"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 tediously laborious proof may be a sign that the writer has been less than felicitous in expressing himself; but more often than not, as we know, it indicates that he has been laboring under limitations which prevented him from translating directly into words or formulas some very simple ideas." (André Weil)
"Analogy cannot serve as proof." (Louis Pasteur)
"Any good theorem should have several proofs, the more the better. For two reasons: usually, different proofs have different strengths and weaknesses, and they generalise in different directions: they are not just repetitions of each other." (Michael F Atiyah)
"Don’t just read it; fight it! Ask your own questions, look for your own examples, discover your own proofs." (Paul R Halmos)
"Empirical evidence can never establish mathematical existence—nor can the mathematician's demand for existence be dismissed by the physicist as useless rigor. Only a mathematical existence proof can ensure that the mathematical description of a physical phenomenon is meaningful." (Richard Courant)
"It is very difficult to write mathematics books today. If one does not take pains with the fine points of theorems, explanations, proofs and corollaries, then it won’t be a mathematics book; but if one does these things, then the reading of it will be extremely boring." (Johannes Kepler)
"Just give me the insights. I can always come up with the proofs!" (Bernhard Riemann)
"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)
"Some facts can be seen more clearly by example than by proof." (Leonard Euler)
"The arguments […] by which you support my theories, are most ingenious, but not founded on demonstrated facts; analogy is no proof." (Louis Pasteur)
"The essential quality of a proof is to compel belief." (Pierre de Fermat)
"There have been only Mathematicians who were able to find some proofs, that is to say some sure and certain reasons." (René Descartes)
"We found a beautiful and most general proposition, namely, that every integer is either a square, or the sum of two, three or at most four squares. This theorem depends on some of the most recondite mysteries of numbers, and it is not possible to present its proof on the margin of this page." (Pierre de Fermat)
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