Men of Mathematics

by E. T. Bell

  Poincaré stated his views on mathematical discovery in an essay first published in 1908 and reproduced in his Science et Méthode. The genesis of mathematical discovery, he says, is a problem which should interest psychologists intensely, for it is the activity in which the human mind seems to borrow least from the external world, and by understanding the process of mathematical thinking we may hope to reach what is most essential in the human mind.

  How does it happen, Poincaré asks, that there are persons who do not understand mathematics? “This should surprise us, or rather it would surprise us if we were not so accustomed to it.” If mathematics is based only on the rules of logic, such as all normal minds accept, and which only a lunatic would deny (according to Poincaré), how is it that so many are mathematically impermeable? To which it may be answered that no exhaustive set of experiments substantiating mathematical incompetence as the normal human mode has yet been published. “And further,” he asks, “how is error possible in mathematics?” Ask Alexander Pope: “To err is human,” which is as unsatisfactory a solution as any other. The chemistry of the digestive system may have something to do with it, but Poincaré prefers a more subtle explanation—one which could not be tested by feeding the “vile body” hasheesh and alcohol.

  “The answer seems to me evident,” he declares. Logic has very little to do with discovery or invention, and memory plays tricks. Memory however is not so important as it might be. His own memory, he says without a blush, is bad: “Why then does it not desert me in a difficult piece of mathematical reasoning where most chess players [Whose “memories” he assumes to be excellent] would be lost? Evidently because it is guided by the general course of the reasoning. A mathematical proof is not a mere juxtaposition of syllogisms; it is syllogisms arranged in a certain order, and the order is more important than the elements themselves.” If he has the “intuition” of this order, memory is at a discount, for each syllogism will take its place automatically in the sequence.

  Mathematical creation however does not consist merely in making new combinations of things already known; “anyone could do that, but the combinations thus made would be infinite in number and most of them entirely devoid of interest. To create consists precisely in avoiding useless combinations and in making those which are useful and which constitute only a small minority. Invention is discernment, selection.” But has not all this been said thousands of times before? What artist does not know that selection—an intangible—is one of the secrets of success? We are exactly where we were before the investigation began.

  To conclude this part of Poincaré’s observations it may be pointed out that much of what he says is based on an assumption which may indeed be true but for which there is not a particle of scientific evidence. To put it bluntly he assumes that the majority of human beings are mathematical imbeciles. Granting him this, we need not even then accept his purely romantic theories. They belong to inspirational literature and not to science. Passing to something less controversial we shall now quote the famous passage in which Poincaré describes how one of his own greatest “inspirations” came to him. It is meant to substantiate his theory of mathematical creation. Whether it does or not may be left to the reader.

  He first points out that the technical terms need not be understood in order to follow his narrative: “What is of interest to the psychologist is not the theorem but the circumstances.

  “For fifteen days I struggled to prove that no functions analogous to those I have since called Fuchsianfunctions could exist; I was then very ignorant. Every day I sat down at my work table where I spent an hour or two; I tried a great number of combinations and arrived at no result. One evening, contrary to my custom, I took black coffee; I could not go to sleep; ideas swarmed up in clouds; I sensed them clashing until, to put it so, a pair would hook together to form a stable combination. By morning I had established the existence of a class of Fuchsian functions, those derived from the hypergeometric series. I had only to write up the results, which took me a few hours.

  “Next I wished to represent these functions by the quotient of two series; this idea was perfectly conscious and thought out; analogy with elliptic functions guided me. I asked myself what must be the properties of these series if they existed, and without difficulty I constructed the series which I called thetafuchsian.

  “I then left Caen, where I was living at the time, to participate in a geological trip sponsored by the School of Mines. The exigencies of travel made me forget my mathematical labors; reaching Coutances we took a bus for some excursion or another. The instant I put my foot on the step the idea came to me, apparently with nothing whatever in my previous thoughts having prepared me for it, that the transformations which I had used to define Fuchsian functions were identical with those of non-Euclidean geometry. I did not make the verification; I should not have had the time, because once in the bus I resumed an interrupted conversation; but I felt an instant and complete certainty. On returning to Caen I verified the result at my leisure to satisfy my conscience.

  “I then undertook the study of certain arithmetical questions without much apparent success and without suspecting that such matters could have the slightest connection with my previous studies. Disgusted at my lack of success, I went to spend a few days at the seaside and thought of something else. One day, while walking along the cliffs, the idea came to me, again with the same characteristics of brevity, suddenness, and immediate certainty, that the transformations of indefinite ternary quadratic forms were identical with those of non-Euclidean geometry.

  “On returning to Caen, I reflected on this result and deduced its consequences; the example of quadratic forms showed me that there were Fuchsian groups other than those corresponding to the hypergeometric series; I saw that I could apply to them the theory of thetafuchsian functions, and hence that there existed thetafuchsian functions other than those derived from the hypergeometric series, the only ones I had known up till then. Naturally I set myself the task of constructing all these functions. I conducted a systematic siege and, one after another, carried all the outworks; there was however one which still held out and whose fall would bring about that of the whole position. But all my efforts served only to make me better acquainted with the difficulty, which in itself was something. All this work was perfectly conscious.

  “At this point I left for Mont-Valérien, where I was to discharge my military service. I had therefore very different preoccupations. One day, while crossing the boulevard, the solution of the difficulty which had stopped me appeared to me all of a sudden. I did not seek to go into it immediately, and it was only after my service that I resumed the question. I had all the elements, and had only to assemble and order them. So I wrote out my definitive memoir at one stroke and with no difficulty.”

  * * *

  Many other examples of this sort of thing could be given from his own work, he says, and from that of other mathematicians as reported in L’Enseignement Mathématique. From his experiences he believes that this semblance of “sudden illumination [is] a manifest sign of previous long subconscious work,” and he proceeds to elaborate his theory of the subconscious mind and its part in mathematical creation. Conscious work is necessary as a sort of trigger to fire off the accumulated dynamite which the subconscious has been excreting—he does not put it so, but what he says amounts to the same. But what is gained in the way of rational explanation if, following Poincaré, we foist off on the “subconscious mind,” or the “subliminal self,” the very activities which it is our object to understand? Instead of endowing this mysterious agent with a hypothetical tact enabling it to discriminate between the “exceedingly numerous” possible combinations presented (how, Poincaré does not say) for its inspection, and calmly saying that the “subconscious” rejects all but the “useful” combinations because it has a feeling for symmetry and beauty, sounds suspiciously like solving the initial problem by giving it a more impressive name. Perhaps this is exactly what Poincaré intended, for h
e once defined mathematics as the art of giving the same name to different things; so here he may be rounding out the symmetry of his view by giving different names to the same thing. It seems strange that a man who could have been satisfied with such a “psychology” of mathematical invention was the complete skeptic in religious matters that Poincaré was. After Poincaré’s brilliant lapse into psychology skeptics may well despair of ever disbelieving anything.

  * * *

  During the first decade of the twentieth century Poincaré’s fame increased rapidly and he came to be looked upon, especially in France, as an oracle on all things mathematical. His pronouncements on all manner of questions, from politics to ethics, were usually direct and brief, and were accepted as final by the majority. As almost invariably happens after a great man’s extinction, Poincaré’s dazzling reputation during his lifetime passed through a period of partial eclipse in the decade following his death. But his intuition for what was likely to be of interest to a later generation is already justifying itself. To take but one instance of many, Poincaré was a vigorous opponent of the theory that all mathematics can be rewritten in terms of the most elementary notions of classical logic; something more than logic, he believed, makes mathematics what it is. Although he did not go quite so far as the current intuitionist school he seems to have believed, as that school does, that at least some mathematical notions precede logic, and if one is to be derived from the other it is logic which must come out of mathematics, not the other way about. Whether this is to be the ultimate creed remains to be seen, but at present it appears as if the theory which Poincaré assailed with all the irony at his command is not the final one, whatever may be its merits.

  Except for a distressing illness during his last four years Poincaré’s busy life was tranquil and happy. Honors were showered upon him by all the leading learned societies of the world, and in 1906, at the age of fifty two, he achieved the highest distinction possible to a French scientist, the Presidency of the Academy of Sciences. None of all this inflated his ego, for Poincaré was truly humble and unaffectedly simple. He knew of course that he was without a close rival in the days of his maturity, but he could also say without a trace of affectation that he knew nothing compared to what is to be known. He was happily married and had a son and three daughters in whom he took much pleasure, especially when they were children. His wife was a great-granddaughter of Étienne Geoffroy Saint-Hilaire, remembered as the antagonist of that pugnacious comparative anatomist Cuvier. One of Poincaré’s passions was symphonic music.

  At the International Mathematical Congress of 1908, held at Rome, Poincaré was prevented by illness from reading his stimulating (if premature) address on The Future of Mathematical Physics. His trouble was hypertrophy of the prostate, which was relieved by the Italian surgeons, and it was thought that he was permanently cured. On his return to Paris he resumed his work as energetically as ever. But in 1911 he began to have presentiments that he might not live long, and on December 9 wrote asking the editor of a mathematical journal whether he would accept an unfinished memoir—contrary to the usual custom—on a problem which Poincaré considered of the highest importance: “. . . at my age, I may not be able to solve it, and the results obtained, susceptible of putting researchers on a new and unexpected path, seem to me too full of promise, in spite of the deceptions they have caused me, that I should resign myself to sacrificing them . . .” He had spent the better part of two fruitless years trying to overcome his difficulties.

  A proof of the theorem which he conjectured would have enabled him to make a striking advance in the problem of three bodies; in particular it would have permitted him to prove the existence of an infinity of periodic solutions in cases more general than those hitherto considered. The desired proof was given shortly after the publication of Poincaré’s “unfinished symphony” by a young American mathematician, George David Birkhoff (1884-).

  In the spring of 1912 Poincaré fell ill again and underwent a second operation on July 9. The operation was successful, but on July 17 he died very suddenly from an embolism while dressing. He was in the fifty ninth year of his age and at the height of his powers—“the living brain of the rational sciences,” in the words of Painlevé.

  * * *

  I. This famous question of the “piriform body,” of considerable importance in cosmogony, was apparently settled in 1905 by Liapounoff, whose conclusion was confirmed in 1915 by Sir James Jeans: they found that the motion is unstable. Few have had the courage to check the calculations. After 1915 Leon Lichtenstein, a fellow-countryman of Liapounoff, made a general attack on the problem of rotating fluid masses. The problem seems to be unlucky; both L’s had violent deaths.

  II. “There is no royal road to Geometry,” as Menaechmus is said to have told Alexander the Great when the latter wished to conquer geometry in a hurry.

  III. Poincaré called some of his functions “Fuchsian,” after the German mathematician Lazarus Fuchs (1833-1902) one of the creators of the modern theory of differential equations, for reasons that need not be gone into here. Others he called “Kleinian” after Felix Klein—in ironic acknowledgment of disputed priority.

  IV. Enquête de “L’Enseignement Mathématique” sur la méthode de travail des mathématiciens. Available either in the periodical or in book form (8 + 137 pp.) by Gauthier-Villars, Paris.

  CHAPTER TWENTY NINE

  Paradise Lost?

  CANTOR

  Mathematics, like all other subjects, has now to take its turn under the microscope and reveal to the world any weaknesses there may be in its foundations.

  —F. W. WESTAWAY

  THE CONTROVERSIAL TOPIC of Mengenlehre (theory of sets, or classes, particularly of infinite sets) created in 1874-1895 by Georg Cantor (1845-1918) may well be taken, out of its chronological order, as the conclusion of the whole story. This topic typifies for mathematics the general collapse of those principles which the prescient seers of the nineteenth century, foreseeing everything but the grand débâcle, believed to be fundamentally sound in all things from physical science to democratic government.

  If “collapse” is perhaps too strong to describe the transition the world is doing its best to enjoy, it is nevertheless true that the evolution of scientific ideas is now proceeding so vertiginously that evolution is barely distinguishable from revolution.

  Without the errors of the past as a deep-seated focus of disturbance the present upheaval in physical science would perhaps not have happened; but to credit our predecessors with all the inspiration for what our own generation is doing, is to give them more than their due. This point is worth a moment’s consideration, as some may be tempted to say that the corresponding “revolution” in mathematical thinking, whose beginnings are now plainly apparent, is merely an echo of Zeno and other doubters of ancient Greece.

  The difficulties of Pythagoras over the square root of 2 and the paradoxes of Zeno on continuity (or “infinite divisibility”) are—so far as we know—the origins of our present mathematical schism. Mathematicians today who pay any attention to the philosophy (or foundations) of their subject are split into at least two factions, apparently beyond present hope of reconciliation, over the validity of the reasoning used in mathematical analysis, and this disagreement can be traced back through the centuries to the Middle Ages and thence to ancient Greece. All sides have had their representatives in all ages of mathematical thought, whether that thought was disguised in provocative paradoxes, as with Zeno, or in logical subtleties, as with some of the most exasperating logicians of the Middle Ages. The root of these differences is commonly accepted by mathematicians as being a matter of temperament: any attempt to convert an analyst like Weierstrass to the skepticism of a doubter like Kronecker is bound to be as futile as trying to convert a Christian fundamentalist to rabid atheism.

  A few dated quotations from leaders in the dispute may serve as a stimulant—or sedative, according to taste—for our enthusiasm over the singular intellectual career of
Georg Cantor, whose “positive theory of the infinite” precipitated, in our own generation, the fiercest frog-mouse battle (as Einstein once called it) in history over the validity of traditional mathematical reasoning.

  In 1831 Gauss expressed his “horror of the actual infinite” as follows. “I protest against the use of infinite magnitude as something completed, which is never permissible in mathematics. Infinity is merely a way of speaking, the true meaning being a limit which certain ratios approach indefinitely close, while others are permitted to increase without restriction.”

  Thus, if x denotes a real number, the fraction 1/x diminishes as x increases, and we can find a value of x such that 1/x differs from zero by any preassigned amount (other than zero) which may be as small as we please, and as x continues to increase, the difference remains less than this preassigned amount; the limit of 1/x, “as x tends to infinity,” is zero. The symbol of infinity is ∞; the assertion 1/∞ = 0 is nonsensical for two reasons: “division by infinity” is an operation which is undefined, and hence has no meaning; the second reason was stated by Gauss. Similarly 1/0 = ∞ is meaningless.

  Cantor agrees and disagrees with Gauss. Writing in 1886 on the problem of the actual (what Gauss called completed) infinite, Cantor says that “in spite of the essential difference between the concepts of the potential and the actual ’infinite/ the former meaning a variable finite magnitude increasing beyond all finite limits (like x in l/x above), while the latter is a fixed, constant magnitude lying beyond all finite magnitudes, it happens only too often that they are confused.”