Recently the article "Foundations of mathematics" in Russian Wikipedia attracted my attention by lots of strange (and often absurd) declarations, in particular, it is written there that David Hilbert (it is not clear, apparently, in some period of his life?) accepted the intuitionistic views.

When discussing this with the Wikipedia authors I understood that a large part of those oddities comes from the Morris Kline book "Mathematics: The Loss of Certainty". As an illustration, at page 250 (Oxford University Press, 1980) he writes that

In metamathematics, Hilbert proposed to use a special logic that was to be free of all objections. The logical principles would be so obviously true that everyone would accept them. Actually, they were very close to the intuitionist principles. Controversial reasoning--such as proof of existence by contradiction, transfinite induction, actually infinite sets, impredicative definitions, and the axiom of choice was not to be used.

Can anybody explain me what this can mean? Is it possible that Hilbert indeed agreed with intuitionists in some moment of his life? If yes, when was that, and when did he change his mind?

Or the explanaltion is that Kline simply does not understand what he describes (and therefore his book can't be treated as a reliable source)?

I would be grateful to people who could cast light on this because from what is written in the Wikipedia article it is seen that the declarations like those from the Kline book generated a series of further interpretations in other "popular texts", which led finally to absolutely absurd conlusions where, for example, Hilbert is presented as a loser, mathematics as a part of science that "abandoned claims for significance of its results", etc.

I can't read this, but I am not a specialist in history of mathematics, and it's difficult for me to understand what can lie behind all this. On the other hand the Wikipedia rules are contradictory, they give a possibility to the people who reached some power in its feudal stairs to abuse this power. So I need help.

EDIT. From the discussion in comments it became clear that the following detail could resolve the main part of my doubts:

Is it true that Hilbert agreed somewhere that the law of excluded middle (and the proofs by contradiction) must be rejected?

This sounds completely implausible.

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    $\begingroup$ IMO, Kline' s book is "non technical" but, in general, interesting and useful. $\endgroup$ Oct 13, 2018 at 15:35
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    $\begingroup$ The question that arises for me is why would anyone be consulting Kline's "popular book" for this, a book that is well-known to grind a lot of axes (and this is mentioned in the many published reviews of his book when it appeared), rather than actual historical studies? It's like consulting Martin Gardner's book Relativity for the Million when wanting to write about some technical nuances of general relativity. (Actually, I've had Gardner's book since the mid 1960s, and it's probably worth looking at!) $\endgroup$ Oct 13, 2018 at 16:18
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    $\begingroup$ @DaveL.Renfro "grinding axes" means saying something controversial? $\endgroup$ Oct 13, 2018 at 16:24
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    $\begingroup$ @MauroALLEGRANZA are there detailed studies of the reliability of Kline's book? They would be more useful for me. $\endgroup$ Oct 13, 2018 at 17:01
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    $\begingroup$ @Sergei Akbarov: For definitions and other information about the phrase "axe to grind", see Origin of “to have an axe to grind”. Regarding reviews of Kline's book, the first 10 hits of this JSTOR search gives 10 reviews. Incidentally, the search does not require JSTOR access (how I did it). As for obtaining the reviews themselves without JSTOR access, many are in well known journals available in many university libraries, for those sufficiently interested to actually delve into the reception of Kline's book. $\endgroup$ Oct 13, 2018 at 17:18

3 Answers 3


Hilbert had a long career and, unsurprisingly, used different logics for different purposes. For his mathematical work, Hilbert is well known as a proponent of classical reasoning, including the law of the excluded middle and the axiom of choice.

For his consistency program, however, Hilbert referred to "finitary" methods. This program is described well in the SEP article. Hilbert did not formally define a logical system for finitism. He explained his motivation for not doing so in his speech "On the infinite" (1925), although his reasoning is still not completely clear to me:

In analyzing an existential statement whose content cannot be expressed by a finite disjunction, we encounter the infinite. Similarly, by negating a general statement, i.e., one which refers to arbitrary numerical symbols, we obtain a transfinite statement. For example, the statement that if a is a numerical symbol, then a + 1 = 1 + a is universally true, is from our finitary perspective incapable of negation. We will see this better if we consider that this statement cannot be interpreted as a conjunction of infinitely many numerical equations by means of `and' but only as a hypothetical judgment which asserts something for the case when a numerical symbol is given.

From our finitary viewpoint, therefore, we cannot argue that an equation like the one just given, where an arbitrary numerical symbol occurs, either holds for every symbol or is disproved by a counter example. Such an argument, being an application of the law of excluded middle, rests on the presupposition that the statement of the universal validity of such an equation is capable of negation.

At any rate, we note the following: if we remain within the domain of finitary statements, as indeed we must, we have as a rule very complicated logical laws. Their complexity becomes unmanageable when the expressions 'all' and 'there exists' are combined and when they occur in expressions nested within other expressions. In short, the logical laws which Aristotle taught and which men have used ever since they began to think do not hold. We could, of course, develop logical laws which do hold for the domain of finitary statements. But it would do us no good to develop such a logic, for we do not want to give up the use of the simple laws of Aristotelian logic. Furthermore, no one, though he speak with the tongues of angels, could keep people from negating general statements, or from forming partial judgments, or from using tertium non datur. What, then, are we to do?


It seems from this speech that Hilbert was at least partially concerned with the law of the excluded middle in the context of finitism as he understood it.

Modern formalizations of finitary reasoning typically do include the law of the excluded middle, although they can be weak in other ways. For example the theory of Primitive Recursive Arithmetic, often associated with finitism, is often presented as a theory with no quantifiers.

Separately, the work of Glivenko and Gödel in the 1930s showed that the law of the excluded middle on its own does not lead to contradiction. For example, Gödel proved that if first-order logic without excluded middle is consistent, then so is first-order logic with the law, and if Heyting Arithmetic without excluded middle is consistent then so is Peano Arithmetic, which consists of Heyting Arithmetic and the law of the excluded middle. In some settings, these results reduced the interest in the law of the excluded middle as a possible source of inconsistency. Of course, people may still use logics without LEM in order to ensure that proofs are more constructive or correspond more closely with algorithms.

Regarding the book "Mathematics: the loss of certainty", I will simply quote the final paragraph of the review from the American Mathematical Monthly:

Finally, Professor Kline does not deal honestly with his readers. He is a learned man and knows perfectly well that many mathematical ideas created in abstracto have found significant application in the real world. He chooses to ignore this fact, acknowledged by even the most fanatic opponents of mathematics. He does this to support an untenable dogma. One is reminded of the story of the court jester to Louis XIV: the latter had written a poem and asked the jester his opinion. "Your majesty is capable of anything. Your majesty has set out to write doggerel and your majesty has succeeded". On balance, such, alas, must be said of this book.

This is unfortunate because other books, such as Kline's "Mathematical Thought from Ancient to Modern Times", do not have the same issues, and "The Loss of Certainty" can unfortunately cast a shadow on those as well.

  • $\begingroup$ Carl, these examples of courtesy look really impressive. Like pictures from another planet: you understand that this does not contradict to physical laws, but at the same time obviously this can't exist in your world. $\endgroup$ Oct 14, 2018 at 20:28
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    $\begingroup$ For a (maybe relevant) comment on tertium non datur in finitistic metamathematics,see p.34 of Hilbert & Bernays' Grundlagen: "The complicated situation we find here with respect to negating judgments from the finitistic standpoint corresponds to Brouwer's thesis of the invalidity of the law of the excluded middle for infinite totalities. Indeed, this invalidity holds for the finitistic standpoint in that we cannot find a finitistic negation of existential and universal judgments satisfying the law of the excluded middle." $\endgroup$ Oct 15, 2018 at 6:28
  • $\begingroup$ I always understood Hilbert's finitism as a requirement that the Gentzen deduction tree must be finite (although, of course, it's impossible to understand and to explain what infinite deduction tree can be). However, Carl, I think from the point of view of the Russian intellectual civil war, the reproach to Kline of hiding some facts is not enough for declaring his book an unreliable source. There must be a direct accusation of falsification. $\endgroup$ Oct 15, 2018 at 8:53

The following statement is attributed to Hilbert:

"Taking the principle of excluded middle from the mathematician would be the same, say, as proscribing the telescope to the astronomer or to the boxer the use of his fists. To prohibit existence statements and the principle of excluded middle is tantamount to relinquishing the science of mathematics altogether."

I do not know the original source of this quotation, but have a look at


https://pdfs.semanticscholar.org/94a8/211d31e5ab6d67114b3451ea7f3e2bb6650b.pdf (p. 24)


I think the quotation is authentic - it is consistent to what we know about Hilbert. In fact, Hilbert felt personally offended by Brouwer and (Hilbert's own student!) Hermann Weyl who supported Brouwer. Let me quote from https://en.wikipedia.org/wiki/Foundations_of_mathematics#Foundational_crisis :

"The main opponent was the intuitionist school, led by L. E. J. Brouwer, which resolutely discarded formalism as a meaningless game with symbols (van Dalen, 2008). The fight was acrimonious. In 1920 Hilbert succeeded in having Brouwer, whom he considered a threat to mathematics, removed from the editorial board of Mathematische Annalen, the leading mathematical journal of the time."

  • $\begingroup$ Paul, yes, I read this and similar things in different places, in particular, in "Mathematical Encyclopedia". That is why what Kline writes is strange for me. But is it possible that Kline is not completetly wrong, and Hilbert changed his opinion in some moments in his life? Can this be that he was closer to intuitionists at the beginning and left them after some time? Or (what is much more plausible for me) Kline just writes stupidities? $\endgroup$ Oct 14, 2018 at 9:38
  • $\begingroup$ I cannot imagine that Hilbert changed his opinion even for "a moment" since that would have counteracted his life's work. When Brouwer came up with his ideas (probably for the first time in 1908 in his article "De onbetrouwbaarheid der logische principes"), Hilbert was in the end of his forties and on a purely psychological level it seems not plausible that he felt any sympathy for radical ideas devalueing everything of what he stood for. However, all that is speculative and a reliable answer could only be given if one is willing to study everything Hilbert has ever written. $\endgroup$
    – Paul Frost
    Oct 14, 2018 at 9:59
  • $\begingroup$ Paul, I suspect there must be reviews of Kline's book where the details like this are discussed. I did not see yet the articles that Dave L. Renfro recommended, do you happen to know a text where what we are speaking about is clarified? $\endgroup$ Oct 14, 2018 at 10:06
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    $\begingroup$ I only found zbmath.org/?q=an%3A0458.03001 $\endgroup$
    – Paul Frost
    Oct 14, 2018 at 11:59
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    $\begingroup$ @SergeiAkbarov: while Kline's book is of quite low quality, I think that your posts may be too aggressive here, and have the effect of turning off some people who might comment, rather than encouraging them to comment. Problems with Wikipedia are not an emergency for this site. $\endgroup$ Oct 14, 2018 at 19:10

I thank @DaveL.Renfro for the key idea to look at the reviews. I found several of them that can be considered as more or less satisfactory answers to what am asking about.

Since there are people who have difficulties with the access to western journals, I believe it will be useful to give here several citations. I give also translations into Russian to make this reading easier for my compatriots. I posted this also in Wikipedia, but my experience with its Russian branch makes me doubt that this information will live there for a long time.

Raymond G. Ayoub, The American Mathematical Monthly, Vol. 89, No. 9 (Nov., 1982), pp. 715-717:

"For centuries, Eucidean geometry seemed to be a good model of space. The results were and still are used effectively in astronomy and in navigation. When it was subjected to the close scrutiny of formalism, it was found to have weaknesses and it is interesting to observe that, this time, it was the close scrutiny of the formalism that led to the discovery (some would say invention) of non-Eucidean geometry. (It was several years later that a satisfactory Eucidean model was devised.)

This writer fails to see why this discovery was, in the words of Kline, a "debacle." Is it not, on the contrary, a great triumph?...

Professor Kline does not deal honestly with his readers. He is a learned man and knows perfectly well that many mathematical ideas created in abstracto have found significant application in the real world. He chooses to ignore this fact, acknowledged by even the most fanatic opponents of mathematics. He does this to support an untenable dogma. One is reminded of the story of the court jester to Louis XIV: the latter had written a poem and asked the jester his opinion. "Your majesty is capable of anything. Your majesty has set out to write doggerel and your majesty has succeeded." On balance, such, alas, must be said of this book."

На протяжении веков евклидова геометрия казалась хорошей моделью пространства. Ее результаты использовались и до сих пор используются в астрономии и навигации. Когда она подверглась пристальному анализу, обнаружилось, что она имеет слабые стороны, и интересно заметить, что именно этот тщательный формальный анализ привел к обнаружению (кто-то сказал бы, к открытию) неевклидовой геометрии. (Для которой несколько лет спустя была разработана удовлетворительная евклидова модель.)

Этот писатель не представляет себе это открытие иначе как, по словам Клайна, «фиаско». Но разве это не великий триумф?..

Профессор Клайн нечестен со своими читателями. Он образованный человек и прекрасно знает, что многие математические идеи, созданные как абстракция, нашли важные применения в реальном мире. Он предпочитает игнорировать этот факт, признанный даже самыми фанатичными противниками математики. И делает это, чтобы поддержать несостоятельную догму. Напомним историю о придворном шуте Людовика XIV: последний написал стихотворение и спросил у шута его мнение: «Ваше величество способно на что угодно. Вашему величеству захотелось написать скверные стишки, Ваше величество преуспело и в этом.» Увы, это должно быть сказано и об этой книге.

John Corcoran, Mathematical Reviews, MR584068 (82e:03013):

"The overall purpose of the book is to advance as a philosophy of mathematics a mentalistic pragmatism which exalts “applied mathematics” and denigrates both “pure mathematics” and foundational studies. Although its thesis is predicated in part on the deep foundational achievements of twentieth century logicians, the basic philosophy is a close cousin of various philosophies which were influential in the nineteenth century. Moreover, as can be seen from the above-listed ideas, the author’s grasp of twentieth century logic is not reliable. Accordingly he finds it surprising (p. 322, 323) that Hilbert, Gödel, Church, members of the Bourbaki school, and other “leaders in the work on foundations affirm that the mathematical concepts and properties exist in some objective sense and that they can be apprehended by human minds”. His only argument against the Platonistic realism of the mathematicians just mentioned is based on his own failure to make the distinction between (human) error and (mathematical) falsehood (p. 324)...

The author does not seem to realize that in order to have knowledge it is not necessary to be infallible, nor does he recognize that loss of certainty is not the same as loss of truth. The philosophical and the foundational aspects of the author’s argument are woven into a comprehensive survey and interpretation of the history of mathematics. One could hope that the argument would be somewhat redeemed by sound historical work, but this is not so. Two of the periods most important for the author’s viewpoint are both interpreted inconsistently. (a) In some passages the author admits the obvious truth that experience and observation played a key role in the development of classical Greek mathematics (pp. 9, 18, 24, 167). But in other passages, he alleges that classical Greek mathematicians scorned experience and observation, founding their theories on “self-evident truths” (pp. 17, 20, 21, 22, 29, 95, 307). (b) In some passages the author portrays the beginning of the nineteenth century as a time of widespread confidence in the soundness of mathematics (pp. 6, 68, 78, 103, 173), but in other passages he describes this period as a time of intellectual turmoil wherein mathematicians entertained grave doubts about the basis of their science (pp. 152, 153, 170, 308)...

One can only regret the philosophical, foundational, and historical inadequacies which vitiate the main argument and which tend to distract attention from the many sound and fascinating observations and insights provided by the book."

Общая цель книги - продвинуть в качестве философии математики менталистический прагматизм, который превозносит «прикладную математику» и очерняет «чистую математику» и фундаментальные исследования. Хотя тезис автора частично основан на глубоких основополагающих достижениях логиков двадцатого столетия, основная его философия - близкая родственница различных философий, существовавших в девятнадцатом веке. Более того, как видно из приведенных выше тезисов, авторское понимание логики двадцатого века несерьезно. Он находит удивительным (стр. 322, 323), что Гильберт, Гёдель, Чёрч, члены школы Бурбаки и другие «лидеры в работе над основаниями» утверждают, что математические концепции и свойства существуют в каком-то объективном смысле и что они могут быть восприняты человеческим разумом. Его единственный аргумент против платонического реализма этих математиков основан на его собственной неспособности провести различие между (человеческой) ошибкой и (математической) ложью (стр. 324)…

Автор, похоже, не понимает, что для того, чтобы иметь знание, нет необходимости быть непогрешимым, и он не признает, что потеря уверенности - это не то же самое, что потеря истины. Философские и основополагающие аспекты авторской идеи вплетены в обширный обзор и интерпретацию истории математики. Можно было бы надеяться, что его аргумент будет в какой-то мере подтвержден убедительным историческим исследованием, но это не так. Два из периодов, наиболее важных с точки зрения автора, интерпретируются противоречиво. (а) В некоторых отрывках автор представляет как очевидную истину, что опыт и наблюдение играли ключевую роль в развитии классической греческой математики (стр. 9, 18, 24, 167). Но в других местах он утверждает, что классические греческие математики презирали опыт и наблюдения, основывая свои теории на «самоочевидных истинах» (стр. 17, 20, 21, 22, 29, 95, 307). (б) В некоторых отрывках автор изображает начало девятнадцатого века как время широко распространенной уверенности в обоснованности математики (стр. 6, 68, 78, 103, 173), но в других местах он описывает этот период как время интеллектуальных потрясений, когда математики испытывали серьезные сомнения относительно основ своей науки (стр. 152, 153, 170, 308)… Можно только сожалеть о философских, основополагающих и исторических недостатках, которые усугубляют главный аргумент и которые, как правило, отвлекают внимание от множества ярких и увлекательных наблюдений и идей, представленных в книге.

Amy Dahan-Dalmédico, Revue d'histoire des sciences, Vol. 36, No. 3/4 (JUILLET-DÉCEMBRE 1983), pp. 356-358:

"Quant aux derniers chapitres sur les grandes tendances des mathématiques contemporaines, ils sont franchement décevants, assez superficiels. Il n'y a pas d'analyse de la mathématique contemporaine (grande période structuraliste, retour au « concret », flux entre les mathématiques et la physique, etc."

Что касается последних глав, посвященных основным тенденциям современной математики, они откровенно разочаровывают, скорее поверхностны. Нет анализа современной математики (великий период структурализма, возврат к «конкретному», поток между математикой и физикой, и т.д.).

Scott Weinstein, ETC: A Review of General Semantics, Vol. 38, No. 4 (Winter 1981), pp. 425-430:

Professor Kline's book is a lively account of a fascinating subject. Its conclusions are, however, overdrawn and in many cases unjustified. The lesson to be learned from twentieth century foundational research is not that mathematics is in a sorry state, but rather the extent to which deep philosophical issues about mathematics may be illuminated, if not settled, by mathematics itself. Gödel's theorems do indeed intimate that there may be limits to what we can come to know in mathematics, but they also demonstrate through themselves the great heights to which human reason can ascend through mathematical thought.

Книга профессора Клайна - живой рассказ об увлекательной теме. Однако его выводы перегружены и во многих случаях необоснованны. Урок, который нужно извлечь из фундаментальной науки двадцатого века, заключается не в том, что математика находится в жалком состоянии, а в том, насколько глубокие философские вопросы о математике могут быть освещены, если не решены, самой математикой. Теоремы Гёделя действительно указывают пределы того, что мы можем узнать в математике, но они также демонстрируют и великие высоты, к которым человеческий разум может подняться через математическую мысль.

Ian Stewart, Educational Studies in Mathematics, Vol. 13, No. 4 (Nov., 1982), pp. 446-447:

"This book is firmly in the tradition that we have come to expect from this author; and my reaction to it is much like my reaction to its predecessors: I think three quarters of it is superb, and the other quarter is outrageous nonsense; and the reason is that Morris Kline really doesn't understand what today's mathematics is about, although he has an enviable grasp of yesterday's...

Morris Kline has said elsewhere that he considers the crowning achievement of twentieth-century mathematics to be the Godel theorem. I don't agree: the Gddel theorem, astonishing and deep as it is, had little effect on the mainstream of real mathematical development. It didn't actually lead into anything new and powerful except more theorems of the same kind. It affected how mathematicians thought about what they were doing; but its effect on what they actually did is close to zero. Compare this to the rise of topology: fifty years of apparently introverted efforts by mathematicians, largely ignoring applied science; polished and perfected and developed into a body of technique of immense and still largely unrealised power; and within the last decade becoming important in virtually every field of applied science: engineering, physics, chemistry, numerical analysis. Topology has far more claim to be the crowning achievement of this century.

But Morris Kline can see only the introversion. It doesn't seem to occur to him that a mathematical problem may require concentrated contemplation of mathematics, rather than the problem to which one hopes to apply the resulting theory, to obtain a satisfactory solution. But if I want to cut down an apple tree, and my saw is too blunt, no amount of contemplation of the tree will sharpen it...

There is good mathematics; there is bad mathematics. There are mathematicians who are totally uninterested in science, who are building tools that science will find indispensable. There are mathematicians passionately interested in science, and building tools for specific use there, whose work will become as obsolete as the Zeppelin or the electronic valve. The path from discovery to utility is a rabbit-warren of false ends: mathematics for its own sake has had, and wil continue to have, its place in the scheme of things. And, after all, the isolation of the topologist who knows no physics is no worse than that of the physicist who knows no topology. Today's science requires specialization from its individuals: the collective activity of scientists as a whole is where the links are forged. If only Morris Kline showed some inkling of the nature of this process, I would take his arguments more seriously. But his claim that mathematics has gone into decline is one based too much on ignorance, and his arguments are tawdry in comparison to the marvellous, shining vigour of today's mathematics. I too would like to see more overt recognition by mathematicians of the importance of scientific problems; but to miss the fact that they do splendid work even in this apparent isolation is to lose the battle before it has begun."

Эта книга продолжает традицию, которую мы ожидаем от этого автора, и моя реакция на нее очень похожа на мою реакцию на его предыдущие книги: я думаю, что три четверти ее превосходны, а оставшаяся четверть - возмутительная чепуха. И причина в том, что Моррис Клайн действительно не понимает сегодняшнюю математику, хотя у него есть завидное понимание вчерашней...

Моррис Клайн сказал в другом месте, что он считает завершающим достижением математики двадцатого века теорему Гёделя. Я не согласен: теорема Гёделя, удивительная и глубокая, мало повлияла на основное направление реального математического развития. На самом деле она не привела ни к чему новому и сильному, кроме теорем того же рода. Она повлияла на то, что математики думают о том, что они делают; но ее влияние на то, что они на самом деле делают, близко к нулю. Сравните это с ростом топологии: пятьдесят лет, по-видимому, интровертированных усилий математиков, в основном игнорирующих прикладную науку, отполированная до совершенства и превращенная в технологию огромная и до сих пор в значительной степени нереализованная энергия, которая в течение последнего десятилетия становится важной практически во всех областях прикладной науки: машиностроение, физика, химия, численный анализ. Топология имеет гораздо больше оснований считаться венцом этого века.

Но Моррис Клайн видит только интроверсию. Ему, похоже, не кажется, что математическая задача может потребовать сосредоточенного созерцания математики, а не проблемы, к которой хотелось бы применить теорию, для получения удовлетворительного решения. Но если я хочу срубить яблоню, и моя пила слишком тупая, никакое созерцание дерева не заострит ее...

Есть хорошая математика, есть плохая математика. Есть математики, совершенно не интересующиеся наукой, но строящие инструменты, которые наука найдет незаменимыми. Есть математики, страстно интересующиеся наукой и строящие инструменты для конкретного использования, чья работа станет такой же устаревшей, как Цеппелин или электронная лампа. Путь от открытия к полезности - это упрямство кролика среди ложных ходов: математика сама по себе имела и будет иметь свое место в схеме вещей. И, в конце концов, изоляция тополога, не знающего физики, не хуже, чем физика, не знающего топологии. Сегодняшняя наука требует специализации от своих адептов: коллективная деятельность ученых в целом - это то, где ссылки подделаны. Если бы Моррис Клайн дал некоторое представление о характере этого процесса, я бы серьезнее отнесся к его аргументам. Но его утверждение, что математика пришла в упадок, основано на слишком большом невежестве, а его аргументы туманны по сравнению с чудесной, сияющей энергией современной математики. Мне тоже хотелось бы, чтобы математики откровеннее признавали проблемы своей науки; но не заметить, что они делают великолепную работу, даже в этой кажущейся изоляции, это проиграть битву, прежде чем она началась.


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