Georg Cantor Quotes (27 quotes) | Quotes of famous people

Georg Cantor: 27 quotes 5 likes

Famous Georg Cantor Quotes

“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.”

Georg Cantor

As quoted in Out of the Mouths of Mathematicians : A Quotation Book for Philomaths (1993) by Rosemary Schmalz.
Context: 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.

“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.”

Georg Cantor

"Über unendliche, lineare Punktmannigfaltigkeiten" in Mathematische Annalen 20 (1882)  Quoted in "Cantor's Grundlagen and the paradoxes of Set Theory" by William W. Tait

“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.”

Georg Cantor

From Kant to Hilbert (1996)
Context: 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”

Georg Cantor

From Kant to Hilbert (1996)
Context: 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].

“The essence of mathematics lies entirely in its freedom.”

Georg Cantor

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!”

Georg Cantor

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."

Georg Cantor Quotes about mathematics

“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.”

Georg Cantor

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.

Georg Cantor

Doctoral thesis (1867); variant translation: In mathematics the art of proposing a question must be held of higher value than solving it.

“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.”

Georg Cantor

"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 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.”

Georg Cantor

As quoted in Mind Tools: The Five Levels of Mathematical Reality (1988) by Rudy Rucker. ~ ISBN 0395468108

“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.”

Georg Cantor

From Kant to Hilbert (1996)

Georg Cantor Quotes

“This view [of the infinite], which I consider to be the sole correct one, is held by only a few.”

Georg Cantor

As quoted in Journey Through Genius (1990) by William Dunham ~
Context: 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!

“My theory stands as firm as a rock; every arrow directed against it will return quickly to its archer.”

Georg Cantor

As quoted in Journey Through Genius (1990) by William Dunham
Context: 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”

Georg Cantor

As quoted in Understanding the Infinite (1994) by Shaughan Lavine
Context: 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.

Sense

“The totality of all alephs cannot be conceived as a determinate, well-defined, and also a finished set.”

Georg Cantor

Letter to David Hilbert (2 October 1897)
Context: 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.

“A set is a Many that allows itself to be thought of as a One.”

Georg Cantor

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.”

Georg Cantor

From Kant to Hilbert (1996)

Success

“However, he was born in Copenhagen, of Jewish parents, of the Portuguese Jewish community there.”

Georg Cantor

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)

Communication , Parent

“Every transfinite consistent multiplicity, that is, every transfinite set, must have a definite aleph as its cardinal number.”

Georg Cantor

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.”

Georg Cantor

Sense

“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.”

Georg Cantor

As quoted in Infinity and the Mind (1995) by Rudy Rucker.

Fear

“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.”

Georg Cantor

"Mitteilungen" (1887-8)

“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.”

Georg Cantor

Letter to Gustac Enestrom, as quoted in Georg Cantor : His Mathematics and Philosophy of the Infinite (1990) by Joseph Warren Dauben ~ ISBN 0691024472

Question , Nature , Dependency

“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.”

Georg Cantor

From Kant to Hilbert (1996)

Nature

“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.”

Georg Cantor

As quoted in Understanding the Infinite (1994) by Shaughan Lavine ~ ISBN 0674921178

Nature

“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.”

Georg Cantor

As quoted in Modern Mathematicians, (1995) by Harry Henderson. ~ ISBN 0816032351