On Proofs (1940-1949)

"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)

"It is difficult, however, to learn all these things from situations such as occur in everyday life. What we need is a series of abstract and quite impersonal situations to argue about in which one side is surely right and the other surely wrong. The best source of such situations for our purposes is geometry. Consequently we shall study geometric situations in order to get practice in straight thinking and logical argument, and in order to see how it is possible to arrange all the ideas associated with a given subject in a coherent, logical system that is free from contradictions. That is, we shall regard the proof of each proposition of geometry as an example of correct method in argumentation, and shall come to regard geometry as our ideal of an abstract logical system. Later, when we have acquired some skill in abstract reasoning, we shall try to see how much of this skill we can apply to problems from real life." (George D Birkhoff & Ralph Beately, "Basic Geometry", 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)

"The fact that the proof of a theorem consists in the application of certain simple rules of logic does not dispose of the creative element in mathematics, which lies in the choice of the possibilities to be examined." (Richard Courant & Herbert Robbins, "What Is Mathematics?", 1941)

"The question of the origin of the hypothesis belongs to a domain in which no very general rules can be given; experiment, analogy and constructive intuition play their part here. But once the correct hypothesis is formulated, the principle of mathematical induction is often sufficient to provide the proof." (Richard Courant & Herbert Robbins, "What Is Mathematics?: An Elementary Approach to Ideas and Methods", 1941)

"Perhaps the extraordinary pervasiveness of number, and the multiplicity of operations which can be performed on number without leading to inconsistency, is not a proof of the ’real existence’ of numbers as such, but a proof of the extreme flexibility of the neural model or calculating machine. This flexibility renders a far greater number of operations possible for it than for any other single process or model." (Kenneth Craik, "The Nature of Explanation", 1943)

"For a logician of a certain sort only complete proofs exist. What intends to be a proof must leave no gaps, no loopholes, no uncertainty whatever, or else it is no proof." (George Pólya, "How to solve it", 1945)

"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." (George Pólya, "How to solve it", 1945)

"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." (George Pólya, "How to solve it", 1945)

"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)