externalobjectoriented" science not only after the invention nonEuclidean geometries, but even after the invention of relativity and quantum theory. 

Engels and Kolmogorov The "prenonEuclidean" 1878 definition of mathematics by Friedrich Engels sounds as follows: "Pure mathematics deals with the space forms and quantity relations of the real world — that is, with material which is very real indeed. The fact that this material appears in an extremely abstract form can only superficially conceal its origin from the external world. But in order to make it possible to investigate these forms and relations in their pure state, it is necessary to separate them entirely from their content, to put the content aside as irrelevant; thus we get points without dimensions, lines without breadth and thickness, a and b and x and y, constants and variables; and only at the very end do we reach the free creations and imaginations of the mind itself, that is to say, imaginary magnitudes. Even the apparent derivation of mathematical magnitudes from each other does not prove their a priori origin, but only their rational connection." (Quoted after Part I, Chapter III from: F. Engels. AntiDühring. Herr Eugen Dühring's Revolution in Science, 1878  published by Marxists Internet Archive). In the former Soviet Union, the wording of this definition could not be changed. Thus, to save it from the progress of mathematics, another solution should have been found. And it was found by extending a la Hegel the meaning of the terms "space forms and quantity relations". For G. W. F. Hegel, quantity was a "cancelled" quality, a property "indifferent" to quality. In this way, the notion of "quantity relations" may subsume all kinds of mathematical structures, abstract algebra, abstract topology etc. included. As put by A. N. Kolmogorov in the same abovementioned 1938/1954 encyclopedia paper: "... as the result of internal needs of M. and the new requirements of natural sciences, the scope of quantity relations and space forms studied in M. is extending extremely; it includes relationships between elements in an arbitrary group, vectors, operators in functional spaces, all the variety of spaces of any number of dimensions etc. If the terms "quantity relations" and "space forms" are understood in such a wide sense, then the abovementioned definition of M. is applicable also to the new, modern stage of its evolution." (Sorry, again, my own English translation.) Now, after 1990 etc., one is feeling somewhat uneasy when remembering these ideas... But, the very direction of thought  wouldn't it be worth of a postmarxist elaboration? 

Three brilliant wrong ideas Once again: is mathematics, indeed, no more than one of the "externalobjectoriented" branches of science? Still, it seems, there is something very special in it, steadily forcing, at least some thinkers to qualify mathematics as almost "orthogonal" to "normal" branches of science. In this way, three brilliant ideas were proposed: 1. Mathematical structures are a separate world (Plato). 2. Mathematical structures are builtin mechanisms of human mind (Kant). 3. Mathematics is a complete set of axioms allowing to derive everything (Hilbert). 

Plato Examples. In geometry, straight lines have zero width, and points have no size at all. Actually, exactly 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. Plato tried to explain the specific position of mathematics by means of his surprising philosophy of two worlds: the "world of ideas" (as perfect as the "world" of the Euclidean geometry) and the "world of things". Each thing is only an imprecise, imperfect implementation of its "idea" (existing independently of the thing itself in the world of ideas). Plato's surprising notion of the nature of mathematical investigation: before a mathematician is born, his soul is living in the world of ideas, and afterwards, doing mathematics, he is simply calling back to his mind what his soul has learned in the world of ideas. I.e. mathematicians do not invent their structures, they simply "call them back"! For me, Plato's fantasies are the first necessary and brilliant step towards understanding the nature of mathematics (and the human knowledge in general). [Added November 19, 2008] Plato was the first modeler in human history! 

Kant The second necessary and brilliant step was due to Immanuel Kant. Fascinated (like as Plato) by the "precision" of the Euclidean geometry, and trying to explain, why we (i.e. people of his time) cannot imagine another kind of geometry, Kant declared it "a form of subjectivity", i.e. a builtin mechanism of the human mind used for arranging of sensations (his famous "synthetic apriori"). (Similarly, arithmetic is built into our intuition of time.) Thus, Kant concluded that Plato's invention of the "world of ideas" was not necessary. But, the invention nonEuclidean geometries showed that Kant's invention of the "synthetic apriori" was not necessary as well... [Immanuel, please, excuse me, if I'm using a simplified model of your philosophy instead of the genuine one. A really useful philosophy should be robust, not "subtle". By worrying about "subtleties" and including them into philosophy, one is taking the risk of "overfitting" (a term used in statistics and data mining), i.e. the risk of modeling "noise". ] 

From theorems to axioms Let's consider one of the first mathematical theorems (6th century BC): Theorem. Every prime number is followed by another prime number. Complete empirical verification is impossible here  there is "an infinity" of natural numbers (whatever it means). How can mathematicians convince themselves that the assertion of the theorem is true? Proof. Having k prime numbers p_{1}, p_{2}, ..., p_{k}, let's consider the number p_{1}p_{2}...p_{k}+1. It is divisible by none of these k numbers, hence, it is divisible by a different prime number. Q.E.D. Why is this convincing (for mathematicians)? By such a proof, one derives one assertion from other ones, no more than that. Where is the end of this reduction chain? Preferably  among the axioms  the assertions that are considered as true by all (mathematicians). In this way, Greeks arrived at the idea of axiomatization. In 19th century, this idea was refined up to its limits: Gottlob Frege and Charles S. Peirce developed the concept of formalization ("total axiomatization"). If we will succeed in formalizing all of mathematics, then what will be the result  a final and complete list of axioms allowing to "derive everything"? As the first step  a brilliant idea! Indeed, at the very beginning of 20th century, David Hilbert conjectured that there must be a complete list of axioms allowing to derive formally all (past and future) mathematical theorems. I.e. there must be a complete list of axioms allowing to define all the mathematical structures (numbers, functions, sets, groups, spaces etc.), instead of merely describing them as something existing independently of the axioms. Complete axioms are not mere descriptions, they are definitions! The third necessary and brilliant step towards understanding the nature of mathematics! [Added November 19, 2008] Hilbert's idea that mathematics is not a pretty free exploring of various axiom systems, but exploring of a unique fundamental axiom system allowing to "derive everything", resembles a kind of platonism ("formalist platonism"?). 

In which sense do mathematical structures exist? At the time, Hilbert's conjecture was understood only by a few. The dominating feeling was: mathematical structures exist independently of axioms, and by means of axioms mathematicians only try to describe these structures. People arrive at the idea of independent existence of basic mathematical structures very early  already when studying mathematics at school. I would propose you the following selftest. Let's consider the sequence of the socalled twin primes: (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), In 1849 Alphonse de Polignac conjectured that this sequence continues infinitely. Today, 150 years later, this (the famous twin prime) conjecture still remains unproved. Do you think that, moving forward along the sequence of natural numbers, we have only two possibilities? Namely, a) Twin pairs appear over and again (i.e. the twin prime conjecture is true). b) 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). Analyze your feelings... Can you can imagine a third possibility? Or, do you think that the infinite sequence of natural numbers is a kind of "road to infinity", and that we can go this road literally step by step? And, this is because we will either: a) reach the last twin pair, and after this  will not meet such pairs any more, or b) meet twin pairs over again ad infinitum. Tertium non datur! Couldn't this be unconscious thinking by analogy with physical roads? But is the infinite sequence of natural numbers completely similar to long physical roads? 

The third possibility Gödel's Incompleteness Theorem. If some system of axioms is formulated precisely enough, and if it allows proving of the simplest properties of natural numbers, then this system cannot be perfect: either it contains contradictions, or it cannot solve some problems that belong to its domain of competence. Kurt Gödel proved this theorem in Vienna, during the summer of 1930. Picture
from Photo gallery by BVI. From a practical point of view, Gödel's theorem makes only a very strong general prediction. But this prediction was confirmed concretely many times starting from 1963, when Paul Cohen (19342007) proved that the axioms of set theory (if they do not contain contradictions) cannot solve the famous continuum problem. The inventor of set theory Georg Cantor arrived at this problem in 1878: can there exist sets of real numbers bigger than the set of natural numbers, but smaller than the set of all real numbers? Cantor conjectured that such sets cannot exist. Thus, it may well happen that our axioms cannot solve also the twin prime problem, i.e. they cannot neither prove the twin prime conjecture, nor disprove it. This represents the third possibility I had in mind: maybe, our mathematical axioms cannot prove neither that the twin prime sequence continues infinitely, nor that it breaks off? Let's assume that this is the case: our axioms are not able, indeed. Do you think that despite of this sad conclusion, "in fact", the sequence of twin primes either continues infinitely, or breaks off? Tertium non datur? Analyze your feelings... Does this mean that our axioms do not describe precisely enough the "real" sequence of natural numbers? If so, then, this elusive sequence, which kind of object could it be? In which sense does it exist? 

Maybe, natural numbers exist in the physical world? Before Einstein, people could think that they exist. The Universe "was" infinite and isotropic, hence, walking in some fixed direction, and counting steps, one will be forced to "use" all natural numbers. Thus, natural numbers "exist in the physical world", and hence, to every definite question about these numbers there must exist a definite answer  "yes", or "no". In particular, such a definite answer must exist to the question about the existing amount of twin primes (the twin prime sequence either breaks off, or continues infinitely  tertium non datur!). Still, modern physics does not support this picture anymore. According to the currently acknowledged cosmological model, there are far less than 10^{1000} elementary particles in the Universe. Thus, although arithmetic is provoking us to imagine a sequence of 10^{1000} particles, such a structure cannot exist in the natural world! Doesn't this mean that the infinite "tail" of the natural number sequence is no more than our fantasy? 

Infinity does not exist in the physical world! Doesn't it, indeed? The problem appears to be somewhat more complicated. Arithmetic is provoking us to imagine not only counting of particles, but also "counting" of sets of particles, sets of sets of particles etc. Having N particles, we have 2^{N} sets of particles, 2^2^{N} sets of sets etc. In this way, even starting with the empty set, we can "obtain" an arbitrarily large set of "things". (This is similar to the traditional way of explaining the "semantics" of set theory: if nothing exists, then there is the empty set o, hence, there is also the set {o} with o as the only member, after this we build the set {o, {o}} etc. ad infinitum...) However, from the physical point of view, this "activity" breaks off very soon because: a) of spontaneous transformations of elementary particles (i.e. we cannot determine with absolute precision "where, when, and what is what"); b) the speed of light is a finite magnitude (i.e. if a set taking up very large space, then the idea of a simultaneous existence of its members is not reasonable); c) the age of the Universe is finite, etc. Perhaps, the following conclusion of Seth Lloyd brings the necessary order into the situation: "All physical systems register and process information. The laws of physics determine the amount of information that a physical system can register (number of bits) and the number of elementary logic operations that a system can perform (number of ops). The Universe is a physical system. The amount of information that the Universe can register and the number of elementary operations that it can have performed over its history are calculated. The Universe can have performed 10^{120} ops on 10^{90} bits ( 10^{120} bits including gravitational degrees of freedom)." S. Lloyd. Computational capacity of the universe. Physical Review Letters, 2002, vol. 88, issue 23, 4 p. (online, extended online version). Thus, during its existence time, as a computer, the Universe could not have produced very much... Anyway, we are forced to conclude that the infinite sequence of natural numbers, though being abstracted from real counting processes of the human practice, cannot be an exact representation of some structure in the physical world. And thus, the infinite "tail" of the natural number sequence is no more than our fantasy, indeed! As put by A. N. Kolmogorov in his lecture "Contemporary views on the nature of mathematics"  the creatures living in a finite world can arrive at the idea of infinity: "... let us imagine an intelligent creature living in a world of finite complexity, [a world] taking only a finite number of physically distinct states and evolutioning in a "discrete time". ... It is possible to explain plausibly how such a creature, according to its structure, unable to cover all the complexity of the world around it and confronted with systems of growing complexity and consisting of very large numbers of elements, will create during the process of its entirely practically and reasonably oriented activities the concept of an infinite sequence of natural numbers." (Sorry, my own English translation.) The original text: 

Formalism If the infinite sequence of the natural numbers is not a "physical structure", then what kind of object could it be? In which sense does it "exist"? And, in which sense do exist the even more complicated mathematical structures: real numbers, functional spaces, algebras, topological spaces, uncountable infinite sets, large cardinals, categories etc.? Are they even "less physical" than natural numbers? First, let's consider the simplest possible answer to these questions: by themselves, mathematical structures do not exist at all, there exist only the axioms defining these structures. Indeed, this should be qualified as the simplest possible solution of all philosophical controversies around mathematics. You may reject it as a "wrong" solution, but you can't deny that it is the simplest possible one! 1) The axioms do exist  one can write them down! 2) Mathematicians explore willingly (and with pleasure) the consequences of any "interesting" axioms, even if they do not know in advance about "things behind them". In the philosophy of mathematics this approach is called formalism. Formalists insist that mathematicians must be allowed to explore any axioms, and that axioms are the only kind of objects defined well enough for discussion. Two persons each using a different system of axioms, could come to an agreement only after they would recognize this fact. (This definition of formalism should not be confused with the caricature of formalism invented by simpleminded opponents: formalism viewing mathematics as a meaningless game with symbols. These simpleminded people believe that every separate symbol must possess its "meaning". In fact, the "meaning" is represented not by symbols, but by the relationships of symbols, i.e. ultimately  by the axioms.) 

Three
prominent formalists This kind of formalism was the starting point for Nikolai Ivanovich Lobachevsky. He decided to explore the consequences of the hypothesis: "given a line and a point outside it, one can draw through the point several different lines that do not intersect the given one". And he could not know in advance, "is this possible, or not" (at the beginning it looked, much more probably, "impossible" ). Working for several years under this "impossible" hypothesis and finding no contradictions, Lobachevsky arrived at the idea of its "possibility". I.e. one may consider two different geometries  the usual Euclidean one, and the "imaginary geometry", and one may ask: which of the two is better as a description of the physical space? And try to use astronomical measurements to decide... See also: Carl Friedrich Gauss in a letter to Franz Adolph Taurinus, Goettingen, November 8, 1824: "... But it seems to me that in spite of the wordmastery of the metaphysicians, we know really too little, or even nothing at all, about the true nature of space to be able to confuse something that seems unnatural with absolutely impossible. If nonEuclidean geometry is the real one and the constant is incomparable to the magnitudes that we encounter on earth or in the heavens then it can be determined aposteriori. I have therefore occasionally for fun expressed the wish that Euclidean geometry not be the real one, for then we would have a priori an absolute measure." Full text: Gauss And NonEuclidean Geometry by Stanley N. Burris (see also the German original at the Göttinger DigitalisierungsZentrum). Janos Bolyai in a letter to his father, November 3, 1823: he had "... created a new, another world out of nothing...". (Quoted after the biography of Janos Bolyai by MacTutor History of Mathematics archive.) 

Formalism: the consequences Of course, none of the fomalists will deny that most of the axioms have their roots in the practical, technical and scientific experience of humans, but the axioms can go much further than this restricted experience: they may extrapolate, idealize, deform, distort etc. As the result, they can produce "structures" that do not have exact analogs in the physical world. From the formalist point of view, the axioms of arithmetic and the axioms of set theory do not describe  they define the sequence of natural numbers. And, if (maybe) these axioms are not able solve the twin prime problem, then we should accept this, or  we can try extending or modifying the axioms. From the formalist point of view, Gödel's Incompleteness Theorem means the inevitability of dialectics in the development process of mathematics. If you have formulated your mathematical axioms precisely enough, then one of the following two disasters is awaiting you inevitably: a) Your axioms will lead to contradictions (then you will be forced to try improving them). b) Your axioms cannot solve some problems that belong to their domain of competence (i.e., most probably, again, you will be forced to try improving them). Thus, a fixed ("frozen") system of axioms cannot be perfect (exactly because of its "frozen" character!), and hence, should be subject of improvement in the future. And hence, contrary to Hilbert, mathematics is NOT a complete set of axioms allowing to "derive everything". What could be better than this colorful and exciting process of running into contradictions and/or undecidable problems, changing the axioms and running into... again? 

[Added September 11, 2009] "...the realm might be illusion. But the experience?" As a rule, working mathematicians are not satisfied with the formalist picture of mathematics. "When I’m working I sometimes have the sense – possibly the illusion – of gazing on the bare platonic beauty of structure or of mathematical objects, and at other times I’m a happy Kantian, marvelling at the generative power of the intuitions for setting what an Aristotelian might call the formal conditions of an object. And sometimes I seem to straddle these camps (and this represents no contradiction to me). I feel that the intensity of this experience, the vertiginous imaginings, the leaps of intuition, the breathlessness that results from “seeing” but where the sights are of entities abiding in some realm of ideas, and the passion of it all, is what makes mathematics so supremely important for me. Of course, the realm might be illusion. But the experience?" Barry Mazur. Mathematical Platonism and its Opposites. NEWSLETTER OF THE EUROPEAN MATHEMATICAL SOCIETY, Issue 68, June 2008, pp. 1718 (online copy). 

Platonism? The formalist solution is only one of the possible solutions of the problem: in which sense do mathematical structures exist? As we have concluded above, the fundamental mathematical structures do not exist in the physical world. But the feeling of most working mathematicians says that these structures "must exist independently of us, humans" (our axioms included). Then, perhaps, they exist in a specific "third reality", to which human mind has access by means of intuition? In the philosophy of mathematics this approach is called platonism. Platonists insist that mathematicians must investigate the unique version of mathematical structures existing in the "third reality". From the platonist point of view, the infinite sequence of natural numbers is a kind of "third reality", and hence, "in it", the twin prime conjecture must be either true, or false. Tertium non datur: in the "third reality", either the sequence of twin pairs breaks off, or it continues ad infinitum. From the platonist point of view, Gödel's Incompleteness Theorem means that a fixed system of axioms cannot give a complete description of the infinite sequence of natural numbers. The situation in the theory of large cardinals seems supporting this illusion... [Added after the lecture  after a remark from the audience.] However, see the last sentence of the following conference abstract: N. V. Belyakin. On one OMEGAinconsistent formalization of set theory. The 9^{th} Asian Logic Conference, 1619 August, 2005, Novosibirsk, Russia. 

The positive role of platonism in mathematics Platonist attitude to mathematical structures is common for most of mathematicians (however, as a rule, they only unwillingly agree to discuss the "meaning" of their "business"). They believe that the objects of their investigations "exist" independently of them (and independently of "us, humans" at all). And the main point: they transfer to their "third reality"  unconsciously and by analogy  many features of the physical reality (first of all  the Law of the Excluded Middle). It seems that, "for us, humans", platonism is the most efficient way of working with imaginary structures. For example, isn't this the best way of working with natural numbers  imagine the sequence of these numbers as a "road to infinity"  and go on searching for the last pair of twin primes. Because, "on the road"  either there is the last pair, or there is not! Mathematicans are "living" for years among their structures as in a specific "world", without much reflecting upon the real meaning of these structures. Working in this way (feeling themselves as platonists, but in fact, working within some definite system of axioms) mathematicians have developed their ability to draw a maximum of conclusions from a given set of premises. This is my solution to the problem of "The Unreasonable Effectiveness of Mathematics in the Natural Sciences" (by Eugene Wigner, Communications on Pure and Applied Mathematics, 13 (1), pp.1–14, 1960). [Added September 11, 2009] And this my answer to Barry Mazur. 

Platonism as a philosophy? Thus, as a method, platonism is extremely efficient  at least in mathematics. Do we need more? As a regular term, "platonism in mathematics" is used since the lecture delivered June 18, 1934 by Paul Bernays: P. Bernays. Sur le platonisme dans les mathematiques. L'enseignement mathematique, Vol. 34 (1935), pp. 5269. (Online English translation by Charles D. Parsons). Bernays considered mathematical platonism as a method that can be  "taking certain precautions"  applied in mathematics. "... platonism reigns today in mathematics", but: "... It is also this transcendent character which requires us to take certain precautions in regard to each platonistic assumption." But how about platonism as a general philosophical idea? If we adopt the platonist hypothesis about the existence of some "third reality", can this give us any practical advantages? The very idea of "third reality" is due to Plato (his absolutely perfect "world of ideas" over the "world of things"). But sceptical Kant did not find sufficient reasons for the introduction of "third reality". Instead, he ascribed mathematical structures to the "second reality", i.e. declared them as builtin mechanisms of human mind. In his famous October 14, 1912 lecture Intuitionism and Formalism, L. E. J. Brouwer proposed to retain Kant's idea only by 50%  declare that, among all the mathematical structures, only the sequence of natural numbers is a builtin mechanism of human mind. As a result of Brouwer's critique, we have now two kinds of mathematical platonism: a) The 100%platonism, or the settheoretical platonism (people believing in the existence of some "unique world of sets" to be investigated). b) The 50%platonism ("subtle platonism"), or platonism "with respect to the natural numbers" (people rejecting the idea of a unique "world of sets", but believing in the existence of a "unique sequence of natural numbers"). The 50%platonists are advocating the same illusion as Kant and intuitionists, but they are ascribing it to Plato's "third reality" instead of the "second" one. This completes our second flow of associations. 
Plato, Kant, Hilbert, Engels, Kolmogorov, formalism, platonism, Wigner, unreasonable effectiveness of mathematics, Bernays, Lloyd, Lobachevsky, Gauss, Bolyai