On Proofs (1970-1979)

"A diagram is worth a thousand proofs."  (Carl E Linderholm, “Mathematics Made Difficult”, 1971)

"Any mathematician endowed with a modicum of intellectual honesty will recognise then that in each of his proofs he is capable of giving a meaning to the symbols he uses." (René F Thom, "Modern mathematics, does it exist?", 1972)

"Historically, hyperbolic geometry was first developed on an axiomatic basis. It arose as a result of efforts to prove the axiom of parallels from the other axioms. Doubt persisted for a long time as to whether this axiom could be deduced from the remaining axioms of Euclidean geometry. In their attempts to prove this axiom, mathematicians used the method of 'proof by contradiction' i.e., they assumed that the axiom of parallels was false and tried, on the basis of this assumption, to obtain a contradiction. All of these attempts were fruitless. True, the theorems obtained by negating the axiom of parallels appeared strange, but they did not contradict one another. The issue was resolved when C. F. Gauss, N. I. Lobachevski and J. Bolyai first stated explicitly that by negating the axiom of parallels one arrives at a new geometry, just as consistent as the usual (Euclidean) geometry." (Isaak Yaglom, "Geometric Transformations", 1973)

"In many cases a dull proof can be supplemented by a geometric analogue so simple and beautiful that the truth of a theorem is almost seen at a glance." (Martin Gardner, "Mathematical Games", Scientific American, 1973)

"Induction is the process of eliciting general laws via observation and the correlation of particular instances. All sciences, including mathematics, make use of the induction method. Now, mathematical induction is applied only by mathematicians in the proof of theorems of a particular kind." (Yakov Khurgin, "Did You Say Mathematics?", 1974)

"Non-standard analysis frequently simplifies substantially the proofs, not only of elementary theorems, but also of deep results. This is true, e.g., also for the proof of the existence of invariant subspaces for compact operators, disregarding the improvement of the result; and it is true in an even higher degree in other cases. This state of affairs should prevent a rather common misinterpretation of non-standard analysis, namely the idea that it is some kind of extravagance or fad of mathematical logicians. Nothing could be farther from the truth. Rather, there are good reasons to believe that non-standard analysis, in some version or other, will be the analysis of the future." (Kurt Gödel, "Remark on Non-standard Analysis", 1974)

"No theory ever agrees with all the facts in its domain, yet it is not always the theory that is to blame. Facts are constituted by older ideologies, and a clash between facts and theories may be proof of progress. It is also a first step in our attempt to find the principles implicit in familiar observational notions." (Paul K Feyerabend, "Against Method: Outline of an Anarchistic Theory of Knowledge", 1975)

"Taking experimental results and observations for granted and putting the burden of proof on the theory means taking the observational ideology for granted without having ever examined it." (Paul K Feyerabend, "Against Method: Outline of an Anarchistic Theory of Knowledge", 1975)

"The conception of the mental construction which is the fully analysed proof as being an infinite structure must, of course, be interpreted in the light of the intuitionist view that all infinity is potential infinity: the mental construction consists of a grasp of general principles according to which any finite segment of the proof could be explicitly constructed." (Michael Dummett, "The philosophical basis of intuitionistic logic", 1975)

"The following four propositions, which appear to the author to be incapable of formal proof, are presented as Fundamental Postulates upon which the entire superstructure of General Systemantics [...] is based [...] (1) Everything is a system. (2) Everything is part of a larger system. (3) The universe is infinitely systematizable, both upward (larger systems) and downward (smaller systems) (4) All systems are infinitely complex. (The illusion of simplicity comes from focusing attention on one or a few variables.)" (John Gall, "General Systemantics: How systems work, and especially how they fail", 1975)

"There is an infinite regress in proofs; therefore proofs do not prove. You should realize that proving is a game, to be played while you enjoy it and stopped when you get tired of it." (Imre Lakatos, "Proofs and Refutations", 1976)

"A proof only becomes a proof after the social act of ‘accepting it as a proof’." (Yuri I Manin, "A Course in Mathematical Logic", 1977)

"Mathematical induction […] is an entirely different procedure. Although it, too, leaps from the knowledge of particular cases to knowledge about an infinite sequence of cases, the leap is purely deductive. It is as certain as any proof in mathematics, and an indispensable tool in almost every branch of mathematics." (Martin Gardner, "Aha! Insight", 1978)

"On the face of it there should be no disagreement about mathematical proof. Everybody looks enviously at the alleged unanimity of mathematicians; but in fact there is a considerable amount of controversy in mathematics. Pure mathematicians disown the proofs of applied mathematicians, while logicians in turn disavow those of pure mathematicians. Logicists disdain the proofs of formalists and some intuitionists dismiss with contempt the proofs of logicists and formalists." (Imre Lakatos, "Mathematics, Science and Epistemology" Vol. 2, 1978)

"A proof is a construction that can be looked over, reviewed, verified by a rational agent. We often say that a proof must be perspicuous or capable of being checked by hand. It is an exhibition, a derivation of the conclusion, and it needs nothing outside itself to be convincing. The mathematician surveys the proof in its entirety and thereby comes to know the conclusion." (Thomas Tymoczko, "The Four Color Problems", Journal of Philosophy , Vol. 76, 1979)

"Science sometimes improves hypothesis and sometimes disproves them. But proof would be another matter and perhaps never occurs except in the realms of totally abstract tautology. We can sometimes say that if such and such abstract suppositions or postulates are given, then such and such abstract suppositions or postulates are given, then such and such must follow absolutely. But the truth about what can be perceived or arrived at by induction from perception is something else again." (Gregory Bateson, "Mind and Nature: A Necessary Unity", 1979)