On Proofs (1950-1959)

"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)

"Many proofs in mathematics have been actually found by extremely roundabout processes. A man starts to prove this theorem and he finds that he wanders all over the map. He starts off and prove a good many results which don’t seem to be leading anywhere and then eventually ends up by the back door on the solution of the given problem." (Claude E Shannon, "Creative Thinking", 1952)

"The diagrams incorporate a large amount of information. Their use provides extensive savings in space and in mental effort. In the case of many theorems, the setting up of the correct diagram is the major part of the proof. We therefore urge that the reader stop at the end of each theorem and attempt to construct for himself the relevant diagram before examining the one which is given in the text. Once this is done, the subsequent demonstration can be followed more readily; in fact, the reader can usually supply it himself." (Samuel Eilenberg & Norman E. Steenrod, "Foundations of Algebraic Topology", 1952)

"The result of the mathematician's creative work is demonstrative reasoning, a proof; but the proof is discovered by plausible reasoning, by guessing. If the learning of mathematics reflects to any degree the invention of mathematics, it must have a place for guessing, for plausible inference." (George Pólya, "Mathematics and plausible reasoning" Vol. 1, 1954)

"We secure our mathematical knowledge by demonstrative reasoning, but we support our conjectures by plausible reasoning. A mathematical proof is demonstrative reasoning, but the inductive evidence of the physicist, the circumstantial evidence of the lawyer, the documentary evidence of the historian, and the statistical evidence of the economist belong to plausible reasoning." (George Pólya, "Mathematics and Plausible Reasoning", 1954)

"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)

"[…] no branch of mathematics competes with projective geometry in originality of ideas, coordination of intuition in discovery and rigor in proof, purity of thought, logical finish, elegance of proofs and comprehensiveness of concepts. The science born of art proved to be an art." (Morris Kline, "Projective Geometry", Scientific America Vol. 192 (1), 1955)

"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)

"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)