"As the objects of abstract geometry cannot be totally grasped by space intuition, a rigorous proof in abstract geometry can never be based only on intuition, but it must be founded on logical deduction from valid and precise axioms. Nevertheless intuition maintains, also in precision geometry, its irreplaceable value that cannot be substituted by logical considerations. Intuition helps us to construct a proof and to gain an overview, it is, moreover, a source of inventions and new mental connections." (Felix Klein, "Elementary Mathematics from a Higher Standpoint" Vol III: "Precision Mathematics and Approximation Mathematics", 1928)
"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, "The Nature of the Physical World", 1928)
"A mathematician is a person who can find analogies between theorems; a better mathematician is one who can see analogies between proofs and the best mathematician can notice analogies between theories." (Stefan Banach, cca. 1930)
“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)
"The search for the most general conditions of validity of a determined statement, if it is ready to reveal its causal proof, doesn’t succeed without a constant reworking of the implemented notions. (Georges Bouligand, "La causalite des theories mathématiques", Actualités Scientifiques et Industrielles 184, 1935)
“A mathematical proof should resemble a simple and clear-cut constellation, not a scattered cluster in the Milky Way.“ (Godfrey H Hardy, “A Mathematician’s Apology”, 1940)
“Without the strictest deductive proof from admitted assumptions, explicitly stated as such, mathematics does not exist.” (Eric T Bell, “The Development of Mathematics”, 1940)
"Heuristic reasoning is reasoning not regarded as final and strict but as provisional and plausible only, whose purpose is to discover the solution of the present problem. We are often obliged to use heuristic reasoning. We shall attain complete certainty when we shall have obtained the complete solution, but before obtaining certainty we must often be satisfied with a more or less plausible guess. We may need the provisional before we attain the final. We need heuristic reasoning when we construct a strict proof as we need scaffolding when we erect a building."
"In mathematics as in the physical sciences we may use observation and induction to discover general laws. But there is a difference. In the physical sciences, there is no higher authority than observation and induction but In mathematics there is such an authority: rigorous proof."
"The cookbook gives a detailed description of ingredients and procedures but no proofs for its prescriptions or reasons for its recipes; the proof of the pudding is in the eating. [...] Mathematics cannot be tested in exactly the same manner as a pudding; if all sorts of reasoning are debarred, a course of calculus may easily become an incoherent inventory of indigestible information." (George Pólya, "How to solve it", 1945)
"Euclid taught me that without assumptions there is no proof. Therefore, in any argument, examine the assumptions.” (Eric T Bell, Mathematics Magazine, 1949)
"Euclid taught me that without assumptions there is no proof. Therefore, in any argument, examine the assumptions.” (Eric T Bell, Mathematics Magazine, 1949)
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