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Kurt Gödel
12 quotes
“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).”
Kurt Gödel (1906–1978) first seen 1931 wikiquote, CC BY-SA 4.0 source
Recorded in: 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).”
Kurt Gödel (1906–1978) first seen 1967 wikiquote, CC BY-SA 4.0 source
Recorded in: as quoted, from a 1967 letter to Hao Wang, by Hao Wang in From Mathematics to Philosophy, 1974, p. 8
“But every error is due to extraneous factors (such as emotion and education); reason itself does not err.”
Kurt Gödel (1906–1978) first seen 1972 wikiquote, CC BY-SA 4.0 source
Recorded in: Attributed as a remark of 29th November 1972, in Incompleteness (2005) by Rebecca Goldstein
“Either mathematics is too big for the human mind, or the human mind is more than a machine.”
Kurt Gödel (1906–1978) said by 1978 at the latest wikiquote, CC BY-SA 4.0 source
Recorded in: As quoted in Topoi : The Categorial Analysis of Logic (1979) by Robert Goldblatt, p. 13
“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].”
Kurt Gödel (1906–1978) said by 1978 at the latest wikiquote, CC BY-SA 4.0 source
Recorded in: 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.”
Kurt Gödel (1906–1978) said by 1978 at the latest wikiquote, CC BY-SA 4.0 source
Recorded in: 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.”
Kurt Gödel (1906–1978) said by 1978 at the latest wikiquote, CC BY-SA 4.0 source
Recorded in: 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.”
Kurt Gödel (1906–1978) said by 1978 at the latest wikiquote, CC BY-SA 4.0 source
Recorded in: 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.”
Kurt Gödel (1906–1978) said by 1978 at the latest wikiquote, CC BY-SA 4.0 source
Recorded in: 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.”
Kurt Gödel (1906–1978) said by 1978 at the latest wikiquote, CC BY-SA 4.0 source
Recorded in: As quoted in The Outer Limits of Reason: What Science, Mathematics, and Logic Cannot Tell Us (MIT Press) 2013 by Yanofsky, Noson S
“The more I think about language, the more it amazes me that people ever understand each other at all.”
Kurt Gödel (1906–1978) said by 1978 at the latest wikiquote, CC BY-SA 4.0 source
Recorded in: Reflections on Kurt Gödel, MIT Press, Hao Wang 1987, page 95, according to Karl Menger
“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 Gödel (1906–1978) first seen 1938 wikiquote, CC BY-SA 4.0 source
Recorded in: Kurt Godel Collected Works: Volume II: Publications 1938-1974, S. Feferman et al., editors (1990)