Mathematical Proofs – Definitions

“A proof is a construction that can be looked over, reviewed, verified by a rational agent. We often say that a proof must be perspicuous or capable of being checked by hand. It is an exhibition, a derivation of the conclusion, and it needs nothing outside itself to be convincing. The mathematician surveys the proof in its entirety and thereby comes to know the conclusion.” (Thomas Tymoczko, “The Four Color Problems”, Journal of Philosophy , Vol. 76, 1979)

“A proof tells us where to concentrate our doubts. […] An elegantly executed proof is a poem in all but the form in which it is written.” (Morris Kline)

“A proof is any completely convincing argument.” (Errett Bishop)

“A proof is a description, like driving instructions.” (Arie Hinkins, “Proofs of the Cantor-Bernstein Theorem”, 2013)

“A proof in mathematics is a psychological device for convincing some person, or some audience, that a certain mathematical assertion is true. The structure, and the language used, in formulating that proof will be a product of the person creating it; but it also must be tailored to the audience that will be receiving it and evaluating it. Thus there is no ‘unique’ or ‘right’ or ‘best’ proof of any given result. A proof is part of a situational ethic.” (Steven G Krantz, “The Proof is in the Pudding”, 2007)

“[…] a proof is a device of communication. The creator or discoverer of this new mathematical result wants others to believe it and accept it.” (Steven G Krantz, “The Proof is in the Pudding”, 2007)

“Heuristically, a proof is a rhetorical device for convincing someone else that a mathematical statement is true or valid.” (Steven G Krantz, “The Proof is in the Pudding”, 2007)

“[…] proof is central to what modern mathematics is about, and what makes it reliable and reproducible.” (Steven G Krantz, “The Proof is in the Pudding”, 2007)

“A proof in logic and mathematics is, traditionally, a deductive argument from some given assumptions to a conclusion. Proofs are meant to present conclusive evidence in the sense that the truth of the conclusion should follow necessarily from the truth of the assumptions. Proofs must be, in principle, communicable in every detail, so that their correctness can be checked.” (Sara Negri  & Jan von Plato, “Proof Analysis”, 2011)

“A mathematical proof is a sequence of sentences that convey a mathematical argument.” (Donald Bindner & Martin Erickson, “A Student’s Guide to the Study, Practice and Tools of Modern Mathematics”, 2011)

“A theorem is simply a sentence expressing something true; a proof is just an explanation of why it is true.”  (Matthias Beck & Ross Geoghegan, “The Art Of Proof”, 2011)

“Proof is an idol before whom the pure mathematician tortures himself. In physics we are generally content to sacrifice before the lesser shrine of Plausibility.” (Sir Arthur S Eddington)