Georg Cantor citations
Date de naissance: 19. février 1845
Date de décès: 6. janvier 1918
Georg Cantor est un mathématicien allemand, né le 3 mars 1845 à Saint-Pétersbourg et mort le 6 janvier 1918 à Halle . Il est connu pour être le créateur de la théorie des ensembles.
Il établit l'importance de la bijection entre les ensembles, définit les ensembles infinis et les ensembles bien ordonnés. Il prouva également que les nombres réels sont « plus nombreux » que les entiers naturels. En fait, le théorème de Cantor implique l'existence d'une « infinité d'infinis ». Il définit les nombres cardinaux, les nombres ordinaux et leur arithmétique. Le travail de Cantor est d'un grand intérêt philosophique et a donné lieu à maintes interprétations et à maints débats.
Cantor a été confronté à la résistance de la part des mathématiciens de son époque, en particulier Kronecker.
Poincaré, bien qu'il connût et appréciât les travaux de Cantor, avait de profondes réserves sur son maniement de l'infini en tant que totalité achevée. Les accès de dépressions récurrents de Cantor, de 1884 à la fin de sa vie, ont été parfois attribués à l'attitude hostile de certains de ses contemporains, mais ces accès sont souvent à présent interprétés comme des manifestations d'un probable trouble bipolaire.
Au XXIe siècle, la valeur des travaux de Cantor n'est pas discutée par la majorité des mathématiciens qui y voient un changement de paradigme, à l'exception d'une partie du courant constructiviste qui s'inscrit à la suite de Kronecker. Dans le but de contrer les détracteurs de Cantor, David Hilbert a affirmé : « Nul ne doit nous exclure du Paradis que Cantor a créé. » Wikipedia
Citations Georg Cantor
de David Hilbert
From Kant to Hilbert (1996)
Contexte: As for the mathematical infinite, to the extent that it has found a justified application in science and contributed to its usefulness, it seems to me that it has hitherto appeared principally in the role of a variable quantity, which either grows beyond all bounds or diminishes to any desired minuteness, but always remains finite. I call this the improper infinite [das Uneigentlich-unendliche].
„I have never proceeded from any Genus supremum of the actual infinite. Quite the contrary, I have rigorously proved that there is absolutely no Genus supremum of the actual infinite. What surpasses all that is finite and transfinite is no Genus; it is the single, completely individual unity in which everything is included, which includes the Absolute, incomprehensible to the human understanding. This is the Actus Purissimus, which by many is called God.“
As quoted in Out of the Mouths of Mathematicians : A Quotation Book for Philomaths (1993) by Rosemary Schmalz.
Contexte: I have never proceeded from any Genus supremum of the actual infinite. Quite the contrary, I have rigorously proved that there is absolutely no Genus supremum of the actual infinite. What surpasses all that is finite and transfinite is no Genus; it is the single, completely individual unity in which everything is included, which includes the Absolute, incomprehensible to the human understanding. This is the Actus Purissimus, which by many is called God.
I am so in favor of the actual infinite that instead of admitting that Nature abhors it, as is commonly said, I hold that Nature makes frequent use of it everywhere, in order to show more effectively the perfections of its Author. Thus I believe that there is no part of matter which is not — I do not say divisible — but actually divisible; and consequently the least particle ought to be considered as a world full of an infinity of different creatures.
As quoted in Journey Through Genius (1990) by William Dunham ~
Contexte: This view [of the infinite], which I consider to be the sole correct one, is held by only a few. While possibly I am the very first in history to take this position so explicitly, with all of its logical consequences, I know for sure that I shall not be the last!
„Mathematics is in its development entirely free and is only bound in the self-evident respect that its concepts must both be consistent with each other, and also stand in exact relationships, ordered by definitions, to those concepts which have previously been introduced and are already at hand and established.“
From Kant to Hilbert (1996)
Contexte: Mathematics is in its development entirely free and is only bound in the self-evident respect that its concepts must both be consistent with each other, and also stand in exact relationships, ordered by definitions, to those concepts which have previously been introduced and are already at hand and established. In particular, in the introduction of new numbers, it is only obligated to give definitions of them which will bestow such a determinacy and, in certain circumstances, such a relationship to the other numbers that they can in any given instance be precisely distinguished. As soon as a number satisfies all these conditions, it can and must be regarded in mathematics as existent and real.
„My theory stands as firm as a rock; every arrow directed against it will return quickly to its archer.“
As quoted in Journey Through Genius (1990) by William Dunham
Contexte: My theory stands as firm as a rock; every arrow directed against it will return quickly to its archer. How do I know this? Because I have studied it from all sides for many years; because I have examined all objections which have ever been made against the infinite numbers; and above all because I have followed its roots, so to speak, to the first infallible cause of all created things.
As quoted in Understanding the Infinite (1994) by Shaughan Lavine
Contexte: The transfinite numbers are in a certain sense themselves new irrationalities and in fact in my opinion the best method of defining the finite irrational numbers is wholly dissimilar to, and I might even say in principle the same as, my method described above of introducing transfinite numbers. One can say unconditionally: the transfinite numbers stand or fall with the finite irrational numbers; they are like each other in their innermost being; for the former like the latter are definite delimited forms or modifications of the actual infinite.
„The totality of all alephs cannot be conceived as a determinate, well-defined, and also a finished set.“
Letter to David Hilbert (2 October 1897)
Contexte: The totality of all alephs cannot be conceived as a determinate, well-defined, and also a finished set. This is the punctum saliens, and I venture to say that this completely certain theorem, provable rigorously from the definition of the totality of all alephs, is the most important and noblest theorem of set theory. One must only understand the expression "finished" correctly. I say of a set that it can be thought of as finished (and call such a set, if it contains infinitely many elements, "transfinite" or "suprafinite") if it is possible without contradiction (as can be done with finite sets) to think of all its elements as existing together, and to think of the set itself as a compounded thing for itself; or (in other words) if it is possible to imagine the set as actually existing with the totality of its elements.
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Variant translation: The essence of mathematics is in its freedom.
From Kant to Hilbert (1996)
„Had Mittag-Leffler had his way, I should have to wait until the year 1984, which to me seemed too great a demand!“
Letter (1885), written after Gösta Mittag-Leffler persuaded him to withdraw a submission to Mittag-Leffler's journal Acta Mathematica, telling him it was "about one hundred years too soon."
As quoted in Infinity and the Mind (1995) by Rudy Rucker. ~ ISBN 0691001723
„If there is some determinate succession of defined whole real numbers, among which there exists no greatest, on the basis of this second principle of generation a new number is obtained which is regarded as the limit of those numbers, i. e. is defined as the next greater number than all of them.“
From Kant to Hilbert (1996)
Er ist aber in Kopenhagen geboren, von israelitischen Eltern, die der dortigen portugisischen Judengemeinde.
Of his father. In a letter written by Georg Cantor to Paul Tannery in 1896 (Paul Tannery, Memoires Scientifique 13 Correspondance, Gauthier-Villars, Paris, 1934, p. 306)
„Every transfinite consistent multiplicity, that is, every transfinite set, must have a definite aleph as its cardinal number.“
Letter to Richard Dedekind (1899), as translated in From Frege to Gödel : A Source Book in Mathematical Logic, 1879-1931 (1967) by Jean Van Heijenoort, p. 117
„There is no doubt that we cannot do without variable quantities in the sense of the potential infinite. But from this very fact the necessity of the actual infinite can be demonstrated.“
"Über die verschiedenen Ansichten in Bezug auf die actualunendlichen Zahlen" ["Over the different views with regard to the actual infinite numbers"] - Bihand Till Koniglen Svenska Vetenskaps Akademiens Handigar (1886)
„The old and oft-repeated proposition "Totum est majus sua parte" [the whole is larger than the part] may be applied without proof only in the case of entities that are based upon whole and part; then and only then is it an undeniable consequence of the concepts "totum" and "pars". Unfortunately, however, this "axiom" is used innumerably often without any basis and in neglect of the necessary distinction between "reality" and "quantity", on the one hand, and "number" and "set", on the other, precisely in the sense in which it is generally false.“
"Über unendliche, lineare Punktmannigfaltigkeiten" in Mathematische Annalen 20 (1882) <!-- pp 113-121 --> Quoted in "Cantor's Grundlagen and the paradoxes of Set Theory" by William W. Tait
„The fear of infinity is a form of myopia that destroys the possibility of seeing the actual infinite, even though it in its highest form has created and sustains us, and in its secondary transfinite forms occurs all around us and even inhabits our minds.“
As quoted in Infinity and the Mind (1995) by Rudy Rucker.
„The potential infinite means nothing other than an undetermined, variable quantity, always remaining finite, which has to assume values that either become smaller than any finite limit no matter how small, or greater than any finite limit no matter how great.“