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Georg Ferdinand Ludwig Philipp Cantor (3 March 1845 – 6 January 1918) was a Russian-born German mathematician and philosopher of Danish and Austrian descent, most famous as the creator of set theory, and of Cantor's theorem which implies the existence of an "infinity of infinities."
- In re mathematica ars proponendi quaestionem pluris facienda est quam solvendi.
- In mathematics the art of asking questions is more valuable than solving problems.
- Doctoral thesis (1867); variant translation: In mathematics the art of proposing a question must be held of higher value than solving it.
- In mathematics the art of asking questions is more valuable than solving problems.
- I realize that in this undertaking I place myself in a certain opposition to views widely held concerning the mathematical infinite and to opinions frequently defended on the nature of numbers.
- Grundlagen einer allgemeinen Mannigfaltigkeitslehre [Foundations of a General Theory of Aggregates] (1883)
- 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."
- 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) Quoted in "Cantor's Grundlagen and the paradoxes of Set Theory" by William W. Tait
- 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)
- In order for there to be a variable quantity in some mathematical study, the domain of its variability must strictly speaking be known beforehand through a definition. However, this domain cannot itself be something variable, since otherwise each fixed support for the study would collapse. Thus this domain is a definite, actually infinite set of values. Hence each potential infinite, if it is rigorously applicable mathematically, presupposes an actual infinite.
- "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 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.
- "Mitteilungen" (1887-8)
- Er ist aber in Kopenhagen geboren, von israelitischen Eltern, die der dortigen portugisischen Judengemeinde.
- 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)
- 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.
- Letter to David Hilbert (2 October 1897)
- 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
- 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 Out of the Mouths of Mathematicians : A Quotation Book for Philomaths (1993) by Rosemary Schmalz.
- 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
- 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 actual infinite arises in three contexts: first when it is realized in the most complete form, in a fully independent otherworldly being, in Deo, where I call it the Absolute Infinite or simply Absolute; second when it occurs in the contingent, created world; third when the mind grasps it in abstracto as a mathematical magnitude, number or order type.
- As quoted in Mind Tools: The Five Levels of Mathematical Reality (1988) by Rudy Rucker. ~ ISBN 0395468108
- That from the outset they expect or even impose all the properties of finite numbers upon the numbers in question, while on the other hand the infinite numbers, if they are to be considered in any form at all, must (in their contrast to the finite numbers) constitute an entirely new kind of number, whose nature is entirely dependent upon the nature of things and is an object of research, but not of our arbitrariness or prejudices.
- Letter to Gustac Enestrom, as quoted in Georg Cantor : His Mathematics and Philosophy of the Infinite (1990) by Joseph Warren Dauben ~ ISBN 0691024472
- 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 ~ ISBN 0471500305
- 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 Journey Through Genius (1990) by William Dunham
- What I assert and believe to have demonstrated in this and earlier works is that following the finite there is a transfinite (which one could also call the supra-finite), that is an unbounded ascending ladder of definite modes, which by their nature are not finite but infinite, but which just like the finite can be determined by well-defined and distinguishable numbers.
- As quoted in Understanding the Infinite (1994) by Shaughan Lavine ~ ISBN 0674921178
- 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.
- As quoted in Understanding the Infinite (1994) by Shaughan Lavine
- I entertain no doubts as to the truths of the transfinites, which I recognized with God’s help and which, in their diversity, I have studied for more than twenty years; every year, and almost every day brings me further in this science.
- As quoted in Modern Mathematicians, (1995) by Harry Henderson. ~ ISBN 0816032351
From Kant to Hilbert (1996)
- Quotes of Cantor in From Kant to Hilbert : A Source Book in the Foundations of Mathematics (1996) by William Bragg Ewald ~ ISBN 0198505361
- 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].
- Infinity, in its first form (the improper-infinite) presents itself as a variable finite [veranderliches Endliches]; in the other form (which I call the proper infinite [Eigentlich-unendliche]) it appears as a thoroughly determinate [bestimmtes] infinite.
- What I declare and believe to have demonstrated in this work as well as in earlier papers is that following the finite there is a transfinite (transfinitum)--which might also be called supra-finite (suprafinitum), that is, there is an unlimited ascending ladder of modes, which in its nature is not finite but infinite, but which can be determined as can the finite by determinate, well-defined and distinguishable numbers.
- Mathematics, in the development of its ideas, has only to take account of the immanent reality of its concepts and has absolutely no obligation to examine their transient reality.
- 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.
- The essence of mathematics lies entirely in its freedom.
- Variant translation: The essence of mathematics is in its freedom.
- 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.
- No one shall expel us from the Paradise that Cantor has created.
- David Hilbert, in "Über das Unendliche" [On the Infinite], in Mathematische Annalen 95 (1925)
- Variant translation: No one shall expel us from the Paradise that Cantor has created for us.
- In 1874 the German mathematician Georg Cantor made the startling discovery that there are more irrational numbers than rational ones, and more transcendental numbers than algebraic ones. In other words, rather than being oddities, most real numbers are irrational; and among irrational numbers, most are transcendental.
- Eli Maor, e: The Story of a Number (1994)
- Why was Cantor so vehemently opposed to infinitesimals? In his valuable essay, "The Metaphysics of the Calculus," Abraham Robinson suggests that Cantor already had enough problems trying to defend transfinite numbers. It seems likely that, consciously or otherwise, Cantor deemed it politically wise to go along with orthodox mathematicians on the question of infinitesimals. Cantor's stance might be compared to that of a pro-marijuana Congressional candidate who advocates harsh penalties for the sale or use of heroin.
- Rudy Rucker, in Infinity and the Mind (2005), p. 80
- After being relegated to an obscure mid-tier university, blocked from leading journals and openly mocked by his peers, including his former mentor, the late 19th century German mathematician found refuge for his groundbreaking work on infinities in, of all places, the Roman Catholic Church … Catholic theologians welcomed Cantor's ideas, which provided a workable way of understanding mathematical infinities, as evidence that humans could grasp the infinite and could also, therefore, have a greater understanding of God, himself infinite.
What a welcome relief this must have been to the chronically depressed Cantor! As John D. Barrow writes in The Infinite Book: A Short Guide to the Boundless, Timeless and Endless, Cantor "started to tell his friends that he had not been the inventor of the ideas about infinity that he had published. He was merely a mouthpiece, inspired by God to communicate parts of the mind of God to everyone else."- Jenn Shreve, in "What we definitely know about endlessly possible", a review of The Infinite Book : A Short Guide to the Boundless, Timeless and Endless by John D. Barrow
- I discovered the works of Euler and my perception of the nature of mathematics underwent a dramatic transformation. I was de-Bourbakized, stopped believing in sets, and was expelled from the Cantorian paradise.
- Brief biography at MacTutor archive (University of St Andrews, Scotland)
- Set theory by Thomas Jech at the Stanford Encyclopedia of Philosophy
- Selections from Cantor's philosophical writing.
- Text of the 1891 diagonal proof.
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