In Mathematics the Art of Asking Questions Is More Valuable Than Solving Problems Cantor Source

In mathematics the art of request questions is more valuable than solving problems.

Georg Ferdinand Ludwig Philipp Cantor (iii 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'due south theorem which implies the being of an "infinity of infinities."

Quotes [edit]

Had Mittag-Lefflerhad his way, I should have to await until the year 1984, which to me seemed too great a demand!

There is no doubt that we cannot do without variable quantities in the sense of the potential infinite. Simply from this very fact the necessity of the actual infinite tin be demonstrated.

Each potential infinite, if it is rigorously applicative mathematically, presupposes an actual space.

The to the lowest degree particle ought to exist considered as a world full of an infinity of different creatures.

My theory stands as firm every bit a rock; every arrow directed confronting it will return quickly to its archer.

• In re mathematica ars proponendi quaestionem pluris facienda est quam solvendi.
  • In mathematics the fine art of request questions is more valuable than solving problems.
    • Doctoral thesis (1867); variant translation: In mathematics the fine art of proposing a question must be held of college value than solving it.

• 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 style, I should accept to wait until the year 1984, which to me seemed too great a demand!
  • Letter (1885), written afterward Gösta Mittag-Leffler persuaded him to withdraw a submission to Mittag-Leffler'southward journal Acta Mathematica, telling him it was "most i hundred years too soon."

• The onetime and oft-repeated suggestion "Totum est majus sua parte" [the whole is larger than the part] may exist applied without proof only in the case of entities that are based upon whole and function; then and only then is it an undeniable result of the concepts "totum" and "pars". Unfortunately, nevertheless, this "axiom" is used innumerably often without any footing and in neglect of the necessary distinction between "reality" and "quantity", on the 1 hand, and "number" and "set", on the other, precisely in the sense in which information technology is more often than not false.
  • "Über unendliche, lineare Punktmannigfaltigkeiten" in Mathematische Annalen 20 (1882) Quoted in "Cantor's Grundlagen and the paradoxes of Set Theory" past William W. Tait

• There is no doubt that nosotros cannot do without variable quantities in the sense of the potential infinite. But from this very fact the necessity of the actual space tin exist demonstrated.
  • "Über dice verschiedenen Ansichten in Bezug auf die actualunendlichen Zahlen" ["Over the different views with regard to the bodily infinite numbers"] - Bihand Till Koniglen Svenska Vetenskaps Akademiens Handigar (1886)

• In order for in that location to exist a variable quantity in some mathematical written report, the domain of its variability must strictly speaking be known beforehand through a definition. However, this domain cannot itself exist something variable, since otherwise each fixed back up for the study would collapse. Thus this domain is a definite, really infinite set of values. Hence each potential infinite, if it is rigorously applicable mathematically, presupposes an actual space.
  • "Über die verschiedenen Ansichten in Bezug auf die actualunendlichen Zahlen" ["Over the different views with regard to the actual space 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 whatsoever finite limit no thing how small, or greater than any finite limit no matter how great.
  • "Mitteilungen" (1887-eight)

• Er ist aber in Kopenhagen geboren, von israelitischen Eltern, die der dortigen portugisischen Judengemeinde.
  • However, he was built-in in Copenhagen, of Jewish parents, of the Portuguese Jewish community in that location.
  • 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 gear up. 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 almost important and noblest theorem of fix theory. 1 must but sympathise the expression "finished" correctly. I say of a gear up that information technology can be thought of equally finished (and telephone call such a set, if it contains infinitely many elements, "transfinite" or "suprafinite") if it is possible without contradiction (as can be washed with finite sets) to recollect of all its elements equally existing together, and to retrieve of the set up itself as a compounded thing for itself; or (in other words) if it is possible to imagine the prepare as actually existing with the totality of its elements.
  • Letter of the alphabet to David Hilbert (ii Oct 1897)

• Every transfinite consistent multiplicity, that is, every transfinite set, must have a definite aleph as its cardinal number.
  • Alphabetic character to Richard Dedekind (1899), every bit translated in From Frege to Gödel : A Source Volume in Mathematical Logic, 1879-1931 (1967) by Jean Van Heijenoort, p. 117

• I have never proceeded from whatever 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 then in favor of the actual space that instead of admitting that Nature abhors it, as is normally said, I hold that Nature makes frequent use of it everywhere, in order to bear witness more effectively the perfections of its Writer. Thus I believe that there is no part of matter which is non — I exercise 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 up is a Many that allows itself to be idea of every bit a Ane.
  • As quoted in Infinity and the Heed (1995) by Rudy Rucker. ~ ISBN 0691001723

The fear of infinity is a form of myopia that destroys the possibility of seeing the actual space, even though information technology in its highest form has created and sustains us, and in its secondary transfinite forms occurs all around us and even inhabits our minds.

• The fear of infinity is a class of myopia that destroys the possibility of seeing the actual space, even though information technology in its highest form has created and sustains us, and in its secondary transfinite forms occurs all around us and even inhabits our minds.
  • Equally quoted in Infinity and the Mind (1995) past Rudy Rucker.

• The actual infinite arises in three contexts: first when it is realized in the most consummate class, in a fully contained otherworldly being, in Deo, where I call information technology the Absolute Infinite or simply Absolute; 2d when information technology occurs in the contingent, created globe; third when the mind grasps it in abstracto equally a mathematical magnitude, number or gild type.
  • As quoted in Mind Tools: The Five Levels of Mathematical Reality (1988) by Rudy Rucker. ~ ISBN 0395468108

• That from the get-go they wait or fifty-fifty impose all the properties of finite numbers upon the numbers in question, while on the other hand the space 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 enquiry, merely non 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 ane, is held by only a few. While possibly I am the very kickoff 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!
  • Equally quoted in Journey Through Genius (1990) by William Dunham ~ ISBN 0471500305

• My theory stands as firm as a stone; every arrow directed confronting it will return apace to its archer. How practise I know this? Because I have studied information technology from all sides for many years; because I take examined all objections which accept ever been fabricated against the space numbers; and in a higher place all because I accept followed its roots, so to speak, to the kickoff 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 before works is that following the finite there is a transfinite (which i could as well call the supra-finite), that is an unbounded ascending ladder of definite modes, which by their nature are non finite but infinite, but which only like the finite can be adamant by well-divers and distinguishable numbers.
  • As quoted in Understanding the Space (1994) by Shaughan Lavine ~ ISBN 0674921178

• The transfinite numbers are in a certain sense themselves new irrationalities and in fact in my stance the all-time method of defining the finite irrational numbers is wholly dissimilar to, and I might even say in principle the same as, my method described higher up of introducing transfinite numbers. One tin 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 similar the latter are definite delimited forms or modifications of the actual infinite.
  • Equally 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 assistance and which, in their multifariousness, I have studied for more than xx years; every year, and about every twenty-four hour period brings me further in this science.
  • As quoted in Modern Mathematicians, (1995) past Harry Henderson. ~ ISBN 0816032351

From Kant to Hilbert (1996) [edit]

Quotes of Cantor in From Kant to Hilbert : A Source Volume 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 awarding in scientific discipline and contributed to its usefulness, information technology seems to me that information technology has hitherto appeared principally in the part of a variable quantity, which either grows beyond all premises or diminishes to whatever desired minuteness, but always remains finite. I call this the improper space [das Uneigentlich-unendliche].

• Infinity, in its first course (the improper-infinite) presents itself as a variable finite [veranderliches Endliches]; in the other form (which I call the proper space [Eigentlich-unendliche]) it appears equally a thoroughly determinate [bestimmtes] infinite.

• What I declare and believe to have demonstrated in this work equally well as in earlier papers is that post-obit the finite there is a transfinite (transfinitum)--which might likewise be chosen supra-finite (suprafinitum), that is, there is an unlimited ascending ladder of modes, which in its nature is non finite merely space, only which can be determined every bit can the finite past determinate, well-divers 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 gratuitous and is only jump 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 paw and established. In item, in the introduction of new numbers, it is only obligated to requite 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 equally a number satisfies all these conditions, information technology tin and must be regarded in mathematics as existent and existent.

• The essence of mathematics lies entirely in its freedom.
  • Variant translation: The essence of mathematics is in its freedom.

• If at that place is some determinate succession of defined whole real numbers, among which at that place exists no greatest, on the ground of this 2d principle of generation a new number is obtained which is regarded equally the limit of those numbers, i.e. is divers every bit the next greater number than all of them.

Quotes about Cantor [edit]

• No i shall miscarry u.s. from the Paradise that Cantor has created.
  • David Hilbert, in "Über das Unendliche" [On the Space], in Mathematische Annalen 95 (1925)
  • Variant translation: No one shall miscarry us from the Paradise that Cantor has created for us.

• In 1874 the German mathematician Georg Cantor made the startling discovery that in that location are more irrational numbers than rational ones, and more transcendental numbers than algebraic ones. In other words, rather than being oddities, about real numbers are irrational; and among irrational numbers, most are transcendental.
  • Eli Maor, due east: The Story of a Number (1994)

• Why was Cantor and so vehemently opposed to infinitesimals? In his valuable essay, "The Metaphysics of the Calculus," Abraham Robinson suggests that Cantor already had plenty issues trying to defend transfinite numbers. Information technology seems likely that, consciously or otherwise, Cantor accounted it politically wise to get 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 auction or use of heroin.
  • Rudy Rucker, in Infinity and the Mind (2005), p. lxxx

• Later on existence relegated to an obscure mid-tier university, blocked from leading journals and openly mocked past 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 building … Catholic theologians welcomed Cantor's ideas, which provided a workable mode of understanding mathematical infinities, as evidence that humans could grasp the infinite and could also, therefore, have a greater understanding of God, himself space.
  What a welcome relief this must accept 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 non been the inventor of the ideas nigh 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 incessantly possible", a review of The Space Book : A Curt 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.
  • Alexander Stepanov, in "Bjarne Stroustrup: Evolving a language in and for the existent globe: C++ 1991-2006" in ACM HOPL-III (June 2007)

External links [edit]

• Brief biography at MacTutor annal (University of St Andrews, Scotland)
  • A history of ready theory
• Prepare 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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Source: https://en.wikiquote.org/wiki/Georg_Cantor

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