"It was not until the nineteenth century, chiefly through... Gauss, Bolyai, Lobachevsky, and Riemann, that the impossibility of deducing the parallel axiom from the others was demonstrated. This outcome was of the greatest intellectual importance. ...[I]t called attention... to the fact that a proof can be given of the impossibility of proving certain propositions within a given system. ...Gödels paper is a proof of the impossibilty of formally demonstrating certain important propositions in number theory. ...[T]he resolution of that parallel axiom question forced the realization that Euclid was not the last word on the subject of geometry, since new systems of geometry can be constructed... incompatible with those adopted by Euclid. ...[I]t gradually became clear that the proper business of pure mathematicians is to derive theorems from postulated assumptions, and that it is not their concern whether the axioms are actually true."
A mathematical proof is a deductive argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms, along with the accepted rules of inference. Proofs are ex
"Paul Erdős, although an atheist, spoke of an imaginary book, in which God has written down all the most beautiful mathematical proofs. When Erdős wanted to express particular appreciation of a proof, he would exclaim "This ones from the Book!". This viewpoint expresses... that mathematics, as the... foundation on which the laws of our universe are built, is a natural candidate for what has been personified as God..."
Mathematical proof"Erdős was a genius at finding brilliantly simple proofs of deep results, but, until recently, very much of his work was ignored..."
Mathematical proof"Proofs are for the mathematician what experimental procedures are for the experimental scientist: in studying them one learns of new ideas, new concepts, new strategies—devices which can be assimilated for ones own research and be further developed."
Mathematical proof"Under the same assumptions made in the Chord papers, the [SIGCOMM] version of the protocol is not correct, and not one of the properties claimed invariant in [PODC] is actually invariantly true of it."
Mathematical proof"Perhaps we should discard the myth that mathematics is a rigorously deductive enterprise... hand-waving is intrinsic. We try to minimize it and we can sometimes escape it, but not always, if we want to discover new theorems."
Mathematical proof"are expressed using no numbers or other symbolic formalisms. Though the nitty-gritty details of the proof are formidably technical, the proofs overall strategy, delightfully, is not. ...They belong to a branch of mathematics known as formal logic or mathematical logic, a field which was viewed, prior to Gödels achievement, as mathematically suspect."
Mathematical proof