Men of Mathematics (106 page)

The last started a whole department of mathematics, the investigation of
periodic orbits:
given a system of planets, or of stars, say, with a complete specification of the initial positions and relative velocities of all members of the system at a stated epoch, it is required to determine under what conditions the system will return to its initial state at some later epoch, and hence continue to repeat the cycle of its motions indefinitely. For example, is the solar system of this recurrent type, or if not, would it be were it isolated and not subject to perturbations by external bodies? Needless to say the general problem has not yet been solved completely.

Much of Poincaré's work in his astronomical researches was qualitative rather than quantitative, as befitted an intuitionist, and this characteristic led him, as it had Riemann, to the study of analysis situs. On this he published six famous memoirs which revolutionized the subject as it existed in his day. The work on analysis situs in its turn was freely applied to the mathematics of astronomy.

We have already alluded to Poincaré's work on the problem of rotating fluid bodies—of obvious importance in cosmogony, one brand of which assumes that the planets were once sufficiently like such bodies to be treated as if they actually were without patent absurdity. Whether they were or not is of no importance for the mathematics of the situation, which is of interest in itself. A few extracts from Poincaré's own summary will indicate more clearly than any paraphrase the nature of what he mathematicized about in this difficult subject.

“Let us imagine a [rotating] fluid body contracting by cooling, but slowly enough to remain homogeneous and for the rotation to be the same in all its parts.

“At first, very approximately a sphere, the figure of this mass will become an ellipsoid of revolution which will flatten more and more, then, at a certain moment, it will be transformed into an ellipsoid with three unequal axes. Later, the figure will cease to be an ellipsoid and
will become pear-shaped until at last the mass, hollowing out more and more at its ‘waist,' will separate into two distinct and unequal bodies.

“The preceding hypothesis certainly can not be applied to the solar system. Some astronomers have thought that it might be true for certain double stars and that double stars of the type of Beta Lyrae might present transitional forms analogous to those we have spoken of.”

He then goes on to suggest an application to Saturn's rings, and he claims to have proved that the rings can be stable only if their density exceeds 1/16 that of Saturn. It may be remarked that these questions were not considered as fully settled as late as 1935. In particular a more drastic mathematical attack on poor old Saturn seemed to show that he had not been completely vanquished by the great mathematicians, including Clerk Maxwell, who have been firing away at him off and on for the past seventy years.

*  *  *

Once more we must leave the banquet having barely tasted anything and pass on to Poincaré's voluminous work in mathematical physics. Here his luck was not so good. To have cashed in on his magnificent talents he should have been born thirty years later or have lived twenty years longer. He had the misfortune to be in his prime just when physics had reached one of its recurrent periods of senility, and he was so thoroughly saturated with nineteenth century theories when physics began to recover its youth—after Planck, in 1900, and Einstein, in 1905, had performed the difficult and delicate operation of endowing the decrepit roué with its first pair of new glands—that he had barely time to digest the miracle before his death in 1912. All his mature life Poincaré seemed to absorb knowledge through his pores without a conscious effort. Like Cayley, he was not only a prolific creator but also a profoundly erudite scholar. His range was probably wider than ever Cayley's, for Cayley never professed to be able to understand everything that was going on in applied mathematics. This unique erudition may have been a disadvantage when it came to a question of living science as opposed to classical.

Everything that boiled up in the melting pots of physics was grasped instantly as it appeared by Poincaré and made the topic of several purely mathematical investigations. When wireless telegraphy was invented he seized on the new thing and worked out its mathematics.

While others were either ignoring Einstein's early work on the (special) theory of relativity or passing it by as a mere curiosity, Poincaré was already busy with its mathematics, and he was the first scientific man of high standing to tell the world what had arrived and urge it to watch Einstein as probably the most significant phenomenon of the new era which he foresaw but could not himself usher in. It was the same with Planck's early form of the quantum theory. Opinions differ, of course; but at this distance it is beginning to look as if mathematical physics did for Poincaré what Ceres did for Gauss; and although Poincaré accomplished enough in mathematical physics to make half a dozen great reputations, it was not the trade to which he had been born and science would have got more out of him if he had stuck to pure mathematics—his astronomical work was nothing else. But science got enough, and a man of Poincaré's genius is entitled to his hobbies.

*  *  *

We pass on now to the last phase of Poincaré's universality for which we have space: his interest in the rationale of mathematical creation. In 1902 and 1904 the Swiss mathematical periodical
L'Enseignement Mathématique
undertook an enquiry into the working habits of mathematicians. Questionnaires were issued to a number of mathematicians, of whom over a hundred replied. The answers to the questions and an analysis of general trends were published in final form in 1912.
^IV
Anyone wishing to look into the “psychology” of mathematicians will find much of interest in this unique work and many confirmations of the views at which Poincaré had arrived independently before he saw the results of the questionnaire. A few points of general interest may be noted before we quote from Poincaré.

The early interest in mathematics of those who were to become great mathematicians has been frequently exemplified in preceding chapters. To the question “At what period . . . and under what circumstances did mathematics seize you?” 93 replies to the first part were received:
35
said before the age of ten; 43 said eleven to fifteen; 11 said sixteen to eighteen;
3
said nineteen to twenty; and the lone laggard said twenty six.

Again, anyone with mathematical friends will have noticed that
some of them like to work early in the morning (I know one very distinguished mathematician who begins his day's work at the inhuman hour of five a.m.), while others do nothing till after dark. The replies on this point indicated a curious trend—possibly significant, although there are numerous exceptions: mathematicians of the northern races prefer to work at night, while the Latins favor the morning. Among night-workers prolonged concentration often brings on insomnia as they grow older and they change—reluctantly—to the morning. Felix Klein, who worked day and night as a young man, once indicated a possible way out of this difficulty. One of his American students complained that he could not sleep for thinking of his mathematics. “Can't sleep, eh?” Klein snorted. “What's chloral for?” However, this remedy is not to be recommended indiscriminately; it probably had something to do with Klein's own tragic breakdown.

Probably the most significant of the replies were those received on the topic of inspiration versus drudgery as the source of mathematical discoveries. The conclusion is that “Mathematical discoveries, small or great . . . are never born of spontaneous generation. They always presuppose a soil seeded with preliminary knowledge and well prepared by labor, both conscious and subconscious.”

Those who, like Thomas Alva Edison, have declared that genius is ninety nine per cent perspiration and only one per cent inspiration, are not contradicted by those who would reverse the figures. Both are right; one man remembers the drudgery while another forgets it all in the thrill of apparently sudden discovery but both, when they analyze their impressions, admit that without drudgery and a flash of “inspiration” discoveries are not made. If drudgery alone sufficed, how is it that many gluttons for hard work who seem to know everything about some branch of science, while excellent critics and commentators, never themselves make even a small discovery? On the other hand, those who believe in “inspiration” as the sole factor in discovery or invention—scientific or literary—may find it instructive to look at an early draft of any of Shelley's “completely spontaneous” poems (so far as these have been preserved and reproduced), or the successive versions of any of the greater novels that Balzac inflicted on his maddened printer.

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.