La lettre de la Preuve ISSN Septembre/Octobre Shortcuts in Proof. Jean-Paul Delahaye Universite des Sciences et Technologies de Lille

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1 La lettre de la Preuve ISSN Septembre/Octobre 2000 Shortcuts in Proof Jean-Paul Delahaye Universite des Sciences et Technologies de Lille How can we know if a proof is correct? People often imagine that it suffices for a mathematician to make the effort to read it carefully, line by line, after having taken note of the definitions and known results that might be of use. If certain questions are unresolved, as to be sure some are, we would known precisely which ones, and the work of the researcher would consist uniquely of resolving these classified enigmas. Certainty would reign throughout mathematics. This situation, if it were true, would make mathematics categorically the opposite of physics. In physics, nothing can definitively establish a theory, which is nothing more than a hypothesis that can always be called into question by additional evidence. We know thus that one cannot prove a physical law, which is a general assertion based on observations and experiments which are no more than particular assertions. Logicians support the idealized vision of mathematics by maintaining that since the beginning of the century codified systems for writing proofs, called formal systems, have been available. If a proof is written with one of these systems a computer can verify its correctness without human intervention. The correctness of a formal proof, say the logicians, is mechanically verifiable and requires no intelligence. Actually, a multitude of reasons complicate this ideal tableau and smudge the excessively beautiful image of a mathematical science without moods, without disputes, without errors, where unanimity holds on every question and where discords can be settled with computers as arbiters. We are going to present the difficulties involved in writing mechanically verifiable formal proofs. These difficulties do not signify that codified systems for the writing of proofs are useless or unsatisfactory -- no one today would think of bypassing them, and it's a mistake to evoke Gödel's theorems on incompleteness as arguments against formalism. The difficulties simply signify that the use of logical formalism is much more delicate than it looks, and that mathematics overflows with a complexity which one cannot in practice lock up in the aseptic boxes of the logicians' formal systems. Page 1 sur 8

2 In the introduction to his grand treatise on mathematics, Nicolas Bourbaki explains this point of view, which is at the heart of the modern conception of mathematics, very lucidly: "holding always present, as a sort of horizon, the possibility of a total formalization, our Treatise aims at perfect rigor.[ ] From the fact that we seek to hold ourselves constantly as close to a formalized text as seems possible without insupportable length, verification is in principle simplified; errors (inevitable in an enterprise of this nature) can be located without excessive loss of time, and the risk of their nullifying a chapter or an entire Book remains very small." Some of the figures in this article reproduce proofs without words which complement those already offered in this journal in February, These proofs illustrate a strong idea: aside from the presentation of proofs in vernacular or formal language, a silent, geometric exposition of mathematical proofs is occasionally possible. The real proofs which mathematicians write are very different from the formal proofs for one main reason: the generalized use of abbreviations. Among the abbreviations there are some benign ones which introduce absolutely no ambiguity. A mathematician will indicate, for instance, that to save space she will write a 7 instead of a x a x a x a x a x a x a, or sin(5)(x) instead of sin(sin(sin(sin(sin (x))))) ; the parentheses around the 5 eliminate the confusion between sin(5)(x) and sin5(x). If abbreviations of this sort were the only ones used, it would be easy to translate a human proof into a formal proof (and thus to verify it mechanically.) One would require the computer systematically to replace each abbreviated notation by its complete version and even though this practice would elongate the text it would not seriously disturb the computer in charge of checking it. Shortcuts and abbreviations Unfortunately, most abbreviations are ambiguous and only the context makes it possible to determine their meaning. The notation dx for example, can represent the product of the variable d and the variable x or else the differential of x used in integral calculus. The mathematician Littlewood cites an anecdote where a professor who was writing the polynomial ax 4 + bx 3 + cx 2 + dx + e commented on the formula by saying that "e is not necessarily, but can be, the base for the natural logarithm." The sign + designates whole number addition or matrix addition or all sorts of other operations (it is not unusual to find in the same mathematical text several + signs being used, each referring to a different concept.) The notation i, depending on the text, may designate a natural number, or the complex number whose square is -1, or something else entirely. Frequently also there are indications which introduce an ambiguity: "The parameter k will be fixed for the rest of the paragraph, and in place of f k (x) we will write f(x)." Mathematical texts are filled with these shortcuts which make translation into formal language difficult. Mathematicians are conscious of it, as this quotation from Nicolas Bourbaki (Algèbre linéaire, chapter II, p.53) attests: "The notation u ƒ v may thus cause confusion, and it will be necessary to deduce from the context whether it indicates a tensor product or a linear application." A thousand other examples could be given. Page 2 sur 8

3 Gaps everywhere All of these abbreviations could, with attentive and meticulous effort, be tracked down and eliminated from the text of a proof. Abbreviations in the structure itself of proofs, on the other hand, are much more serious. Mathematical texts are nonetheless riddled with them. One can understand the meaning of "the reader can verify easily that the units digit of is 6" or "When (x+4x 5 ) 12 is expanded, the coefficient of x 20 is 1056; " taking a sheet of paper the reader (assumed to be a patient one) will reassure himself of the justice of what is affirmed, and the piece of the proof which was lacking &emdash; a calculation &emdash; will thus be reconstituted. Occasionally the calculation which would permit the verification is impossible without the use of computers equipped with some particular software. Imagine what you would have to do to check the statement " 9.98% of the first million decimals of the number p are 7's." One must either have confidence in the author who made the statement or else spend hours, or even days, to verify it. As long as it is just a matter of calculations, filling in the hole present in human proof does not seem totally insurmountable. On the other hand, how can one help worrying when the mathematician, after giving the proof of the theorem where b is positive, writes "the case where b is negative is easy to handle" or "is analogous?" Will it work? How can one help being downright anxious when in the text of a proof one finds the laconic and yet very common "left as an exercise for the reader"? Here are some examples of that type, taken from the Linear Algebra book of the Bourbaki treatise. First the classic "It is easy to verify that " (chapter 1, p. 107.) A sentence like that does not seem to provide the reader with much guidance. No more, in fact, than the one which, if you run into an obstacle in the omitted proof, is likely to reduce you to despair: "The proof is even simpler for " (Chapter II, p.85.) The inevitable "It is clear that " is, to be sure, part of the proof arsenal of the treatise (Chapter III, p. 158.) When details are given they are not necessarily speaking ones and the reader is not sure of being able to manage with the cryptic ones in the proof of proposition 3 of Chapter II, p. 39, which end with "The conclusion thus results from definition 1 and from II, p. 10, prop. 5(ii)." More delicious yet is the astonishing one-line proof of a statement which occupies four lines: "This follows from E, III, p.29, prop. 12 and E,III, p.42, cor.1" (Chapter III, p. 87.) Even the theorems may be omitted Non-mathematicians may be astonished to find that short-cuts, even in the best treatises, can extend beyond the proofs. Thus, occasionally, the definitions and statements themselves are omitted. Still in the treatise of N. Bourbaki is the admirable: "We will leave to the reader the task of formulating the analogous definitions and remarks for systems of left-hand linear equations" (chapter II, p. 146), and also "We leave to the reader the task of defining a projective system of rings and verifying that " (chapter II, p. 146.) Why not leave to the reader the task of writing the next chapter entirely on her own? These omissions are not without danger, and many mathematicians will tell you that, even if they seem inevitable, they remain perilous. In fact, it is not uncommon for errors to appear in mathematical texts precisely in those areas whose details Page 3 sur 8

4 have been left incomplete by a mathematician who is hurried and sincerely persuaded that the proof is possible using the succinct indications which he has offered by way of an argument. These simplifications authorized by "left for the reader" and others :"by way of an example we will give the proof for the number 5" are dangerous, because everyone who writes mathematical texts knows that often they themselves have not written out on a separate sheet of paper the details that they are suggesting that their reader reconstruct alone. They have thought about it, they are deeply convinced that "it'll work", but is that enough? Yet how could one deny oneself these simplifications, since Bourbaki himself recognizes that in mathematical texts "the use of resources of rhetoric is necessary" (Bourbaki, Introduction, p.7)? Encouraged by the pleasure of the economy and elegance of conciseness, concerned also not to pass for an imbecile by giving details on a genuinely easy passage, every mathematician abbreviates proofs, especially when addressing highlevel mathematicians. Thus, rather oddly, the risks are maximal in papers presenting research results. In a course for students the mathematician attempts to restrain the penchant for short sketches of proofs, without however resisting the temptation of "case 1 is trivial" and especially not of the blow of "left as an exercise." These laconic pirouettes are the despair of mediocre students, unable to plug up these allegedly simple gaps, and they leave a sensation of guilt with the hurried reader who dispenses with the suggested work even though in the introduction the author has perfidiously indicated that it is important to treat such exercises carefully to guarantee a good assimilation of the material. A famous gap One of the most famous gaps in a mathematical proof originated in one of those imprudent shortcuts. It is well known that on Wednesday, June 23, 1993, at the end of three days of a conference Andrew Wiles concluded a public presentation of Fermat's Last Theorem (which he proved as a special case of more general results.) No one could verify on the spot the far too complex proof, of which only the major lines had been explained orally. The proof, which took about two hundred pages, was submitted to a committee of experts consisting of six specialists who were to spend months working on it. Numerous times they had to ask questions of A. Wiles in order to get the explanations which would permit them to gather the meaning of what was written. If it had been a formal proof the mathematical experts would have had no need to get in contact with A. Wiles. A. Wiles himself claimed to have verified each step of the proof twice before confiding it to his colleagues. To each question from the experts he moreover offered a response which removed the difficulty which had been brought up. Little by little, A. Wiles made his document comprehensible, which, if mathematics conformed to the idealized description at the beginning of this article, it should have been from the start. They advanced slowly towards total validation until the day when a response did not totally satisfy an expert, nor did the completion of the response the next day: there was a hole in the proof which, despite all the care A. Wiles took, he had not seen. The "bug" resulted from the use, outside of the precise conditions in which it had been proved, of a procedure which A. Wiles, in a hasty generalization, had used without doing all of the verifications required by this case. What followed is well known: more than two years were required to correct the error and arrive, following a different route from the one proposed in 1993, at a Page 4 sur 8

5 complete proof accepted by the experts. Let us note, to reassure the anxious, that once this agreement had been reached among the experts, the proof was confirmed by other experts. Today there remains no doubt whatever about its validity, that is, about the possibility of writing a formal proof of Fermat's Theorem. The length of the proof prohibits, for the moment, this complete formal transcription, but there is absolutely no controversy about its existence. The difficulty cited in the November issue concerning the transcription of A. Wiles' proof in the formalism of Peano's elementary arithmetic (more limited than that of the theory of sets which serves as an implicit reference today) does not in the least taint the unanimity which reigns over the validity of A. Wiles' proof, and unless the theory of sets is imagined contradictory, the victory is definitive. Controversies Today when there is a controversy in mathematics over the validity of a proof, it is thus true, when all is said and done, that it will not last long, and the reason it won't is the possibility of approaching more and more closely to what would be a complete, formal proof of the result. The relation between formal proofs and real proofs can thus be summed up by saying: formal proofs are only rarely written by mathematicians because they are too long, but it is nonetheless they which are the ultimate recourse in the evaluation of a real proof whose validity is in doubt. In effect, it is by approaching step by step towards what would be a formal proof of the result under discussion that one arrives at a definite agreement: either one encounters no obstacles on the route which leads there (which one rarely follows to its end) and the real proof is considered to be correct, or else an irreparable hole appears and the proof is insufficient, thus nonexistent The elimination of controversies in mathematics is relatively new, because, before the progress in logic at the beginning of the century, they appeared regularly, and grave uncertainties divided the minds, even in mathematics. This was the case at the end of the seventeenth century when differential calculus was discovered and the idea of a limit was only imperfectly mastered (it was mastered by A Cauchy, and by K. Weierstrass, ) At the moment when topology was being perfected, other prolonged controversies took place, like that which the philosopher Imre Lakatos details in his book on the subject of Euler's formula v - s + f = 2 (v is the number of vertices, s the number of sides and f the number of faces) of which it was difficult to determine the exact domain of validity. At the beginning of the century came one of the last big controversies in mathematics. It was on foundations, but it didn't deal with the accuracy or inaccuracy of particular proofs, but rather was a discussion on the modes of reasoning which one could legitimately use (notions of set theory, axiom of choice, non-constructive reasoning, etc.) Today, thanks to formal systems, the controversy on the foundations of mathematics has died out and is no more than a problem for philosophers about which few mathematicians worry. On all important questions there is unanimous agreement, which sets mathematics apart as a science. The doubts about Gödel's Theorem which were expressed by some mathematicians just after its appearance and the serious incomprehension which causes doubts among amateurs about this or that result (for example the uncountability of the real numbers or the impossibility of squaring the circle) should be taken no more seriously than the attempts to realize perpetual motion Page 5 sur 8

6 which are an obsession of the Sunday inventors. Yes, in mathematics today there is unanimity about correctness, and not one controversy is in process. Progress in formalization Even though few mathematicians bother from day to day with the complete formalization of their proofs, the progress made on these questions is nonnegligible. Poincaré made fun of the fact that the writing of 1 in the formal system of the Principia of Russell and Whitehead (the first complete formal system to permit the writing of an important volume of mathematics) required a large number of symbols, which he found absurd. Fascinated by the treatise of Mathematics by N. Bourbaki, whose opening pages consist of a detailed description of the formal system on which the treatise is thereafter based, a friend amused himself by writing explicitly the sequence of symbols representing the set of parts of a set without using a single abbreviation. The whole of it, transcribed onto a roll of paper, was three meters in length. Today's formal systems do not present all these ridiculous faults, because with the goal of manipulating them, people have conceived more efficient notations with computers. The techniques brought to bear on this occasion also serve to prove the validity of computer programs, and it is there that an entire active research domain exists where logic comes to the aid of computer science. Formalized infinitesimals Other progress in logic has astonished mathematicians. People despaired of justifying by a precise formalism the methods of calculus, which do in fact function well and which are based on the manipulation of the infinitesimal. Analysis in the eighteenth and nineteenth centuries had moreover rid mathematics of these worrisome infinites which appeared untamable. A. Robinson's formal system of non-standard Analysis, in the nineteen sixties, succeeded in the exploit of offering a regulated use of the infinitesimal. Anyone who wants to make use of her physicist intuition to make calculations now has available a formalism which will be able to validate her calculations, hitherto considered non-rigorous. The known logical results concerning non-standard Analysis are remarkable, and one of them indicates that any property not mentioning the infinitesimal proved in the formalized framework of non-standard Analysis can also be proved without use of the infinitesimal in the usual set theoretic formalism. This result justifies a posteriori the choice made historically by mathematicians to use infinitesimals, but, knowing it, it permits anyone now who finds them useful to give free rein to his taste. Heuristic proofs and "almost proofs" Along side of the notion of formal proof which permits everyone to arrive at agreement about what has really been proved, there exist a multitude of proofs which are non-formal and resolutely non-formalizable (at least with systems known today) which provide near-certainty. These so-called heuristic proofs are strange, because anyone who follows them winds up convinced of the truth of the propositions in question. Nonetheless, as in the case of the earlier proofs making use of the infinitesimal, mathematicians refuse to accept them. Even though they Page 6 sur 8

7 "make what is true visible," as long as no good formal systems are available (analogous to those of A. Robinson for non-standard Analysis) it will be impossible to transform them into formal proofs, which means quite simply that in the eyes of mathematicians they prove nothing. An example of a heuristic proof which is difficult to transform into a formal proof is indicated in figure 5. Almost, but not really A different example of an almost proved result is that of odd perfect numbers, a result which has resisted proof for 20 centuries. A perfect number is one which is equal to the sum of its own distinct divisors. The number 6, which is divisible by 1,2 and 3 is a perfect number, because 6 = It is the smallest perfect number, and the next three are 28, 496 and Do there exist any odd perfect numbers? The question was first posed more than two millennia ago. No one has ever found one, but no one has ever proved that none exists. Despite this lack of proof, everyone now believes that no odd perfect number exists, because a huge number of properties which such a number N would have to have if one exists have been (rigorously) proved. - An odd perfect number N, if one exists, has at least 8 different prime factors, and if it is a multiple of 3, at least 11 (P. Hagis, 1983) - An odd perfect number N, if one exists, is such that the sum of the inverses of its prime factors is between and (D. Suranarayan, P. Hagis, 1983) - The largest power of a prime number which divides an odd perfect number, if one exists, is larger than (G.L. Cohen, 1988) - An odd perfect number N, if one exists, is larger than (R. Brent, G.L. Cohen, H. te Riele, 1989) - One of the factors of an odd perfect number N, if one exists, is larger than 10 6 (P. Hagis, G.L.Cohen, 1998) another is larger than 10 4, and yet another is larger than 10 3 (1999, D. Iannucci) Note that certain of these results have just been proved within the past few months. It seems as if no number could possibly satisfy all these conditions at the same time (but that has not been rigorously established), and thus the belief that no odd perfect number exists is stronger than ever. This feeling and the idea that one is within a few millimeters of the final result do not constitute a formalizable mathematical proof, and thus, as is always the case, there is absolutely no controversy on this point: no matter how strong our feeling may be that no odd imperfect number exists, no one would claim that it has been proved. References Bourbaki N., Éléments de mathématique, Paris. Lakatos I. (1984) Preuves et réfutations. Paris : Éditions Hermann. Nelsen R. (1993) Proofs Without Words : Exercices in Visual Thinking, The Page 7 sur 8

8 Nelsen R. (1993) Proofs Without Words : Exercices in Visual Thinking, The Mathematical Association of America, 1993, ISBN Dieudonné J. (1987) Pour l'honneur de l'esprit humain, Hachette, Salanskis J.-M. (1999) Le constructivisme non standard, Presse universitaires du Septentrion, ISBN , (À propos de l'analyse non standard). Singh S. (1998) Le dernier théorème de Fermat, Éditions J.-C. Lattès, Translation Virginia Warfield Jean-Paul.Delahaye@lifl.fr First published in "Pour la Science" n 268, February 2000 Pour la Science 2000 Books by Jean-Paul Delahaye, Editions Belin/Pour la science : "Logique, informatique et paradoxes", 1995 "Le fascinant nombre Pi", 1997 "Jeux mathématiques et mathématiques des jeux", 1999 Back to the Newsletter Page 8 sur 8