Proving Is Convincing and Explaining

REUBEN HERSH PROVING IS CONVINCING AND EXPLAINING ABSTRACT. In mathematical research, the purpose of proof is to convince. The test of whether something is a proof is whether it convinces qualified judges. In the classroom, on the other hand, the purpose of proof is to explain. Enlightened use of proofs in the mathematics classroom aims to stimulate the students' understanding, not to meet abstract standards of "rigor" or "honesty." I. WHAT IS PROOF? This is one question we mathematics teachers and students would normally never think of asking. We've seen proofs, we've done proofs. Proof is what we've been watching and doing for 10 or 20 years! In the Mathematical Experience (Davis and Hersh, 1981, pp. 39-40) this almost unthinkable question is asked of the Ideal Mathematician (the I.M. - the most mathematician-like mathematician) by a student of philosophy: "What is a mathematical proof?" The I.M. responds with examples - the Fundamental Theorem of This, the Fundamental Theorem of That. But the philosophy student wants a general definition, not just some examples. The I.M. tells the philosophy student about proof as it's portrayed in formal logic: permutation of logical symbols according to certain formal rules. But the philosophy student has seen proofs in mathematics classes, and none of them fit this description. The Ideal Mathematician crumbles. He confides that formal logic is rarely employed in proving theorems, that the real truth of the matter is that a proof is just a convincing argument, as judged by competent judges. The philosophy student is appalled by this betrayal of logical standards. The I.M. puts an end to the conversation with these final words: "Everybody knows what a proof is. Read, study, and you'll catch on. Unless you don't." The Mathematical Experience has been in print for ten years. No philosopher or mathematician has yet taken up the Ideal Mathematician's challenge: "If not me, then who?" (Meaning: Who has a better right than I to decide what is a proof?) II. PROOF AMONG PROFESSIONAL MATHEMATICIANS In mathematical practice, in the real life of living mathematicians, proof is convincing argument, as judged by qualified judges. How does this notion of proof Educational Studies in Mathematics 24: 389-399, 1993. (~) 1993 Kluwer Academic Publishers. Printed in the Netherlands. 390 REUBEN HERSH differ from proof in the sense of formal logic? Firstly, formal proof can exist only within a formalized theory. Formal proof has to be expressed in a formal vocabulary, founded on a set of formal axioms, reasoned about by formal rules of inference. But the passage from an informal, intuitive theory to a formalized theory inevitably entails some loss of meaning or change of meaning. The informal by its nature has connotations and alternative interpretations that are not in the formalized theory. Consequently, any result that is proved formally may be challenged: "How faithful are this statement and proof to the informal concepts we are actually interested in?" Secondly, for many mathematical investigations, full formalization and complete formal proof, even if possible in principle, may be impossible in practice. They may require time, patience, and interest beyond the capacity of any human mathematician. Indeed, they can exceed the capacity of any available or foreseeable computing system. The attempt to verify formal proofs by computer introduces new sources of error: random errors caused by fluctuations of the physical characteristics of the machine, and inevitable human errors in design and production of software and hardware (logic and programming). For most non-trivial mathematics, the hope of formal proof remains only hypothetical. When, as in the four-color theorem of Appel and Haken (see below), machine computation does execute part of the proof, some mathematicians reject the proof because the details of the machine computation are hidden (inevitably). More than whether a conjecture is correct, mathematicians want to know why it is correct. We want to understand the proof, not just be told it exists. (See Paul Halmos' complaint below about the Appel-Haken theorem). Some discussions of proof talk about mathematics only as it's presented in journals and textbooks. There, proof functions as the last judgment, the final word before a problem is put to bed. But the essential mathematical activity is finding the proof, not checking after the fact that it is indeed a proof. At the stage of creation, proofs are often presented in front of a blackboard, hopefully and tentatively. The detection of an error or omission is welcomed as a step toward improving the proof. Lakatos' (1976) book is a fascinating presentation of this aspect of proof. At the time of her conversation with the Ideal Mathematician our philosophy student had never witnessed proof at this level. If she had, she might not have been so shocked by the I.M.'s confession. ("Proof is an argument that convinces the experts.") G. H. Hardy (1929, p. 18), the most eminent English mathematician of his day, wrote: "I have myself always thought of a mathematician as in the first instance an observer, who gazes at a distant range of mountains and notes down his observations. His object is simply to distinguish clearly and notify to others as many different peaks as he can. There are some peaks which he can distinguish easily, while others are less clear. He sees A sharply, while of B he can obtain only transitory glimpses. At last he makes out a ridge which leads from A and, following it to its end, he discovers that it culminates in B. B is now fixed in his vision, and from this point he can proceed to further discoveries. In other cases CONVINCING AND EXPLAINING 391 perhaps he can distinguish a ridge which vanishes in the distance, and conjectures that it leads to a peak in the clouds or below the horizon. But when he sees a peak, he believes that it is there simply because he sees it. If he wishes someone else to see it, he points to it, either directly or through the chain of summits which led him to recognize it himself. When his pupil also sees it, the research, the argument, theproofis finished." Here the "chain of summits" is the chain of statements in a proof, connecting previously known facts (peaks) to new ones. Hardy may use a chain of summits in order to see a new peak. But once he sees the new peak, he believes in it because he sees it, no longer needing any "chain". All real-life proofs are to some degree informal. A piece of formal argument - a calculation - is meaningful only as part of an informal proof, to complete or to verify some informal reasoning. The formal-logic picture of proof is a fascinating topic for study in logic. It is not a truthful picture of real-life mathematical proof. Some writers say that informal proofs are convincing if in principle they can be turned into formal proofs. To accept that a formal proof could be written, without seeing it written, is an act of faith. Is such a formalization possible for all the vast collections of accepted theorems in today's mathematics? Perhaps it is possible. But firm belief in that possibility is an act of faith. (See Davis and Hersh, 1981, pp. 57-73). III. THREE MEANINGS OF "PROOF" The root meaning of the English word "prove" is: (A) Test, try out, determine the true state of affairs. (As in Aberdeen Proving Ground, galley proof, "the proof of the pudding," etc.). It comes from the Latin probare, and is cognate to "probe", "probation", "probable", "probity". In mathematics, "proof" has two meanings, one in common practise, the other specialized in mathematical logic and in philosophy of mathematics. The first mathematical meaning, the "working" meaning, is: (B) An argument that convinces qualified judges. The second mathematical meaning, the "logic" one, is: (C) A sequence of transformations offormal sentences, carried out according to the rules of the predicate calculus. What is the relation between these three meanings, one colloquial and two mathematical? The logical definition C is intended to be faithful to the everyday meaning B, but more precise. But there has never been and can never be a strict demonstration that definition C really is identical or even similar to what mathematicians do when they prove. This is a universal limitation on any mathematical model of real-world phenomena. As a matter of principle it is impossible to give a mathematical proof that a mathematical model is faithful to reality. All one can do is test the model against experience. Certainly no one has proved that the logic model of proof is incorrect. If a counterexample is ever found, logicians will adapt logic to accommodate it. On the other hand, no one has ever attempted to show that 392 REUBENHERSH mathematicians' daily practise is faithfully described by the predicate calculus (first-order logic). To the casual observer, they do not seem similar, as the philosophy student said to the Ideal Mathematician. Some have said that logic has nothing to do with discovering mathematics, only with verifying it. But no one has shown that the actual practise of verification by mathematicians is faithfully described by the predicate calculus. Some might say that there is no need for such a demonstration, for if a mathematician doesn't follow the rules of the predicate calculus, then his/her reasoning is incorrect, and needn't be considered at all. I take the opposite standpoint: what mathematicians at large sanction and accept is correct. Their work is the touchstone of logic, not vice versa.

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