"Mathematical proofs are out of the reach of topical arguments; and are not to be attacked by the equivocal use of words or declaration, that make so great a part of other discourses, - nay, even of controversies.” (John Locke, “An Essay Concerning Human Understanding”, 1690)
"[…] it is an error to believe that rigor in the proof is the enemy of simplicity." (David Hilbert, [Paris International Congress] 1900)
"Proof is an idol before whom the pure mathematician tortures himself." (Sir Arthur S Eddington, "The Nature of the Physical World", 1928)
"A modern mathematical proof is not very different from a modern machine, or a modern test setup: the simple fundamental principles are hidden and almost invisible under a mass of technical details." (Hermann Weyl, "Unterrichtsblätter für Mathematik und Naturwissenschaften", 1932)
"A mathematical proof is demonstrative reasoning [...]" (George Pólya, "Mathematics and Plausible Reasoning", 1954)
"The result of the mathematician's creative work is demonstrative reasoning, a proof; but the proof is discovered by plausible reasoning, by guessing." (George Pólya, "Induction and Analogy in Mathematics", 1954)
"Rigorous proofs are the hallmark of mathematics, they are an essential part of mathematics’ contribution to general culture." (George Pólya, "Mathematical Discovery", 1962)
"[...] 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)
"A proof is a construction that can be looked over, reviewed, verified by a rational agent." (Thomas Tymoczko, "The Four Color Problems", Journal of Philosophy , Vol. 76, 1979)
"Proof serves many purposes simultaneously […] Proof is respectability. Proof is the seal of authority. Proof, in its best instance, increases understanding by revealing the heart of the matter. Proof suggests new mathematics […] Proof is mathematical power, the electric voltage of the subject which vitalizes the static assertions of the theorems." (Reuben Hersh, "The Mathematical Experience", 1981)
"People might suppose that a mathematical proof is conceived as a logical progression, where each step follows upon the ones that have preceded it." (Roger Penrose, "The Emperor’s New Mind", 1989)
"A mathematical proof is a chain of logical deductions, all stemming from a small number of initial assumptions ('axioms') and subject to the strict rules of mathematical logic." (Eli Maor, "e: The Story of a Number", 1994)
"Mathematics rests on proof - and proof is eternal." (Saunders Mac Lane,"Reponses to …", Bulletin of the American Mathematical Society Vol. 30 (2), 1994)
"Proofs are not impersonal, they express the personality of their creator/discoverer just as much as literary efforts do. If something important is true, there will be many reasons that it is true, many proofs of that fact." (Gregory Chaitin, "Meta Math: The Quest for Omega", 2005)
"Within mathematics, a proof is an intellectual structure in which premises are conveyed to their conclusions by specific inferential steps. Assumptions in mathematics are called axioms, and conclusions theorems. This definition may be sharpened a little bit. A proof is a finite series of statements such that every statement is either an axiom or follows directly from an axiom by means of tight, narrowly defined rules." (David Berlinski, "Infinite Ascent: A short history of mathematics", 2005)
"[…] a proof is a device of communication." (Steven G Krantz, "The Proof is in the Pudding", 2007)
"A proof is a series of steps based on the (adopted) axioms and deduction rules which reaches a desired conclusion." (Cristian S Calude et al, "Proving and Programming", 2007)
"A proof is part of a situational ethic." (Steven G Krantz, "The Proof is in the Pudding", 2007)
"Heuristically, a proof is a rhetorical device for convincing someone else that a mathematical statement is true or valid." (Steven G Krantz, "The Proof is in the Pudding", 2007)
"[…] proof is central to what modern mathematics is about, and what makes it reliable and reproducible." (Steven G Krantz, "The Proof is in the Pudding", 2007)
"So the theorems and propositions are the new heights of knowledge we achieve, while the proofs are essential as they are the mortar which attaches them to the level below. Without proofs the structure would collapse." (Sidney A Morris, "Topology without Tears", 2007)
"A proof is like a piece of theatre or music, with moments of high drama where some major shift takes the audience into a new realm."
"[…] proof is the key ingredient of the emotional side of mathematics; proof is the ultimate explanation of why something is true, and a good proof often has a powerful emotional impact, boosting confidence and encouraging further questions ‘why’." (Alexandre V Borovik, "Mathematics under the Microscope: Notes on Cognitive Aspects of Mathematical Practice", 2009)
"A mathematical proof is a watertight argument which begins with information you are given, proceeds by logical argument, and ends with what you are asked to prove." (Sydney A Morris, "Topology without Tears", 2011)
"A proof in logic and mathematics is, traditionally, a deductive argument from some given assumptions to a conclusion. Proofs are meant to present conclusive evidence in the sense that the truth of the conclusion should follow necessarily from the truth of the assumptions. Proofs must be, in principle, communicable in every detail, so that their correctness can be checked." (Sara Negri & Jan von Plato, "Proof Analysis", 2011)
"A proof is simply a story. The characters are the elements of the problem, and the plot is up to you." (Paul Lockhart, "Measurement", 2012)
"A mathematical proof is like a battle, or if you prefer a less warlike metaphor, a game of chess. Once a potential weak point has been identified, the mathematician’s technical grasp of the machinery of mathematics can be brought to bear to exploit it." (Ian Stewart, "Visions of Infinity", 2013)
"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)
"Proofs are to mathematics what spelling (or even calligraphy) is to poetry. Mathematical works do consist of proofs, just as poems do consist of words." (Vladimir Arnold)
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