Mathematical proof

“Proof is the idol before whom the pure mathematician tortures himself.”

“Another roof, another proof.”

“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 one's 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...”

“It really is worth the trouble to invent a new symbol if we can thus remove not a few logical difficulties and ensure the rigour of the proofs. But many mathematicians seem to have so little feeling for logical purity and accuracy that they will use a word to mean three or four different things, sooner than make the frightful decision to invent a new word.”

“are expressed using no numbers or other symbolic formalisms. Though the nitty-gritty details of the proof are formidably technical, the proof's 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 Gödel's achievement, as mathematically suspect.”

“In 1920 logic was mostly a philosopher's garden. There were also a few mathematicians there, cultivating the logical roots of the mathematical tree. Today, Theory, Set Theory, and Proof Theory, logic's major subdisciplines, have become full-fledged branches of mathematics.”

“We could call it "proof from n to n + 1" or still simpler "passage to the next integer." Unfortunately, the accepted technical term is "mathematical induction." This name results from a random circumstance. ...Now, in many cases... the assertion is found experimentally, and so the proof appears as a mathematical complement to induction; this explains the name.”

“Those who have written the history of geometry have thus far carried the development of this science. Not much later than these is Euclid, who wrote the 'Elements,' arranged much of Eudoxus' work, completed much of Theaetetus's and brought to irrefragable proof propositions which had been less strictly proved by his predecessors.”

“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 one's own research and be further developed.”

“An oral tradition makes it possible to indicate the line segments with the fingers; one can emphasize essentials and point out how the proof was found. All of this disappears in the written formulation... as soon as some external cause brought about an interruption in the oral tradition, and only books remained, it became very difficult to assimilate the work of the great predursors, and next to impossible to pass beyond it.”

“Comparatively few of the propositions and proofs in the Elements are [Euclid's] own discoveries. In fact, the proof of the "Theorem of Pythagoras" is the only one directly ascribed to him.”

“The analytic method is not conclusive, unless all operations involved in it are known to be reversible. To remove all doubt, the Greeks, as a rule added to the analytic process a synthetic one, consisting of a reversion of all operations occurring in the analysis. Thus the aim of analysis was to aid in the discovery of synthetic proofs or solutions.”

“Proofs have gaps and are... inherently incomplete and sometimes wrong. ...There is another reason ...Humans err. ...and others do not necessarily notice our mistakes. ...This suggests an important reason why "more elementary" proofs are better... The more elementary... the easier it is to check, and the more reliable its verification.”

“Erdős was a genius at finding brilliantly simple proofs of deep results, but, until recently, very much of his work was ignored...”

“There are... masterpieces of... exposition... Two examples... are Weil's Number Theory for Beginners... and Artin's . Mathematics can be done scrupulously.”

“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.”