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Georg Cantor

Born: March 3, 1845; Saint Petersburg, Russian Empire
Died: January 6, 1918 (aged 72); Halle, Province of Saxony, German Empire

Georg Ferdinand Ludwig Philipp Cantor was a German mathematician. He created set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence between the members of two sets, defined infinite and well-ordered sets, and proved that the real numbers are more numerous than the natural numbers. In fact, Cantor's method of proof of this theorem implies the existence of an "infinity of infinities". He defined the cardinal and ordinal numbers and their arithmetic. Cantor's work is of great philosophical interest, a fact he was well aware of

Cantor's theory of transfinite numbers was originally regarded as so counter-intuitive – even shocking – that it encountered resistance from mathematical contemporaries such as Leopold Kronecker and Henri Poincaré and later from Hermann Weyl and L. E. J. Brouwer, while Ludwig Wittgenstein raised philosophical objections

Cantor, a devout Lutheran, believed the theory had been communicated to him by God. (Dauben, Joseph (2004) [1993], "Georg Cantor and the Battle for Transfinite Set Theory".)

Florensky P. A. Essay on the ideas of G. Kantor

But isn't it important to raise one more question, namely, the question of G. Kantor's work and the hidden engines of his activity? It seems to me, yes, and I probably will not be mistaken if I want to characterize G. Kantor as a typical Jew, “an Israelite in whom there is no guile” (Jhn 1:47), and through and through the peculiar spirit of his work, as the spirit of the best representatives of the nation. Perhaps, even more than that, G. Kantor is the finalizer of Jewry, he, so to speak, puts an end to i, as if summing up, formulating and accurately defining in logical terms the centuries-old ideas of his nation…

I will try to show this, although I am aware of how difficult such a task is and I foresee a possible failure: biographical data on Cantor have not been published anywhere and therefore the actual material is extremely scarce. One has to interpolate by intuition, but having created an idea of his personality, it is extremely difficult to prove the legitimacy of one's view.

The rings of steel chain mail clinging to each other are tightly bound, with which Kantor protects himself from poisonous, but for him toy arrows of the modern worldview. He is invulnerable behind his tightly interwoven network of theorems. The theorems flow harmoniously from each other. Each word is minted and so fitted into its place that in Cantor's compact exposition, not a single letter can be thrown out so as not to violate the integrity of everything. But when the thought is finished in such a way, when the exposition is brought into a classical form, reminiscent of the antique, then it is difficult to see behind this hard shell what, in fact, created it and for what purpose. Only sometimes indomitable flames break out from within; fiery tongues rush over the steel net, and only those burned by them can guess what kind of fire was the impulse of creativity. But how to single out these flames, inextricably linked to the whole? They fade and fade away, leaving only a feeling of warmth if we tear them away from their glowing source. It is possible to demonstrate them only for poetry, but not for my essay. Better not to try to catch the elusive. It’s better to say directly: this is what I saw and, I hope, everyone who wants to get acquainted with the works of Kantor can see…

G. Kantor was a Jew by birth. The family of Portuguese Jews, from which he later descended, split into two branches; one of them emigrated to Denmark and produced the well-known historian of mathematics M. Kantor, the other moved to Russia and here, in 1845, Georg Ferdinand-Louis-Philip Kantor was born in St. Petersburg. At the age of eleven (in 1856) he left for Germany; there he received his upbringing, education and, finally, a professorship. He is still a professor at Halle. I deliberately emphasize these data of the formulary list, because Cantor's Jewish origin provides a key to understanding his work.

First of all, this is reflected in amazing endurance, in the intensity of work and in the ability to wait and endure. For ten years, Kantor harbors ideas without letting them know in print. Of course, if it were in the 16th century, then there would be nothing special. But whoever is familiar with modern literature and knows how many people tremble for priority in some trifles, how sometimes an idea that is completely ill-conceived, underdeveloped, unprocessed gets into a journal, cannot but be surprised at Kantor's restraint. He renounced fame and glory, which he undoubtedly had every reason to count on, neglected the vain desire to write something like that by which reputations are created, and walked past fashionable issues (an imaginary variable, etc.).

Lonely and incomprehensible, he sits in his Halle and ponders, stubbornly ponders whether there are any errors and inconsistencies in his ideas, whether his teaching leads to pantheism, which, obviously, disgusts his Jewish soul. Thought leads to the need to recognize the Transfinite, and on the other hand, its religious consciousness needs completeness, form, personality; the wavering and indefinite ghost of the God of the monists, faceless and formless, is unacceptable to him. Is there a contradiction here with the recognition of the Transfinite? But here the painful question is answered in the negative. No, Transfinite not only does not contradict theism, but is even required by it. "The tradition, always highly revered"* (his words) by Cantor, is preserved and reinforced; but this was achieved not by tendentious attempts to pervert the truth, logically given, not by overt or covert manipulations, but at the cost of persistent reflection and steady progress. Positivism is a cloudy layer through which one must pass; retreat and flight will not help, and the swamp fire will follow the fleeing one. The postulate of the coincidence of the results of religious experience and scientific data received additional confirmation. The contradictions turned out to be imaginary. Little of; It turned out that the idea of ​​transfinite already presupposes the idea of ​​the Absolute, and having recognized the first, we have no right to reject the second.

Then Kantor publishes his first works, which quickly fell from a cornucopia one after another in a whole series of journals. The dam was broken by the pressure of internal force, internal fullness gushed forth in a powerful stream. Kantor literally pours out. His articles have a striking non-literary character. He forgets what he writes, he creates on paper, because he cannot but create, but this is not for the public.

Of course, the public does not understand him. What does he need? For philosophers he is a "philosophizing" mathematician, for mathematicians he is a metaphysician, for the indifferent he is suspiciously religious, no matter how there are tricks; for theologians it is supposedly dangerous: “do not these speculations lead to pantheism?” — that's the theologians' back thought. Even for Cantor's admirers “it is difficult to predict the future of such a new (récente) work, which, however, is of more interest from the point of view of philosophy than the future benefit of the development of mathematics.” This is what Tannery, one of the first to break the bon ton and to speak in a well-mannered society, which was shocked by such too original innovations, about the work of Cantor. “An attempt so bold,” says Tannery further, “that it is difficult to follow, may lead to unexpected results. It is interesting to know to which philosophical school [such] a thinker as G. Kantor belongs…” This whole quotation is characterized by a tone of indecision and embarrassment; it seems that the person is somewhat confused and cannot figure out his attitude. And this is said by a mathematician who was brought up in the history of mathematics, a man who could see the historical necessity of the emergence of new ideas!

Very slowly these new ideas are being accepted. Mathematicians use transfinite numbers with caution, philosophers try to use them for their own purposes (Wundt), although this is not without misunderstanding and confusion. Theologians who previously stated: "…what G. Cantor called Transfinitum in natura naturata [in created nature (lat.)], cannot be defended in a certain sense", which, however, by their own admission, G. Kantor “apparently not given”, such an understanding (not given by Cantor) will contain the error of pantheism, now these theologians, after a lot of explanations and interpretations from Cantor, are becoming softer and agree, finally, “that in such an understanding, as I see before so far,” one theologian writes to Cantor, “there is no danger to your concept of transfinite for religious truths.”

Then Kantor from "M. G. Cantor" turns into "Eminent savant de Halle" [an outstanding scientist from Halle (fr.)], his ideas become topics for dissertations, and their development creates a whole literature.

Apparently, it would be possible to calm down on this; however, this thinker is not one of those who calm down easily. For more than twenty years he has been struggling, spending a lot of work and effort to give a more thorough presentation, to look at the matter from a new point of view. His scrupulousness is unusual and, in order to correct some, in the reader's opinion, almost imperceptible mistakes, he reworks his memoirs again and again, supplements, explains. At the same time, he studies the questions he touches on historically, sorting through old writers, and one can only wonder where he got the strength to read the vast literature of all ages. He carefully studies mathematicians, theologians and philosophers, especially the ancients, since he himself considers himself the direct successor of Plato and Aristotle. Mystical and scholastic writings attract his special attention, and he studies authors, even whose names are unknown to anyone except learned specialists.

But this hard work is not in vain, and the inner fire burns him. Severe mental illness periodically interrupts his activities. Kantor goes mad, then recovers in order to publish a memoir devoted to the same questions that are dear to him, or to finish what he started before illness, then he falls ill again, perhaps from an excessive exertion of thought, and this is repeated several times. But even now the action has not yet ended, the play has not been played, and we know nothing about the denouement; one can only guess about the sad end, the saddest for such a clear mind as G. Kantor…

When you read it for the first time, you might think that all these crystal-clear thoughts in their clear and childishly unsophisticated presentation, highly reminiscent of Mozart's music, that this complete absence of external effect and ostentatious learning testifies to the ease of the work itself; the theorem follows so harmoniously from the theorem, the proofs are so amazingly brief, the whole train of thought goes so naturally that you involuntarily deceive yourself and want to exclaim: “Why, it’s all so obvious and clear that it was not worth talking about!” And in a fit of all-encompassing joy, you begin to think that all this was already known before, so you quickly become related to Cantor's ideas. But if you read his letters, think about how much time and incredible effort it took Kantor to reach the final formulation, and take into account what gross mistakes the subtlest thinkers made when discussing the same questions, then you see that this is not at all simple and, on the contrary, it required purely Jewish perseverance to overcome all difficulties and think through to the end. As proof, I can refer to the fact that Paul Du-Bois Reymond, who is generally accepted as a subtle thinker, also dealt with these issues almost simultaneously with Cantor, and although, as far as I know, he had the opportunity to get acquainted with Cantor's results (from his first memoirs), however confused the matter more than created anything positive, and he, I repeat, is very subtle. But he did not burn and was not a monoideist, but simply engaged in research, while Kantor went ahead to a certain goal — he believed. At the same time, it is interesting to note how, in all studies, in Cantor the main idea is the idea of ​​actual infinity, while in du Bois Reymond it is potential; one might think that the racial characteristics of both scientists also affect here.

Everyone remembers, of course, the chased characterization of the Jewish people, made by Vl. Solovyov in a number of works. I do not dare to add anything of my own or change here, I cannot even spoil it by exposition. I'm just reminding. If we recall it, even a cursory glance at Kantor's writings is enough to see how vividly he expressed in himself the best traits of Jewry, all the “theocratic virtues” of his people. Here, first of all, two main and most characteristic features are striking. This is, firstly, the extreme development of well-being, self-consciousness and self-activity, inner strength, so to speak, the stability of the individual. Let others around Kantor be carried away by all sorts of fashionable trends, the urgent needs of science, “current affairs.” He perfectly understands what is going on around him, but he himself will not deviate a single hair from his path, from his chosen goal. This goal, however, is not a whim, curiosity or personal desire, and this is the second characteristic feature. This is a command from above, and Kantor, as a Jew, has religiosity to the point of self-sacrifice. If Solovyov says about Moses: “Despite all the temptations of Egyptian theosophy and theurgy, 'by faith Moses was great, refusing to be called the son of Pharaoh's daughter and by faith leave Egypt, not fearing the wrath of the king'” [Соловьев В. С. Еврейство и христианский вопрос], the same mutatis mutandis [having made the appropriate changes (lat.)] I would like to say about Cantor, only the role of magic in this case will be played by all the conventions of the science of his time, the temptations almost without difficulty, magically, to do research from the ready.


And he leaves the house of his fathers — modern science, modern world outlook. It has been going on for more than 30 years, and this is not just a scientific work, just a patient study. First of all, this is a feat of great faith, faith in the ability to create symbols for the infinite.

If we are nothing before the Absolute, then nevertheless we are morally homogeneous with Him, we can comprehend Him, but not directly, but in symbols; we carry in ourselves the transfinite, supra-finite, we, the cosmos, are not something finite, directly opposite to the Divine, we are transfinite, “the middle between everything and nothing.” But then it is necessary to show how this is possible. And now 30 years of work go to the justification of this faith. Kantor does not know where the work will lead; everything speaks against the possibility of such a justification, everyone shakes their heads with a smile, but Kantor does not create idols for himself. Does he leave his work, the soil of the tradition and science that has nurtured him, and rush past all temptations into the unknown, into the desert of pure thought? What is he striving for? To create a "temple", to create symbols for the Infinite. He wants to see the realization of the Divine forces, he wants to make sure that it is possible, and he needs it sooner. He needs to be shown that the idea of ​​the Transfinite is not internally contradictory, that it is legitimate and necessary. Otherwise, there is no moral homogeneity of the cosmos and the Divine, there is not and cannot be an “agreement”, we cannot determine ourselves and act on our own without becoming empty automata, which are pulled by strings.

This is where Kantor's “religious materialism” comes into play to the strongest degree. The illusory nature of the world seems to him as fictional as the world of a dead mechanism. The latest idealism, modern positivism, seems to him monstrously wild, not because of their abstractness, but because they destroy the real, the concrete and the individual, the living individual. He fights passionately against nominalism and academic skepticism because he needs to feel the real and the living; materialism disgusts Cantor no less.

“The more distant the goal, the more strength and courage are needed to believe in it and go towards it. If even now, after the founding and visible spread of Christianity, it is so difficult for a person to advance towards the divine-human goal, how much more difficult was it to serve Christianity two thousand years before Christ? This is the unattainable greatness of these semi-wild nomads, who, near the mountains and tall oaks of Canaan, laid the first stones of the future ecumenical temple! Before them there was nothing clear and definite, all their hopes were in the fog of the distant future, and in the present they had only to obey, only to believe” [Соловьев В. С. История и будущность теократии]. …This trait of devotion to the Higher Will, always preserved in the spirit of the nation, affected and in this case; I think that Cantor's works will be correctly considered as new stones laid on the walls of the temple by the same nation.

If Kantor, as a person, is the most living example of a Jew, then his worldview is in the nature of the same almost to an even greater extent. The idea of ​​a complete infinity, both in the absolute personality — God, and in the human, is the property of the Jews, and this idea is, it seems, the most essential basis for Cantor. While others, the Aryans, recognize only a potential infinity, “evil”, indefinite and unlimited, to his soul the thought of the impossibility of actual infinity seems monstrous. He cannot reconcile with her and is looking for means to justify his faith. Even the most potential infinity is important for him only under the condition of not indefinite growth, not infinity in the literal sense, but under the condition of striving for the limit, for actual infinity, as his ideal goal. One usually looks at potential infinity, at the progression of sub specie finiti [from the point of view of the finite (lat.)] through the eyes of this world. Cantor, on the other hand, looks at it from the other side, from the point of view of its goal, sees sub specie infinitatis [from the point of view of the infinite (lat.)]. He sees “the image of this world pass away” (1 Cor 7:31).

And in this contrast of views, once again, those basic moods clearly appear that create the idea of ​​a human-deity, finite, always remaining finite and qualitatively similar to itself, but desiring to become higher and higher, God-manhood, “becoming Absolute.” This is the same contrast as in the words of the tempting serpent: “you shall be like gods” (Gen 3:5) and the words of Scripture: “you are gods, and the sons of the Most High are all of you…” (Psa 82:6). Probably everyone knows the “Easter song” of the Jews. You remember, of course, resolute perseverance, roughly speaking, almost importunity in prayers to God. This persistence in the request, this theomachism, “I will not let you go until you bless(Gen 32:26), is highly characteristic of the work of Georg Cantor, and I think I can’t better explain the meaning of his activity better than citing the text of this song [Соловьев В. С. История и будущность теократии].

Here it is:

“God Almighty, now is near and soon build Your temple, soon, in our days, as close as possible, now create, now create, now create, now build Your temple! Merciful God, Great God, meek God, most high God, good God, sweetest God, measureless God, God of Israel, in the near future build Your temple, soon, soon in our days, now create, now create, now create, now create, now build your temple soon! Mighty God, living God, mighty God, glorious God, merciful God, reigning God, rich God, glorious God, faithful God, now raise up Thy temple without delay, soon, soon, in our days, without delay, soon, now build, now create, now create, now create, now quickly create your temple!”

Florensky P. A.
О символах бесконечности
(Essay on the ideas of G. Kantor)

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