THE FUTURE OF MATHEMATICS."

By HENRI POINCARÉ,

Member of the Académie des Sciences and the Académie Française, Professor at the Sorbonne.

(Translated by permission from Revue générale des Sciences pures et appliquées, Paris, 19th year, No. 23, December, 1908.)

The true method of forecasting the future of mathematics lies in the study of its history and its present state.

And have we not here, for us mathematicians, a task in some sort professional? We are accustomed to extrapolation, that process which serves to deduce the future from the past and the present and so well know its limitations that we run no risk of being deluded with its forecasts.

In the past there have been prophets incapable of seeing progress, those who have so willingly affirmed that all problems capable of solution have been solved and that nothing remains for future gleaning. Happily the example of the past reassures us. Often enough, already, it has been believed that all problems capable of solution have been solved or at least stated. Then the sense of the word solution becomes broadened and the insolvable problems become the most interesting of all and undreamed-of problems have arisen. To the Greeks a good solution must employ only the rule and compass; later it became that obtained by the extraction of roots; still later that obtained by the use of algebraic or logarithmic functions. These prophets of no advance thus always outflanked, always forced to retreat, have, I believe, been forced out of existence.

As they are dead I will not combat them. We know that mathematics still develops and our task is to find in what sense. Some one replies, "in every sense;" and in part that is true. But, if absolutely true, it would be somewhat startling. Our riches would soon

• Address delivered April 10, 1908, at the general session of the Fourth International Congress of Mathematicians (Rome, April 6-11, 1908); previously published in pamphlet form by and at the expense of the Mathematical Society of Palermo. M. Poincaré was unable to deliver this lecture and M. Darboux graciously undertook the task. To M. Guccia we express our gratitude for the authority which he has courteously extended for its reproduction.

become an incumbrance and their increase produce an accumulation as incomprehensible as all the unknown truth is to the ignorant.

The historian, the physicist himself, must make his selection from among the facts; the brain of the scholar-but a small corner of the universe could never contain this entire universe; so among the countless facts which nature presents, some must be passed by, others retained. It is as true, a fortiori, in mathematics for neither may the mathematician himself gather pellmell all the facts which come before him. Rather it is he-I was going to say his caprice-which creates them. It is he who constructs from the facts a new combination. Nature does not in general bring this to him ready-made.

Doubtless it happens sometimes that the mathematician approaches a problem set by the needs of physics, as when the physicist or the engineer asks of him the calculation of some number in view of an application. Shall we say, we mathematicians, that we must content ourselves to await these commands and, instead of cultivating our science for our pleasure, to have no other care than accommodating ourselves to the tastes of our clients? If there were no other objects for mathematicians than to come to the aid of those who are studying nature it would be from them then that we must await the word of command. Yet is this the right point of view? Certainly not; if we had not cultivated the exact sciences for themselves our mathematical machine would not have been created, and on the day when the word of command came from the physicist we would have been without arms.

Nor do the physicists, before studying some phenomenon, wait until some urgent need of life has made the study a necessity, and they are right; had the scientists of the eighteenth century neglected the study of electricity because in their eyes it was but a curiosity of no practical interest we would not have in the twentieth century either the telegraph, or electro-chemistry, or our electrical machinery. The physicist, when forced to choose, is not guided in his selection solely by utility. What brings about then his selection from among the facts of nature? We can not easily say. The phenomena which interest him are those which may lead to the discovery of some law. Those facts interest him which bear some analogy to many other phenomena, which do not appear as isolated facts but closely grouped with others. An isolated fact can be observed by all eyes; by those of the ordinary person as well as of the wise. But it is the true physicist alone who may see the bond which unites several facts among which the relationship is important though obscure. The story of Newton's apple is probably not true, but it is symbolical; so let us think of it as true. Well, we must believe that many before Newton had seen apples fall, but they made no deduction. Facts are sterile until there are minds capable of choosing between them and

discerning those which conceal something and recognizing that which is concealed; minds which under the bare fact see the soul of the fact.

That is exactly what we do in mathematics; out of the various elements at our disposal we could evolve millions of different combinations, but one of these combinations by itself alone is absolutely void of value. Oftentimes we take much trouble in its construction, but that serves absolutely for naught, unless possibly to give a task for further consideration. But it will be wholly different on the day that that combination takes its place in a class of like results and we have noted this analogy. We are no longer in the presence of a bare fact but of a law. And the true inventor is not the workman who has patiently built some few of these combinations, but he who has shown their relationships, their parentage. The former saw only the mere fact, the other alone felt the soul of the fact. Oftentimes for the indication of this parentage it has served the inventor's purpose to invent a new name and this name becomes creative; the history of science will supply us with innumerable such instances.

The celebrated Viennese philosopher, Mach, states the rôle of science to be the production of economy of thought just as a machine produces economy of labor. And that is very just. The savage counts with his fingers or with his assemblage of pebbles. By teaching the children the multiplication table we spare them later innumerable countings of pebbles. Someone, sometime, has discovered with his pebbles, or otherwise, that 6 times 7 makes 42; it occurred to him to note the fact and he thus spared us the necessity of doing it over again. He did not waste his time even though his calculation was only for his own pleasure; his operation cost him but two minutes; it would have cost two thousands of millions of minutes had a thousand of million of men to recompute it after he had.

The importance of a fact is known by its fruits, that is to say, by the amount of thought which it enables us to economize.

In physics, the facts of great fruitage are those which combine into some very general law, because they then allow us to predict a great number of other facts, and it is just the same with mathematics. I have devoted myself to a complicated calculation and have come laboriously to a result; but I will not feel repaid for my pains if I am not now able to foresee the results of other analogous calculations and to pursue such calculations with sure steps, avoiding the hesitations, the gropings of the first time. I shall not have wasted my time, on the contrary, if these gropings have ended in revealing to me in the problem which I have just treated some hidden relationship with a far more extended class of problems. If at the same time they have shown me resemblances and differences; if, in short, they have made me forsee the possibility of a gen

eralization, then it is not merely a new answer which I have acquired; it is a new force.

An example which comes at once to mind is the algebraic formula which gives us the solution of a class of numerical problems when its letters are replaced by numbers. Thanks to the formula, a single algebraic demonstration spares us the pains of going over the same ground time after time for each new calculation. But this gives us only a very rough illustration. Everyone knows that there are analogies, some most valuable, which can not be expressed by a formula.

If a new result has value it is when, by binding together longknown elements, until now scattered and appearing unrelated to each other, it suddenly brings order where there reigned apparent disorder. It then allows us to see at a glance the place which each one of these elements occupies in the ensemble. This new fact is not alone important in itself, but it brings value to all the older facts which it now binds together. The brain is as weak as the senses, and it would be lost in the complexities of the world were there not harmony in that complexity. After the manner of the shortsighted, we would see only detail after detail, losing sight of each detail before the examination of another, unable to bind them together. Those facts alone are worthy of our attention which bring order into this complexity and so render it comprehensible.

Mathematicians attach great importance to the elegance of their methods and results; nor is this pure dilettanteism. Indeed, what brings to us this feeling of elegance in a solution or demonstration? It is the harmony among the various parts, their happy balancing, their symmetry; it is, in short, all that puts order among them, all that brings unity to them and which consequently gives us a certain command over them, a comprehension at the same time both of the whole and of the parts. But as truly it is that which brings with it a further harvest, for, in fact, the more clearly we comprehend this assemblage, and at a glance, the better we will realize its relationships with neighboring groups, the greater consequently will be our chances of divining further possible generalizations. Elegance may arise from the feeling of surprise in the unexpected association of objects which we had not been accustomed to group together; it occurs frequently from the contrast between the simplicity of the means employed and the complexity of the given problem; we consequently reflect as to the reason of this contrast and almost without fail we find the cause not in pure hazard, but in some unexpected law. In a word, the sentiment of mathematical elegance is naught else than the satisfaction due to some, I know not just what, adaptation between the solution just found and the needs of our mind, and it is because of this adaptation itself that the solution becomes an

instrument to us. This æsthetic satisfaction is therefore connected with the economy of thought. Thus the caryatides of the Erechtheum engender in us the same feeling of elegance, for example, because they carry their heavy load with such grace, or we might say so cheerfully, that they produce in us a feeling of economy of effort.

It is for the same reason that when a somewhat long calculation has led us to a simple and striking result we are not fully satisfied until we have shown that we could have foreseen, if not the whole result, at least its most characteristic details. Why? What is it that prevents our satisfaction with this accomplished calculation giving all which we seemed to desire? It is because our long calculation would not again serve in another analogous case and because we have not used that mode of reasoning, often half intuitive, which would have allowed us to foresee our result. When our process is short we may see at a glance all its steps, so that we may easily change and adapt it to whatever problem of the same nature may occur, and then, since it allows us to foresee whether the solution of the problem will be simple, we can tell at least whether the problem is worth undertaking.

What we have just said suffices to show how vain would be any attempt whatever to replace by any mechanical process the free initiative of the mathematician. To obtain a result of real worth it will not suffice to grind it out or to have a machine for putting our facts in order. It is not alone order but the unexpected order which is of real worth. The machine may grind upon the mere fact, but the soul of the fact will always escape it.

Since the middle of the last century mathematicians have been more and more anxious for the attainment of absolute rigor in their processes; they are right, and that tendency will increase more and more. In mathematics rigor is not everything, but without it there would be nothing; a demonstration which is not rigorous is void. I believe no one will contest this truth. But to take this too literally would bring the conclusion, for example, that before 1820 there was no mathematics. That is surely going too far; then the geometricians assumed willingly what we explain by a prolix discussion. This does not mean that they did not realize their omission, but they passed it over too rapidly, and for greater surety they would have had to go through the trouble of giving this discussion.

But is it necessary to repeat every time this discussion? Those who, first in the field, had to be preoccupied with all this rigor have given us demonstrations which we could try to imitate; but if the demonstrations of the future must be built upon this model our mathematical treatises would become too long, and if I fear this length it is not only because I dread the incumbrance of our libraries, but