As quoted in Out of the Mouths of Mathematicians : A Quotation Book for Philomaths (1993) by Rosemary Schmalz.
Контексте: 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.
Георг Кантор: Цитаты на английском языке
From Kant to Hilbert (1996)
Контексте: 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.
“I call this the improper infinite”
From Kant to Hilbert (1996)
Контексте: 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].
"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
“This view [of the infinite], which I consider to be the sole correct one, is held by only a few.”
As quoted in Journey Through Genius (1990) by William Dunham ~
Контексте: 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!
As quoted in Journey Through Genius (1990) by William Dunham
Контексте: 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.
“The transfinite numbers are in a certain sense themselves new irrationalities”
As quoted in Understanding the Infinite (1994) by Shaughan Lavine
Контексте: 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.
Letter to David Hilbert (2 October 1897)
Контексте: 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.
“The essence of mathematics lies entirely in its freedom.”
Variant translation: The essence of mathematics is in its freedom.
From Kant to Hilbert (1996)
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."
“A set is a Many that allows itself to be thought of as a One.”
As quoted in Infinity and the Mind (1995) by Rudy Rucker. ~ ISBN 0691001723
From Kant to Hilbert (1996)
“However, he was born in Copenhagen, of Jewish parents, of the Portuguese Jewish community there.”
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)
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
"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)
As quoted in Infinity and the Mind (1995) by Rudy Rucker.
"Mitteilungen" (1887-8)
Letter to Gustac Enestrom, as quoted in Georg Cantor : His Mathematics and Philosophy of the Infinite (1990) by Joseph Warren Dauben ~ ISBN 0691024472
From Kant to Hilbert (1996)
As quoted in Understanding the Infinite (1994) by Shaughan Lavine ~ ISBN 0674921178
As quoted in Modern Mathematicians, (1995) by Harry Henderson. ~ ISBN 0816032351
As quoted in Mind Tools: The Five Levels of Mathematical Reality (1988) by Rudy Rucker. ~ ISBN 0395468108
"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)
From Kant to Hilbert (1996)
Grundlagen einer allgemeinen Mannigfaltigkeitslehre [Foundations of a General Theory of Aggregates] (1883)
“In mathematics the art of asking questions is more valuable than solving problems.”
In re mathematica ars proponendi quaestionem pluris facienda est quam solvendi.
Doctoral thesis (1867); variant translation: In mathematics the art of proposing a question must be held of higher value than solving it.
