Kurt Gödel

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Either mathematics is too big for the human mind, or the human mind is more than a machine.

Kurt Gödel (28 April 190614 January 1978) was a logician, mathematician, and philosopher of mathematics.


  • To every ω-consistent recursive class κ of formulae there correspond recursive class signs r, such that neither v Gen r nor Neg (v Gen r) belongs to Flg (κ) (where v is the free variable of r).
    • Proposition VI, On Formally Undecidable Propositions in Principia Mathematica and Related Systems I (1931); Informally, recursive systems of axioms cannot be complete.
  • The completeness theorem, mathematically, is indeed an almost trivial consequence of Skolem 1923a. However, the fact is that, at that time, nobody (including Skolem himself) drew this conclusion (neither from Skolem 1923a nor, as I did, from similar considerations of his own).
  • But every error is due to extraneous factors (such as emotion and education); reason itself does not err.
    • Attributed as a remark of 29th November 1972, in Incompleteness (2005) by Rebecca Goldstein
  • The formation in geological time of the human body by the laws of physics (or any other laws of similar nature), starting from a random distribution of elementary particles and the field is as unlikely as the separation of the atmosphere into its components. The complexity of the living things has to be present within the material [from which they are derived] or in the laws [governing their formation].
    • As quoted in "On 'computabilism’ and physicalism: Some Problems" by Hao Wang, in Nature’s Imagination (1995), edited by J. Cornwall, p. 161-189
  • I like Islam, it is a consistent idea of religion and open-minded.
    • As quoted in A Logical Journey: From Gödel to Philosophy (1996) by Hao Wang, p. 148
  • There are other worlds and rational beings of a different and higher kind. The world in which we live is not the only one in which we shall live or have lived.
    • As quoted in A Logical Journey: From Gödel to Philosophy (1996) by Hao Wang, p. 316
  • Religions are, for the most part, bad-but religion is not.
    • As quoted in A Logical Journey: From Gödel to Philosophy (1996) by Hao Wang, p. 316
  • Ninety percent of [contemporary philosophers] see their principal task as that of beating religion out of men's heads. … We are far from being able to provide scientific basis for the theological world view.
    • As quoted in Logical Dilemmas : The Life and Work of Kurt Gödel (1997) by John W. Dawson Jr.
  • The meaning of the world is the separation of wish and fact. Wish is a force as applied to thinking beings, to realize something. A fulfilled wish is a union of wish and fact. The meaning of the whole world is the separation and the union of fact and wish.
    • As quoted in The Outer Limits of Reason: What Science, Mathematics, and Logic Cannot Tell Us (MIT Press) 2013 by Yanofsky, Noson S
  • Secondly, even disregarding the intrinsic necessity of some new axiom, and even in case it has no intrinsic necessity at all, a probable decision about its truth is possible also in another way, namely, inductively by studying its "success." Success here means fruitfulness in consequences, in particular in "verifiable" consequences, i.e. consequences verifiable without the new axiom, whose proofs with the help of the new axiom, however, are considerably simpler and easier to discover, and make it possible to contract into one proof many different proofs. The axioms for the system of real numbers, rejected by the intuitionists, have in this sense been verified to some extent, owing to the fact that analytic number theory frequently allows one to prove number-theoretical theorems which, in a more cumbersome way, can subsequently be verified by elementary methods. A much higher degree of verification than that, however, is conceivable. There might exists axioms so abundant in their verifiable consequences, shedding so much light upon a whole field, and yielding such powerful methods for solving problems, (and even solving them constructively, as far as that is possible) that, no matter whether or not they are intrinsically necessary, they would have to be accepted at least in the same sense as any well-established physical theory.
    • Kurt Godel Collected Works: Volume II: Publications 1938-1974, S. Feferman et al., editors (1990)
  • The more I think about language, the more it amazes me that people ever understand each other at all.
    • Reflections on Kurt Gödel, MIT Press, Hao Wang 1987, page 95, according to Karl Menger

Quotes about Gödel

  • If a 'religion' is defined to be a system of ideas that contains unprovable statements, then Gödel taught us that mathematics is not only a religion, it is the only religion that can prove itself to be one.
  • Fifty years ago Kurt Gödel... proved that the world of pure mathematics is inexhaustible. No finite set of axioms and rules of inference can ever encompass the whole of mathematics. Given any finite set of axioms, we can find meaningful mathematical questions which the axioms leave unanswered. This discovery... came at first as an unwelcome shock to many mathematicians. It destroyed... the hope that they could solve the problem of deciding by a systematic procedure the truth or falsehood of any mathematical statement. ...Gödel's theorem, in denying ...the possibility of a universal algorithm to settle all questions, gave... instead, a guarantee that mathematics can never die. ...there will always be, thanks to Gödel, fresh questions to ask and fresh ideas to discover.
  • Gödel published comparatively little, but almost always to maximum effect; his papers are models of precision and incisive presentation.
    • Solomon Feferman, in "Gödel's Life and Work", Kurt Gödel: Collected Works, Volume I : Publications 1929-1936 (1986), p. 1
  • In the end we search out the beginnings. Established, beyond comparison, as the most important logician of our times by his remarkable results of the 1930s, Kurt Gödel was also most unusual in the ways of his life and mind. Deeply private and reserved, he had a superb all embracing rationality, which could descend into a maddening attention to detail in matters of everyday life.
    • Solomon Feferman, in "Gödel's Life and Work", Kurt Gödel: Collected Works, Volume I : Publications 1929-1936 (1986), p. 2
  • The progenitor of information theory, and perhaps the pivotal figure in the recent history of human thought, was Kurt Gödel, the eccentric Austriac genius and intimate of Einstein who drove determinism from its strongest and most indispensable redoubt; the coherence, consistency, and self-sufficiency of mathematics.
    Gödel demonstrated that every logical scheme, including mathematics, is dependent upon axioms that it cannot prove and that cannot be reduced to the scheme itself.
    In an elegant mathematical proof, introduced to the world by the great mathematician and computer scientist John von Neumann in September 1930, Gödel demonstrated that mathematics was intrinsically incomplete. Gödel was reportedly concerned that he might have inadvertently proved the existence of God, a faux pas in his Viennese and Princeton circle. It was one of the famously paranoid Gödel's more reasonable fears.
    • George Gilder, in Knowledge and Power : The Information Theory of Capitalism and How it is Revolutionizing our World (2013), Ch. 10: Romer's Recipes and Their Limits
  • Not even mathematics can be considered as a closed and complete system of axioms and theorems. The mathematical world is inexhaustible, no finite set of postulates and deductions will ever be able to give us the answer to all questions. Gödel's theorem, whose statement dates back to about half a century ago, brutally put an end to all attempts to condense mathematics into a list of axioms from which the truth or falsity of each of its assertions should follow. If the same mathematical language that physics uses to describe the world remains intrinsically incomplete, it is not reasonable to expect that the universe can be describable starting from a finite set of natural laws. The incompleteness of mathematics and consequently that of physics is repugnant to many, but it must be said that for the exact sciences, Gödel's theorem is by no means a defeat: on the contrary, it provides us with an intellectual push towards ever broader and more fruitful developments.
  • ... according to what Veblen told me, the association between Einstein and Gödel arose in the following way.
    Veblen felt that he had to look out for Gödel, and spent quite a lot of time talking with him. And then, he thought that he might perhaps get Einstein to take over part of this responsibility. And that seemed to go so extremely well that Veblen removed himself, essentially, from the picture.
    Einstein and Gödel remained very close. They tended to come to the Institute together, and leave the Institute together, very often. Of course, Gödel's interest in the theory of relativity theory undoubtedly goes back to this association with Einstein. ... I don't think he had any interest in physics before that. I know he had some philosophical interests, but I think the specific interest in the theory of relativity, in which he did write some papers and create some results of significance, that goes back to that association.
  • Toward the end of his life, Gödel feared that he was being poisoned, and he starved himself to death. His theorem is one of the most extraordinary results in mathematics, or in any intellectual field in this century. If ever potential mental instability is detectable by genetic analysis, an embryo of someone with Kurt Gödel's gifts might be aborted.
    • Brian L. Silver, in The Ascent of Science (1998)
  • The one man who was, during the last years, certainly by far Einstein's best friend, and in some ways strangely resembled him most, was Kurt Gödel, The great logician. They were very different in almost every personal way — Einstein gregarious, happy, full of laughter and common sense, and Gödel extremely solemn, very serious, quite solitary, and distrustful of common sense as a means of arriving at the truth. But they shared a fundamental quality: both went directly and wholeheartedly to the questions at the very center of things.
    • Ernst G. Strauss, in reminiscences of 1982, as quoted in "Gödel's Life and Work", by Solomon Feferman, in Kurt Gödel: Collected Works, Volume I : Publications 1929-1936 (1986), p. 2
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