SCIENTIFIC COUNTER-REVOLUTION IN MATHEMATICS

As to the known Alan Sokal's hoax:

before to test IQ of "poor humanitarians", it is much better to put a place in order in our own mathematical home.

**See also in Russian**:

**"INDEPENDENT NEWSPAPER", **19 July, 2000. Supplement "IN-SCIENCE", Number 7 (32).

WEB-address: http://science.ng.ru/magnum/2000-07-19/5_mathem.html**
.**

"Left-hemispheric criminality" already during more than a century

controls the possession of "the Queen of all sciences"

Interview with Alexander Zenkin

Not so long ago, the official scientific journal of the Russian Academy of Sciences ("Bulletin of the Russian Academy of Sciences, 1999, # 6, pp. 553-558) published a paper of a well-known mathematician, vice-president of the International Mathematical Union, Academician Vladimir Arnold. The title of the paper was rather unusual, and I could tell, provocative - "Anti-Scientific Revolution and Mathematics". Already one this title induces the "natural feeling" of anxiety and perplexity among the common people who have got used to concern to a science, and especially - to the mathematics, with almost innate deep respect.

I have asked my recent interlocutor Professor, Doctor of physical and mathematical sciences Alexander Zenkin, the Leading Research Scientist of the Computing Center of the Russian Academy of Sciences to comment the paper of the Academician Àrnold, as Àlexander has declared in this interview: "the cognitive computer visualization of mathematical abstractions promises a revolution in the scientific cognition" (see his interview "Multimedia Version of Rock Painting", "Independent Newspaper - The supplement "IN-Science", # 3, March 2000. WEB-Version:

http://science.ng.ru/policy/2000-03-22/1_mmedia.html),

. **– ****Andrey Vaganov**, responsible editor of the "Independent Newspaper" Supplement "IN-Science".

Alexander Zenkin :

The situation is really not quite usual. One of the leading modern mathematicians accuses the mathematics of a dangerous propensity to the abstract thinking or of a so-called "left-hemispheric abstractionism". "In the middle of the XX century. – as Vladimir Arnold writes, in particular, - the possessing a large influence *mafia* of "left-hemispheric mathematicians" has managed to eliminate the geometry from the mathematical education (at first in France, and then also in other countries), by replacing all informal part of this discipline by training in a formal manipulation by abstract concepts ... Such "abstract" description of mathematics is unsuitable neither for education, nor for any practical applications", and, moreover, creates "the modern sharply negative attitude of the society and governments to the mathematics".

The diagnosis is undoubtedly correct and frightening, but ...not new. - More than three centuries ago, the famous bishop J. Berkeley ("famous" especially in the former USSR, as, according to V.I. Lenin's "edict", he was included into the "black list" of class aliens of the dialectic and historical materialism) wrote: *"If the mind of a man from his young years is immersed in a world of abstractions, he loses in mature age an ability to react adequately to the environmental reality"*. One of the creators just of abstract theoretical, formal bases of the modern computer science, J. von Neuman, half a century ago warned that *"an excessive formalization and symbolization of mathematical theory is dangerous for the healthy development of the mathematical science"*.

So, if very not ordinary representatives of the mathematical science put the same unfavorable diagnosis during three centuries, does it turn out that the illness is incurable?

Not entirely so. The matter is that the mathematics has arisen just as a tool of the most common and objective, and so the most abstract and formal descriptions of Nature laws. It is enough to remember the geometry of Euclid with its the most ancient axiomatic system, which without essential modifications has reached up to now and has become the reference point for all modern, formally axiomatic and really scientific constructions. Therefore to object to natural rushing of mathematics to the maximal common, abstract-theoretical description "of an objective reality" means, using a known comparison of D.Hilbert, "to take away a telescope from an astronomer or to prohibit to the boxers to use their fists".

Nevertheless, certainly it is difficult to argue with Arnold and many other mathematicians, who consider that super-abstractionism ("bourbakism", in Arnold's terms) of modern mathematics has led to that that two mathematicians working in neighbor rooms, can not already understand each other.

About thirty years ago, for the sake of "sports interest" I began to collect various "logics" used in modern logical-mathematical treatises. When their amount exceeded the second hundred, it has become clear: if the logic can be selected "on a taste" (or even can be constructed "on a need"), such notion as "science" becomes here simply inappropriate.

Perhaps, the situation somewhat reminds the famous "Babylon" epic: the sounds – symbols of abstract speeches are almost the same, but the sense, if that is present, of everyone is peculiar. What was the end of the First Babylon is described in The Holy Bible...

From my point of view, there is the only exit from the situation…

WANTED: COUNTER-CANTOR-REVOLUTION

Many of you, certainly, heard about revolutionary discoveries in mathematics, for example, the already mentioned above axiomatics of Euclid, or discovery of the differential and integral calculation by Newton and Leibniz, or, at last, the recent solution of the famous problem of Fermat. The historical-revolutionary shocks of the opposite type are also known - great crises in foundations of mathematics connected to the discovery of irrationals, infinitesimals, and famous paradoxes of the set theory. " But that... the *COUNTER*-revolution! And where? - In mathematics!? " - Many people will be surprised.

What is the common feature for great crises in foundations of mathematics, though they are divided by milleniums? If to be brief, the deep-rooted trend of mathematicians to understand the essence of Infinite. I want to note immediately that previously all mathematicians involved in these crises were simultaneously the outstanding philosophers, as, as fairly underlines D.Hilbert, "the clearing up of the essence of infinity falls outside the limits of narrow interests of special sciences and, moreover, … has become necessary for the honor of the human reason itself”. But, as God-language scientists state, Infinite is an attribute of the God, and for a finite man the ambitious encroachment on "relics" can always be related with unsafe consequences... – So, "the clearing up of the essence of infinite" requires not only of unlimited "love to freedom" ("the essence of mathematics is in its freedom", according to Cantor), but also of a wise caution.

What has served as a reason and a beginning of the Third crisis in foundations of mathematics? – It was the cheeky attempt of the not enough known at that time German mathematician Georg Cantor to actualize (by other words – to finalize) the Infinite.

I remind, that from the times of Aristotle the two contradicting (i.e. eliminating each other) concepts of Infinite are distinguished. Namely, if you begin to count

1, 2, 3, and so on, (1)

and state that it is impossible in principle to complete this process, such type of "the lack of an end" of a series (1) is called by its *POTENTIAL infinity*. However, If you agree that the series (1) does not has the last, the greatest element (nobody dared to object against this up to now, glory to the God), but nevertheless, following Cantor, suppose that "as though it can seem to be contradictory, there is nothing absurd in that" *TO DESIGNATE *("to imagine to yourself " - in the Cantor’s original) this series (1) by a certain *SYMBOL*, for example, by Greek symbol w (OMEGA), *TO NAME* this symbol w as an integer, and, by skipping *through the potential infinity *of the series (1) to continue the count "further":

? w , w + 1, w + 2, w + 3, and so on. (2)

Such rather free treatment of the series (1) is named by its actualization, and its infinity (!) "becomes" completed (!?), perfect (!?) or *ACTUAL infinity*.

I note, by the way, that after the infinite series (1) is designated by the symbol w (or by any other one), the character of its infinity becomes to be quite insignificant from the point of view of a possibility to construct the series (2), i.e. the possibility of formal outlining of elements of the series (2) is quite not connected to an actual completeness or incompleteness of a series (1).

As known, already great Aristotle warned: ** "Infinitum Actu Non Datur"**, which is equivalent to the Russian statement: "The concept of actual infinity is internally contradictory", that is why its use in science is inadmissible. As rather durating, almost 2200-year's historical practice has shown, it is not only possible, but also necessary to trust to Aristotle in problems of “the highest logic and philosophical order"!

However, at the very end of the XIX century some mathematicians, rather known at that time, have perceived the mentioned above statement almost literally as a strict mathematical "proof" of legitimacy of *actually*-infinite sets introduction into mathematics, although from a mathematical point of view this reasoning of G.Cantor is incredibly naive and contains of much more "desirable" than of the "real". The triumphal process of the "universal actualization" of infinite sets in mathematics has begun.

PATHOLOGICAL INCIDENT IN HISTORY OF MATHEMATICS

However, the tragical consequences of such rather fast step have not slowed down to have an effect. At first G.Cantor himself (1895) and soon after B.Russel (1902) have opened the broad series of paradoxes (i.e., unresolvable inconsistencies), connected exactly with the actualization of infinite sets. The Third Great crisis of foundations of mathematics has begun, which, according to the opinion of many well-known mathematicians and philosophers, "continues up to now ".

One more, already purely psychological incident is that the discovery of any similar inconsistency in any other science would mean its full discrediting and immediate closing for "all future times". However, a great number of outstanding mathematicians and philosophers of the first half of the twentieth century (such as Russel, Hilbert, Brouwer, et al.) was devoted all their life to "saving" of the Cantor set theory, and consequently, of his ideas of the infinity actualization. Sacrificing thus large "pieces" of the healthy body of the mathematical science… Russel, for example, has brought as a victim to the actual infinity the self-applicability of mathematical notions playing an important role in the usual mathematical analysis, Brouwer – the most fundamental law of logic – the law of the excluded middle, and Hilbert in his famous program of the formalization of all the mathematics actually appealed to refuse at all the semantics, i.e. an informal sense, of mathematical constructions, i.e. to refuse any connection of the mathematical theories with the physical world.

Is there any rational explanation of such a massive "sacrificing"? – More probable, there is an irrational one, similar to those motives, by which the unlucky builders of the First Babylon Tower were guided – “to become equal to the God! ". It was so courageous and tempting idea for many to come in "the open Space" of transfinite Cantor’s "behind-mirror-land", i.e. to leave the borders of usual final natural numbers, which, according to the very deep providencing note of Kronecker, “were created by The God".

I think, it was Brouwer who appeared to be the closest to the rational explanation of such the nontraditional for the classical mathematics "behavior". Eventually, he was forced "to diagnose" all Cantor’s theory as a whole as "a pathological casus in the history of mathematics, from which the future generations of mathematicians simply will be horrified".

Anyway, but this "pathological casus" proceeds already for more than hundred years. How to save "future generations of mathematicians" from forthcoming to them the "trial by horror"? If the root of all problems is in an inconsistency of the concept of the *actual* infinity itself, you see that this dispute of mathematicians and philosophers is carried out from the times of Aristotle and its end is not visible!?

Indeed, mathematicians and philosophers for already more than two thousands of years try to prove or to refute the inconsistency of the concept of actual infinity. But this dispute has an only speculative character, because the infinity has that property that it never was and never can become a subject of any experiment. In what sense? - In direct one, i.e. in the sense that "the device", say, with an inverse count of time “... 3, 2, 1" will never be created in principle, such device that once it shows "0" and the supporters, say, of the actual infinity can report joyfully: " Here we are, we have enumerated all natural numbers of a series (1) and therefore have proved that it is the actually infinite!".

However, the doubtless historical merit of G.Cantor is, that he was the first who has passed from speculative *reasonings* on a possibility or impossibility of the actual infinity to its *practical, logic-mathematical use*! And it means that due to Cantor the concept of the actual infinity *for the first time* has become available for strict formal-logical (certainly, in the sense of classical Aristotle’s logic) and mathematical analysis.

ACUPUNCTURE POINT OF THE MODERN *meta-*MATHEMATICS

Certainly, such a "strict analysis" description demands the multivolume monograph, which will be available for a wide audience interested in the nature of Infinite, and which is not more than those numerous scholastic "tractates", accumulated for many centuries in middle ages cloisters libraries. - "What is the power of a set of all devils, which can be arranged on a tip of a needle", for example?

However, not all is so hopeless. Let's recollect, that already our far ancestors perfectly owned by such a unique and effective therapeutic method, which is called today as a *method of acupuncture*. The essence of this method, as is known, consists in practical use of the following universal cybernetic principle. Namely, in any complicated system (for example, in a man or in a society) there are so-called "bottlenecks", or the attractors, or acupuncture points possessing that unique property that even the weakest actions on such points are capable of causing the essential, and not infrequently (under unprofessional interference) catastrophic modifications in a state and behavior of all complex system (living, or technological, financial, social, political, etc. ones) as a whole. – The closest example is the sad "experience" of the former USSR disintegration.

Exactly this ancient method we shall use, certainly, for the sake of reaching positive, improving effect.

What is an acupuncture point of the modern *meta*-mathematics? – No doubt, it is the famous G.Cantor's theorem concerning the uncountability of the set of all real numbers. This theorem is the only “legitimate" reason, which allows to modern meta-mathematicians to tell thoughtfully about the essential distinction of infinite sets on their power, i.e. on amount of the elements, contained in them. And all other really "practicising" mathematicians heed obediently and not less thoughtfully agree with them. Remove or prohibit only one this Cantor's theorem, and the talk itself about differing infinities will become pointless, and *meta*-mathematics itself will lose any attractiveness even for its own, the most "faithful" adherents.

What is the modern *meta*-mathematics and what is the subject of its studies? – *Meta-*mathematics (or, according to Hilbert, the modern "theory of proof") is the principal source of the modern "bourbakism" and is engaged in that it teaches naive mathematicians to prove correctly their mathematical theorems. To understand what is the role played by the famous theorem of the Cantor in the modern meta-mathematics, I shall remind some curious historical facts, which, however, seldom attract the attention even of professional mathematicians. As is known, Cantor has proved his theorem in the beginning of the nineties of the XIX century. The modern *meta-*mathematics, mathematical logic, and axiomatic set theory have added nothing new to this proof, but really use this theorem as the corner stone. However, these directions became to be independent disciplines somewhere in the thirties of the XX century, i.e., almost a half of century after Cantor had proved the theorem! Consequently, this theorem itself, and its proof have nothing to do with the frightening "bourbakised" methods of "reasonings", which are practiced today in frames of mentioned disciplines.

There is a suspicion that the proof of the theorem of Cantor represents a purely mathematical, but terribly sophisticated composition, which can be recognized by not every owner of the red mathematical diploma. - Unfortunately, in reality, not any professional mathematician will call "mathematical" the paper, in which, as, for example, in Cantor's theorem, only three notions of elementary (i.e., studied at school and easy for understanding of each educated humanitarian) mathematics – the notion of a natural number, the notion of a real number and the notion of a sequence of such numbers - are used.

What remains? Maybe the Cantor’s proof represents a treatise of about 100 pages (plus 1000 hours on the supercomputer), as, for example, the solution of the famous four-colors problem? Or of 1000 pages, as the famous proof of the Great Fermat's Theorem recently announced by the American mathematician A.Wiles? - Nothing of the kind! – The proof of the famous G.Cantor's theorem, on which all modern *meta-*mathematics and the axiomatic set theory are based, occupies only ... 10 lines! I am not stipulated, only TEN LINES written on the language of an everyday life quasi-logic of the XIX century!

I expect that Brouwer has not completed slightly his idea (see above): really, "the future generations will be horrified" ... , but only due to a shame for their mathematical predecessors, who, under the hypnosis of these ten lines only, for the whole hundred years and of their own free will hand over their, according to Gauss "queen of all sciences", into a servicing to artful "bourbakism"... – Indeed, quite fairy science-fiction thriller!

TEN LINES WHICH HAVE SHAKEN

ALL MATHEMATICAL WORLD

Of course, it is hard to believe that for 110 years passed from the moment of publication of this 10-lines proof the two tens of *generations* of professional mathematicians could not separate "grains from scales"...

Alas, the question is not about a simple historical misunderstanding, but, according to Brouwer, about a "pathological incident" in the history of mathematics. I think that the brought up to absurdity, especially in the XX century, differentiation of science areas (a so-called linguistic "professionalism") has played here not the last role. Up to that "two by two" is my "territory", where I speak my native language, and "three by three" is another "territory", where already another language is used, and where I should not already "have my own judgment". Strangely enough, this dangerous illness is also a direct, today already social-ecological, consequence of the Great Industrial Revolution of the last three centuries: one of the greatest factors of industrial progress, ** the principle of a labor division** for the sake of an increase of its effectiveness "for the welfare of ... " has as its consequence at first –

If not to plunge deeply into social-psychological "jungles" of this process, then ... philosophers once have decided that the G. Cantor's theorem is a field of professional mathematics, i.e., a zone forbidden for philosophy. On the other hand, 99 % of really working mathematicians, i.e., of such mathematicians, whose achievements are checked eventually by a number or by a practice, once have decided that the G. Cantor's theorem is a *meta*-mathematics, and since then they never came in this area. So, the mathematics has got what it has - the theorem of Cantor plus the "total bourbakization" of any common sense both of the science and of the mathematical education, according to the opinion of dear Vladimir Arnold to which both I and many other mathematicians are forced to join with a melancholy.

NEW "LIAR" IN FOUNDATIONS OF *META*-MATHEMATICS

If the theorem of Cantor is incorrect, what is the reason for such amazing survival of this "pathological incident"? It is even more surprising because the *meta-*mathematical intellectuals usually have IQ certainly well above the average level. The matter is that 10 lines of Cantor’s proof contain 7 (seven!) very nontrivial logic mistakes. I am sure, that if there were only one-two such mistakes, it would be the most likely not necessary to discuss today that "bourbakism" problem. But when in an “area" of ten lines, seven logic errors, bound in an inconceivable tangled skein of almost plausible reasonings, “are placed”, there is no wonder that this quasi-logical charade remained to be unsolved for more than hundred years.

Here is one of such mistakes. Seven centuries before Christ, the ancient Greek sage Epimenid has invented a famous paradox of the "Liar": - "I assert, that I am a liar ". Am I a liar? If I am a liar, I lie when I assert that I am a liar; therefore I am not a liar. But if I am not a liar, I speak the truth, when I assert that I am a liar; therefore I am a liar.

As it is testified by our unbiassed historical science, the common mind of the mankind, including naturally its science, already for more than 2600 years can not find an answer to this "children's" question: " Who am I eventually, a liar or not a liar? ".

Shortly and symbolically this reasoning can be written as (here **L** = "Liar"):

IF "**L**", THEN "**NOT-L**", but IF "**NOT-L**", THEN "**L** ".

But what does the Cantor's proof talk about? – "IF {**B:**} a given sequence contans all real numbers, THEN … {**NOT-B:**} the given sequence does not contan all real numbers". So, the Cantor's proof is ... a half of the classical paradox of the "Liar"-type:

IF "**B**", THEN “**NOT-B**".

Any normal man, not deprived of a feeling of humor and of the "left-right"-hemispheric symmetry of a brain, immediately has a question: "Whether is it possible to complete this "half" up to the full paradox?". It appears that it is possible! And we come to a rather unexpected for the modern *meta-*mathematics conclusion: the famous Cantor's proof is simply ... not completed by the author. But if it will be completed according to the laws of *classical* logic and *classical* mathematics, we obtain a new paradox of the"Liar" type! Thus, the Cantor's theorem and also the whole modern *meta-*mathematics based on that theorem ... are constructed on the "Liar". It is a rather doubtful foundation for the "science", which, according to D.Hilbert, pretends to a role of "the Proof Theory" of the modern (and also of all classical) mathematics. As if till now the naive mathematicians would had no idea, how they should prove their theorems.

What is the sense of the future counter-revolution in mathematics and mathematical education?

Any revolution, as we all know well, destroys that was created before. Therefore, the counter-revolution should recover the best from what has not been destroyed by the last revolution. The revolution concerning the implantation of Georg Cantor's "transfinite" ideas into the consciousness of *meta*-mathematicians could not destroy the common sense of the classical mathematics and classical Aristotle's logic. And exactly they should be rehabilitated as to the consecrated by milleniums practice right to be a strong foundation for a stable development of the science and the practical, in particular, pedagogical activity of the mankind based on that science. - It is the only.

As to details, I recommend to mathematicians (*meta*-mathematicians are
asked don't worry, - they had more than hundred years to understand and to solve
the problem) my paper "Principle
of Time-Sharing and Analysis of One Class of Quasi-Finite Probable Reasonings
(With the G.Cantor's Theorem on Uncountability As an Example)", published
in the journal "Bulletin of the Russian Academy of Sciences", 1997, volume 356,
# 6, pp. 733-735. And to philosophers I recommend a more popular, but not less
strict account in the paper "Mistake of Georg Cantor", published in the journal
"Voprosy Filosofii" ("Problems of Philosophy", 2000, # 2, pp. 45-48. All questions
and notices can be sent to the e-mail address: alexzen@com2com.ru

THE HISTORY IS REPEATED TWICE…

In the previous interview (see above) I have stated a thesis that "the cognitive computer visualization of mathematical abstractions promises a revolution in scientific cognition" Is there an inconsistency here with the discussed counter-revolution in mathematics?

It is obvious that from the formal point of view any, even *meta-*mathematical, axiomatics by definition will not bear a neighborhood of a revolution with a counter-revolution. But if is serious, revolution in scientific cognition, which is promised by cognitive “visualization of mathematical abstraction" and inevitable counter-Cantor-revolution in foundations of modern mathematics do not contradict, but are supplementary to each other.

The matter is that the V.Arnold's paper mentioned above contains one essential mathematical shortage. It (the paper) is nonconstructive, as it does not contain a criterion, which could be used to distinguish a normal, healthy, natural "abstractionism" of mathematics from the *meta-*mathematical "bourbakism". I think it is not accidentally as, apparently, to specify such criterion is basically impossible. Therefore it is necessary to eliminate the deep reasons of the illness called "bourbakism”, rather than "to treat" its external symptoms.

Therefore I see two ways to prevent the "left-hemispheric criminality", mentioned by V.Arnold.

The first way is radically comic: an elimination of the reasons generating
this kind of "criminality", i.e., the mentioned above way of anti-Cantor counter-revolution.
"Is the humor pertinent here? " - the educated reader can ask, meaning, say,
the reaction of Gotlob Frege on the famous letter of Bertrand Russel (1902),
in which the last informed the former on the discovery of the mathematical version
of the "Liar" paradox. I think that humor is pertinent for the three reasons.
Firstly, in a struggle with the psychological "scientific" prejudices the humor
is more preferable than gloomy rationalism. Secondly, together with the new
"Liar" in the Cantor’s set theory I propose also a universal medicine for all
paradoxes (see A.A.Zenkin, "New
approach to analysis of paradoxes problem". – Voprosy Filosofii (*Problems
of Philosophy*), 2000, no. 10, pp. 81-93), and healing of mathematics from
a heavy intellectual illness, anyway, is not an occasion for social-culturological
tragedies. At last, thirdly, the very concrete purely mathematical humor is
that the famous theorem of Cantor "proves" not only the existence of various
infinities, but "moreover" absolutely strictly proves also… the famous Aristotle's
thesis: *"Infinitum actu non datur"*, i.e., proves … the impossibility
of the existence of various infinities.

The second way is not less constructive: the truth should be drawn with the help of the cognitive computer visualization technology and should be presented to "an unlimited circle" of spectators in the form of color-musical cognitive images of its immanent essence (see http://www.com2com.ru/alexzen). If it is really the Truth, and if my neighbor is not a colour-blind person, we (and all other people around) shall see the same. And nobody, at all desire, will be ever able, using as a cover a "bourbakism" camouflage, to pose a falsehood as a truth, and an empty place as an outstanding scientific achievement.

And in a summary. Let us do not forget that mathematics, according to Gauss, is still really “the queen of all sciences, and theory of numbers is "the queen of mathematics", and that according to Kronecker the usual final natural numbers "were created by the God, but all the rest is a human-being work". These mathematicians were really great and knew what to bequeath to the offsprings.

New York, 16 May, 2000.

Professor Alexander Zenkin

Doctor of Physical and Mathematical Sciences,

Leading Research Scientist of the Computing Center

of the Russian Academy of Sciences,

Member of the AI-Association and the Philosophical Society of the Russia,

Full-Member of the International Federation of Artists and of the Creative Union of the Russia Artists.

Creator of the "Artistic "PI"-Number Gallery":

http://www.com2com.ru/alexzen/gallery/Gallery.html

e-mail: alexzen@com2com.ru

WEB-Site http://www.com2com.ru/alexzen

= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =

"*Infinitum Actu Non Datur*" - Aristotle.

"*Drawing is a very useful tool against the uncertainty of words*" - Leibniz.