On Proofs (1950 - 1974)

“We could compare mathematics so formalized to a game of chess in which the symbols correspond to the chessmen; the formulae, to definite positions of the men on the board; the axioms, to the initial positions of the chessmen; the directions for drawing conclusions, to the rules of movement; a proof, to a series of moves which leads from the initial position to a definite configuration of the men.” (Friedrich Waismann, “Introduction to Mathematical Thinking: The Formation of Concepts in Modern Mathematics”, 1951)

“You have to guess the mathematical theorem before you prove it: you have to guess the idea of the proof before you carry through the details. You have to combine observations and follow analogies: you have to try and try again. The result of the mathematician’s creative work is demonstrative reasoning, a proof; but the proof is discovered by plausible reasoning, by guessing” (George Polya, “Mathematics and plausible reasoning” Vol. 1, 1954)

"We speak in terms of ‘acceptance’, ‘confidence’, and ‘probability’, not ‘proof’. If by proof it is meant the establishment of eternal and absolute truth, open to no possible exception or modification, then proof has no place in the natural sciences." (George G Simpson, “Life: An Introduction to Biology”, 1957)

"A mathematical proof, as usually written down, is a sequence of expressions in the state space. But we may also think of the proof as consisting of the sequence of justifications of consecutive proof steps - i.e., the references to axioms, previously-proved theorems, and rules of inference that legitimize the writing down of the proof steps. From this point of view, the proof is a sequence of actions (applications of rules of inference) that, operating initially on the axioms, transform them into the desired theorem." (Herbert A Simon, "The Logic of Heuristic Decision Making", [in "The Logic of Decision and Action"], 1966)

"It is sometimes said of two expositions of one and the same mathematical proof that the one is simpler or more elegant than the other. This is a distinction which has little interest from the point of view of the theory of knowledge; it does not fall within the province of logic, but merely indicates a preference of an aesthetic or pragmatic character." (Karl Popper, "The Logic of Scientific Discovery", 1959)

"The most natural way to give an independence proof is to establish a model with the required properties. This is not the only way to proceed since one can attempt to deal directly and analyze the structure of proofs. However, such an approach to set theoretic questions is unnatural since all our intuition come from our belief in the natural, almost physical model of the mathematical universe." (Paul J Cohen, "Set Theory and the Continuum Hypothesis", 1966)

"It is characteristic of science that the full explanations are often seized in their essence by the percipient scientist long in advance of any possible proof." (John Desmond Bernal, "The Origin of Life", 1967)

"A diagram is worth a thousand proofs."  (Carl E Linderholm, “Mathematics Made Difficult”, 1971)

"In many cases a dull proof can be supplemented by a geometric analogue so simple and beautiful that the truth of a theorem is almost seen at a glance." (Martin Gardner, "Mathematical Games", Scientific American, 1973)

See also:
Proofs I, II, III, IV, V, VII, VIII, IX
Theorems I, II, III, IV, V, VI, VII, VIII, IX, X