Popular Science Monthly/Volume 82/March 1913/Henri Poincare as an Investigator

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MARCH, 1913



IT has not seemed to me appropriate, nor would there be time, nor should I be able, to enter into an exhaustive study of the life-work of a master-mind like Jules Henri Poincaré. Indeed, to analyze his contributions to astronomy needs a Darwin; to report on his investigations in mathematical physics needs a Planck; to expound his philosophy of science needs a Royce; to exhibit his mathematical creations in all their fullness needs Poincaré. Let it suffice that he was the pride of France, not only of the aristocracy of scholars, but of the nation. He was inspired by the genius of France, with its keen discernment, its eternal search for exact truth, its haunting love of beauty. The mathematical world has lost its incomparable leader, and its admiration for the magnitude of his achievements will be tempered only by the vain desire to know what visions he had not yet given expression to. Investigators of brilliant power for years to come will fill out the outlines of what he had time only to sketch. His vision penetrated the universe from the electron to the galaxy, from instants of time to the sweep of space, from the fundamentals of thought to its most delicate propositions.

In his funeral oration, Painlevée, speaking for the Académie des Sciences, said:[2]

He was only twenty-four years of age, when after four years of silent and sustained reflection, he began the series of mathematical publications which leaves us in doubt whether to admire most its surprising profundity or its surprising fecundity.

Whether he attacked the ascension, step by step, of the truths of arithmetic discontinuity, or unloosed the tangle of geometric form, or followed the subtlest windings and caprices of the continuous laws that join quantities together, there is not one of his works which has not the masterly touch, not one of his fifteen hundred publications which does not show the lion's claw. At the age of twenty-seven, the Faculty of Sciences offered this young conqueror its chair of physical mechanics. At thirty-three the Academy of Sciences opened its door, an example soon followed by the learned academies of the entire world; for there was no body of scientists in Europe or America which did not feel that it honored itself in adjoining the cooperation of Henri Poincaré. But the mathematical sciences were for this illustrious analyst only a manifold and prodigious measuring instrument admirably adapted to the comparative study of the phenomena of the universe. This instrument he set himself to use, and what skill he displayed! At the age of thirty, he astonished the physicists by his critique of the general principles of their science; that was but the beginning of bold speculations which led him year by year up to the very edge of the unknown, to the constitution of matter, to the paradoxical mechanics that sprung up after the unexpected discovery of the mysterious radioactivity. Yet this was only part of his activity: geodesy, cosmogony, astronomy, philosophy of science, he included them all, penetrated all, explored all. His celestial mechanics would be glory enough. It was this that revealed him first to a wide public. King Oscar II. of Sweden, Mæcenus of science, enlightened and generous, in 1887 opened an international competition in mathematics. In 1889, at the end of the contest, France learned with joy that the great gold medal, supreme prize of this new tournament, had been awarded to one of her children, a young scientist thirty-five years of age, for a marvelous study of the mechanical stability of our universe; and the name of Henri Poincaré was famous. Gentlemen, the Theban hero dying after two victories said: "I leave two immortal daughters." This hero of the world of thought who has just succumbed, leaves in the ideal world, as real as the other world, an immortal posterity which will guide the future researches of men. Indeed his life will remain an example as harmonious in its faultless lines as the orbits of those stars whose eternal past and eternal future he desired to know.

To this eulogy of Professor Painlevé certainly I could add nothing, and it does not seem necessary to enumerate the many other honors of Poincaré's. I shall undertake only to consider briefly his conception of science in its chief phases, and in the light of this conception to consider at more length in particular his ideas of research. As an investigator his opinions carry extraordinary weight, as he was a subtle philosopher and a skilled psychologist. We may treat three phases of scientific activity as distinct, pure science, industrial science and what we may call euthenic science.

In speaking of the death of Brouardel,[3] who did much for the study of hygiene, and had helped in preventing three invasions of cholera, without disturbing commerce, Poincare said before the Académie des Sciences:

In this direction scientists can scarcely count on the satisfaction of discovering general laws, exterior as it were to space and time, but there are other joys and above all that of doing good immediately to humanity and correcting evils without forcing the remedy to wait.

The scientist is accustomed to conquer truth only by degrees; for him all certainty should be bought by long hesitations, by perpetually feeling his way. He suspects what comes too easily, and accepts it only after submitting it to numerous and diverse proofs. The man who must act can not embarrass himself by such scruples. He cares little for a truth which must wait so long, because it may arrive too late, and after the moment for action has passed. He must make rapid conquests; sometimes these are not the most durable nor those we should esteem. He also has to avoid reefs which we know not, we for whom time does not count, and sometimes we are tempted to say a true scientist ought not to risk them; how much better on the contrary to congratulate ourselves that there are men skilful enough to avoid them.

Towards pure science his attitude was almost adoration. It is best set forth by extracts from his "Value of Science" and "Science and Method":

The search for truth should be the goal of our activity; it is the only end worthy of it. . . . When I speak here of truth doubtless I mean primarily scientific truth, but I wish to speak also of moral truth, one of whose aspects is what we call Justice. . . . To find one as well as to find the other, it is necessary to struggle to the utmost to free ourselves from the bonds of prejudice and passion, to attain absolute sincerity.

The best expression of the harmony of nature is Law. Law is one of the most recent conquests of the human mind. Man demands that his gods prove their existence by miracles, but the eternal marvel is that there are not miracles all the time. And the world is divine because it is harmonious. Were it ruled by caprices what could ever prove it due to aught but chance?

But does this harmony which the human intellect believes it finds in nature exist outside the intellect? Doubtless not; a reality completely independent of the mind that conceives it, sees it, feels it, is an impossibility. What we call objective reality is, in the last analysis, what is common to many thinkers and could be common to all; this common part, we shall see, can be only the harmony expressed by mathematical laws.

So we conclude that this harmony is the sole objective reality, the sole truth we can ever attain, and if I add that the universal harmony of the world is the source of all beauty, it becomes comprehensible how we should prize the slow and painful progress by which we learn little by little to know it.

The scientist does not study nature because it is useful; he studies it because it pleases him, and it pleases him because it is beautiful. Were nature not beautiful, it would not be worth knowing, life would not be worth living. I do not mean here, of course, that beauty which impresses the senses, the beauty of qualities and appearances; not that I despise it—far from it; but that has nought to do with science; I mean that subtler beauty of the harmonious order of the parts which pure intellect appreciates. This it is which gives a body, a skeleton as it were, to the fleeting appearances that charm the senses, and without this support the beauty of these fugitive dreams would be but imperfect, because it would be unstable and evanescent. On the contrary intellectual beauty is self-sufficient and for its sake, rather than for the good of humanity, does the scientist condemn himself to long and tedious labors.

In connection with this view of the scientist in his own domain, I desire to quote also from the preface of the recent second German edition of "Value of Science," which expresses his attitude towards industrial science:

Science has always had to contend with skeptics and scoffers who were quite ready to draw conclusions from relative failures and temporary inactivity, and to note the confessions of scientists who admit that the field of science is bounded, but fail to add that inside its own realm it is supreme.

He who views scientific work from the outside is often amazed to see yesterday's truth so easily become to-morrow's error. He believes then, that our conquests are over-confident, that the principles so proudly paraded are only novelties, and he does not see that beneath these necessary changes of form scientific truth is always one and the same. It remains eternally unchanged and only the clothing in which we deck it out changes with the fashion.

Fortunately science is needed in applications, and this silences the skeptic. If he desires to use some new discovery, and convinces himself of its success, he must indeed admit that it is more than an idle dream. We thus perceive the blessing which lies in the development of industry.

I do not wish to say that science is created for its applications, far from it; one must love it for its own sake; but the recognition of its applications protects us from the skeptic.

Poincaré's conception of science can be summed up in these terms: Science consists of the invariants of human thought.

In the field of investigation, the important thing for Poincaré was the discovery of the real relation between isolated facts. The important facts are those that suggest relations. We select facts from this standpoint. The world of relations was as real to him as the world of phenomena, and so far as we know the real relations, in whatever language we express these relations, just so far we know the actual world, the objective world. Even absolute space and absolute time do not exist, these two are relations furnished by our own minds.[4] Thus the term energy, and our notion as to the existence of energy, may change in the course of time, but the persistent relation that gives us our present notion of energy is real and does not change. It may be true, as Herschel said, that in the twinkling of an eye a molecule solves a differential equation which if written out in full would belt the globe, but the molecule knows nothing of the equation—that is created by the mind, and as the modern discontinuous physics develops, it may be that we shall have to use difference equations rather than differential equations. But the differential equation expresses certain persistent relations between phenomena, and is thus real, and is the replica of an objective reality. The differential equation means that the phenomenon is one such that each state is the result of the immediately preceding state; the new integro-differential equation of Volterra means that the state is due to all the preceding states; the difference equation means that the states follow each other abruptly; and integro-difference equations would mean that they depend on all preceding states discontinuously. Each is able to account for certain relations in the states. In the same sense the word atom is the name for a set of relations, and though it may change and the atom itself become a solar system, yet what we really mean by the atom is permanent and represents an objective reality. We are witnesses too of an evolution in science and mathematics from the continuous to the discontinuous. In mathematics it has produced the function defined over a range rather than a line—a chaos, as it were, of elements—and the calculable numbers of Borel. In physics it has produced the electron, the magneton, and the theory of quanta,[5] about which Poincare said shortly before his death:

A physical system is capable of only a finite number of distinct states; it abruptly jumps from one state to another without passing through the intermediate states.

In biology we have the corresponding theory of mutations. Yet despite this apparent reduction of old ideas into dust, contradictory to our hopes of its permanence; as Poincaré put it: this is right and the other is not wrong. They are in harmony, only the language varies; both set forth certain true relations.

Just as Maxwell and Kelvin were able to invent mechanical models of the ether, so Poincaré is perhaps the most profound genius the world has ever known in devising the more subtle machinery of thought to represent the relations he found not only between numbers and geometric figures, but between the phenomena of physics. His mind seemed to create new structures of this kind continually, finding expression for the most intricate relations. Nowadays this is the same as saying that he was a mathematician, for this ideal world of relations is the one with whose structure mathematics is concerned, and which mathematics claims sovereignty over, verifying Gauss's assertion: "Mathematics is Queen of all the Sciences."

In the address of Masson when Poincaré was made one of the forty immortals, he said:

You were born, you have lived, you will live, and you will die a mathematician; the vital function of your brain is to invent and to resolve more cases in mathematics; everything about you relates to that. Even when you seem to desert mathematics for metaphysics, the former furnishes the examples, the reasoning, the paradoxes. It is in you, possesses you, harries you, dominates you; in repose, your brain automatically pursues its work, without your being aware of it—the fruit forms, grows, ripens, and falls, and you have yourself told us of your wonder at finding it in your hand so perfect. You furnish an admirable example of the mathematical type. Since Archimedes it is classical but legendary. Rarely will historian have found an occasion so fit to note in life its external characters, and in place of relating your works, rather is not this the occasion to see how mathematical genius manifests itself, whether it is the result of atavism, or the product of a special culture, at what moment and under what conditions it sees light, at what epoch of life it is most active and brilliant?

Fortunately the answer to Masson's question is to be found in Poincare's own writings, and it becomes the more interesting when taken in connection with his further thesis that the method of research in mathematics is precisely that of all pure science. This method I desire to consider at some length, for I conceive that such a consideration will be entirely appropriate in this place.

The first research mentioned by Rados in the report of the committee to the Hungarian Academy in 1905, when Poincaré was awarded the first Bolyai prize as the most eminent mathematician in the world, is the series of investigations relating to automorphic functions. These functions enable us to integrate linear differential equations with rational algebraic coefficients, just as elliptic functions and abelian functions enable us to integrate certain algebraic differentials. With regard to these researches, Poincaré tells us that for a fortnight he had tried without success to demonstrate their non-existence. He investigated a large number of formulæ with no results. One evening, however, he was restless and got to sleep with difficulty; ideas surged out in crowds and seemed to crash violently together in the endeavor to form stable combinations. The next morning he was in possession of the particular set of automorphic functions derived from the hypergeometric series; he had only to verify the calculations. Having thus found that functions did exist of this kind, he conceived the idea of representing these functions as the quotients of two series, analogous to the theta series in elliptic functions. This he did purely by the analogy, and arrived at theta-fuchsian functions. Having occasion to take a journey, mathematics was laid aside for a time, but in stepping into an omnibus at Coutances, the idea flashed over him that the transformations which he had used to define these automorphic functions were identical with certain others he had used in some researches in non-euclidean geometry. Returning home he took up some questions in the theory of arithmetic forms, and with no suspicion that they were related to the fuchsian functions or the geometric transformation, he worked for some time with no success. But one day while taking a walk, the idea suddenly came to him that the arithmetic transformations he was using were essentially the same as those of his study in non-euclidean geometry. From this fact he saw at once by the connections with the arithmetic forms that the fuchsian functions he had discovered were only particular cases of a more general class of functions. He laid siege now systematically to the whole problem of the linear differential equations and the fuchsian functions and reached result after result, save one thing which seemed to be the key stone of the whole problem was stubborn. He was compelled to go away again to perform military duty, and his mind was full of other things. But one day while crossing the boulevard the solution of the last difficulty suddenly appeared and upon verification was found to be correct.

In this account of the birth and growth of mathematical development, which he assures us is practically the same as for all such developments, it is obvious that the central notion is that of generalization. Elliptic, abelian and theta functions are in turn generalized into a new class of transcendents. Inversion of differentials is generalized into inversion of differential equations. This notion of generalization we need to inspect a little closely. Mathematical generalization consists of two types of thought, often not discriminated, and often scarcely to be discriminated from each other. One type consists in so stating a known theorem that it will be true of a wider class than in its first statement, and the predicate asserted has a wider significance. In such generalization the first statement of the theorem becomes a mere particular case of the second statement. Examples will occur readily to every one. There are two forms of this type: in one, many known cases are brought together under one law; in the other form, the law thus found is made to apply to other known cases, perhaps never before suspected to be related to the first set. It is the guiding threads of analogy that usually bring about these forms of generalization. This kind of generalizing power Poincare had in high degree. In his memoir on "Partial Differential Equations of Physics "[6] he says:

If one looks at the different problems of the integral calculus which arise naturally when he wishes to go deep into the different parts of physics, it is impossible not to be struck by the analogies existing. Whether it be electrostatics, or electrodynamics, the propagation of heat, optics, elasticity or hydrodynamics, we are led always to differential equations of the same family; and the boundary conditions though different, are not without some resemblances. . . . One should therefore expect to find in these problems a large number of common properties.

Also in his "Nouvelles Méthodes de la Mécanique Céleste" he says:

The ultimate aim of celestial mechanics is to solve the great question whether Newton's law alone will explain all astronomical phenomena.

In his address awarding Poincare the gold medal of the Royal Astronomical Society, G. H. Darwin[7] said:

The leading characteristic of M. Poincare 's work appears to be the immense wideness of the generalizations, so that the abundance of possible illustrations is sometimes almost bewildering. This power of grasping abstract principles is the mark of the intellect of the true mathematician; but to one accustomed rather to deal with the concrete the difficulty of completely mastering the argument is sometimes great.

In the account of the creation of the fuchsian functions we see this power of finding examples of his generalizations, that is to say, of applying them. By these functions he could solve differential equations, he could express the coordinates of algebraic curves as fuchsian functions of a parameter, he could solve algebraic equations of any order. Humbert put it succinctly thus: "Poincaré handed us the keys of the world of algebra." Again, from the simplification of the theory of algebraic curves he was able to reach results which led to a generalization of the fuchsian functions to the zetafuchsian functions, which he afterward used in differential equations, the starting point of the problem. He applied the theory of continuous groups to hypercomplex numbers and then applied hypercomplex numbers to the periods of abelian integrals and the algebraic integration of differential equations of certain types. He applied fuchsian functions to the theory of arithmetic forms and opened a wide development of the theory of numbers. He applied fundamental functions to the potential theory of surfaces in general, showing how the Green's function could be constructed for any surface, permitting the solution of the problem. He developed' integral invariants, which persist through cycles of space and time. He dared to apply the kinetic theory of gases and the theory of radiant matter to the Milky Way itself, suggesting that probably we are a speck in a spiral nebula. He analyzed mathematically the rings of Saturn into a swarm of satellites, and the spectroscope confirmed his conclusions, a piece of work ranking with the mathematical discovery of Neptune. He found a generalization for figures of equilibrium of the heavenly bodies, discovering an infinity of forms and pointing out the stable transition shapes from one type to another, of which the piriform was quite new; at the same time throwing light on the problems of cosmogony. He applied trigonometric series, divergent series, and even the theory of probabilities, to show that the stability or instability of our universe has never been demonstrated, but that if probability is measured by continuous functions only, the universe is most probably stable.

There is no essential difference between generalizations of this type in whatever realm they appear. It is generalization to see that projective geometry merely states the invariancies of the projective group, and elementary geometry is a collection of statements about the invariants of the orthogonal group. Expansions in sines and cosines, or Legendrian polynomials, or Bessel functions are particular cases of expansions in fundamental functions, and these arise from the inversion of definite integrals. It is also generalization to reduce the phenomena of light to a wave-theory, then the phenomena of light, electricity and magnetism to ether-properties. It is generalization to reduce physics and physical chemistry to the study of quanta of energy, and, I might add, to reduce all the physical sciences to a study of the kinematics of four-dimensional space. When we say natural law, we mean generalization of this type. The laws of science are generalizations of the relations between phenomena. According to Poincaré there are three classes of hypotheses in science: (1) Natural hypotheses, which are the foundations of the mathematical treatments, such as action decreases with the distance, small movements follow a linear law, effect is a continuous function of the cause, physical phenomena are discontinuous functions; (2) Neutral hypotheses, which enable us to formulate our ideas, and are neither verifiable nor unverifiable, such as the hypothesis of atoms or of a continuous medium; (3) Generalizations, invariantive relationships, which are valuable, may be verified by experiment and lead to real progress. In "Science and Hypothesis" his thesis is, that science consists of observed facts organized according to these three classes of hypotheses. In "Value of Science" the thesis is, that the objective value of science consists in the laws, that is, in the generalizations, discovered. In "Science and Method" the thesis is, that the discovery of laws is by methods substantially the same as those of mathematical investigation, deducing from a significant particular a wide-reaching generalization, selecting our facts because of their significance.

This type of generalization, however, is only a part of the mathematical generalization. It might in broad terms be characterized as the purely scientific type. The second type, which might be broadly characterized as the purely mathematical type, is that in which there is a distinct widening of the field of a conception, usually by the addition of new mathematical entities. Examples are the irrational numbers, negative numbers, imaginary numbers, quaternions and hypercomplex numbers in general. The name imaginary indicates the fact that the actual existence of these was once open to question in the minds of some. Other examples are the non-euclidean geometries, the non-archimedean continuity, transfinite numbers, space of four and of N dimensions. The ideal numbers of Kummer and the geometric numbers of Minkowski are generalizations mainly of this type. It is not possible to separate sharply this kind of generalization from the other, and it would often be difficult to say whether a given mathematical investigation belongs primarily to the one kind or the other. However, when an investigation does not merely utilize material that is already known, but introduces new objects for study whose properties are not known, we can classify it under the second type. Usually the second type arises from inversion processes. We have given certain properties to find the class of things satisfying them. If they do not exist we create them. Whether we consider that the new objects have (in mathematics) been created or discovered, is merely a matter of psychologic point of view. For example, in one of Poincaré's last papers[8] he explains the apparently irreconcilable difference of opinion which there is among mathematicians regarding the existence of a definable infinity as due to the difference in the psychology of the two classes. One, the idealistic, feeling that everything we define is due to the mind, and finite; the other, the realistic, feeling that there is an external world which may well contain an infinity. The idealistic class, to which Poincaré belonged, would consider that these extensions to which we referred are in a sense creations.

It is scarcely necessary to enumerate the creations of Poincaré. They are many, for he was gifted with extraordinary originality. The account given above of the creation of the fuchsian functions is an example of one of his most important. It opened an immense field of investigation. He created a type of arithmetic invariants expressible as series or definite integrals, which opened a new field in theory of numbers. His investigations of ordinary differential equations which are not linear, such as those in dynamics and the problem of N bodies, created an extensive class of new functions which (I believe) are yet without special names, as well as suggesting the existence of classes of functions for which we have, as yet, even no means of expression. The investigations of asymptotic expansions opened paths to dizzy heights. Fundamental functions in partial differential equations also open a region now under development. We may say that the most marvelous of his creations rise from the general field of differential equations. We might cite further his researches in analysis situs, the realm of the invariants of a battered continuity. His double residues and studies in functions of many real variables are creations from which will spring a noble progeny. Even the lectures in which he presented the results of others scintillate with original thoughts.

To generalize in mathematics and science it is not enough simply to get together facts or ideas and to put them into new combinations. Most of these combinations would be useless. The real investigator does not form the useless combinations at all, but unconsciously rejects the unprofitable combinations. It is as if he were an examiner for a higher degree; only the candidates who have passed the lower degrees ever appear before him at all. Often domains far distant furnish the useful combinations, as in the account given of the genesis of the fuchsian functions, the theory of arithmetic forms through the roundabout route of non-euclidean geometry furnished the generalization of the first fuchsian functions to the complete class. This was of the first type. But how are those of the second type born?

We come thus to the heart of the matter. Merely to say that we discover laws is not sufficient. How do we discover extensions? How devise new formulas? Make new constructions? The answer to this question is, for Poincaré, found in psychology. It is necessary to get together many facts, but this does not insure that we shall build with them any more than that a collection of beams and stones will make a cathedral. Mere haphazard construction does not produce the cathedral either. To reach the end it is necessary to have the end in view from the beginning. It is not only necessary to choose a route, but we must see that it is the route to be chosen. This implies a power of the mind which Poincaré calls intuition. It is that power which enables us to perceive the plan of the whole, to seize the unity in the matter at hand. This power is necessary not only to the investigator, but it is also necessary—in less degree, perhaps—to him who desires to follow the investigation. Why is it, he asks, that any one can ever fail to understand mathematics? Here is a subject constructed step by step with infallible logic, yet many do not comprehend it at all. Not on account of poor memory—that may lead to errors in calculation, but has little to do with comprehension of the subject. Sylvester, for example, was notorious for his inability to remember even what he himself had proved. It is not due to lack of the power of attention, for while concentration is necessary in the development of a demonstration, or in following a piece of logic, it does riot give this appreciation of mathematics. A mathematical demonstration is a series of inferences, but it is above all a series of inferences in a certain order. The important thing is the order, just as in chess the mere moving according to the rules is not enough, it is the plan of the game that counts. If one appreciates it, this order, this plan, this unity, this harmony, he need have no fear of a poor memory, nor need he weary his concentration. The student deficient in this power may learn demonstrations by heart, he may assent to each step as logically proved, yet he will know little of the theorem itself. Those who possess this kind of insight which reveals hidden relations, this divining power for the discovery of mines of gold, may hope to become investigators, creators. Those who do not have it must find it or give up the task. The great educational question of the day is the problem of the development of the intuition. If we learn to cultivate this spirituelle flower it will open all doors of invention and discovery of Jaws. It is an interesting problem for even the grade teacher. If it be true, as Boris Sidis and others have claimed, that there are superior methods of education (which seem really to lie along this line) then they must become the methods of future education. We will begin to educate for genius. One thing seems evident, that too prolonged adherence to the methods of rigid reasoning leads to sterility. In mathematics at least both logic and intuition are indispensable, one furnishes the architect's plan of the structure, the other bolts it and cements it together. Logic, says Poincaré, is the sole instrument of certitude, intuition of creation. Yet even the steps of a logical deduction are planned in their entirety by the intuition. In discussing the partial differential equations of physics[9] and their solutions, he points out that one often has to content himself with the guidance he can get by physical considerations. An example of this was the use made by Klein of electrical considerations in handling Dirichlet's problem on a Riemann surface. In the physical aspect of the problem this would usually be sufficient, for the physical data are at the best only approximate. The mathematical necessities of convergence demand, however, that the problems be handled purely analytically and deductively. In one of his lectures he compares the process with the formation of a sponge. When we find it fully formed it is only a delicate lace-work of needles of silica. But the really interesting thing is the form it has taken, and this can be fully understood only by knowing the life-history of the sponge which has impressed its form, its will, so to speak, on the silica. In the same way a logical development of a theorem can really be understood only through a study of its living development. Need we point out the significance of this to the research student? Just as a painter who would become great must sit at the feet of a master and see his creations grow on the canvas, so the student does well to watch a master at work on scientific creations. This is the good he gets at the university. No compendium of results of the great creators will suffice. Nor is a too detailed study of the history of a problem, or too extensive a list of its bibliography, of assistance to the intuition. These might assist the later logical development, but not the inventive power. Poincaré rarely did more than to acquaint himself with the present status of a problem he desired to consider. It is evident too that the intuition is sui generis, and guidance of it in the seminar must simply stimulate, not undertake to determine its form. The investigator must set his own problem and work it out in his own way. The director of research should furnish favorable surroundings and set forth the matter of his lectures in as genetic form as possible, as for example, Poincaré's and Klein's masterly courses. But he should not prescribe forms of development, nor methods of attack for the novitiate.

The types of intuition are numerous. We leave to the psychologist their enumeration and description. For example, we should expect a visualist to think in pictures, for in this direction his imagination would be vivid. Such a mind would make use of diagrams and mechanical forms to embody his ideas. We think at once of Faraday and his lines of force, of Kelvin and his models of the ether. Poincaré compares Bertrand and Hermite, schoolmates educated at the same time in the same way. Bertrand when speaking was always in motion, apparently trying to paint his ideas. Hermite seemed to flee the world, his ideas were not of the visible kind. Weierstrass thought in artificial symbols, Riemann in pictures and geometric constructions. Poincare is spoken of as belonging to the audile type, for he remembered sounds well. He seems from his memoirs and papers, however, also to be equally of the visual and the symbolic types. He valued words highly, and his style is a mountain brook descending from rarefied heights, its clear current here falling over rocks, there gliding smoothly down. His thought is a penetrating ray that illuminates the deepest recesses of the wilderness of phenomena.

But in any case, whether one be analyst, physicist, biologist or psychologist, the characteristic trait of the intuition is the direct appreciation of relationships between the objects of thought, which unite them into a complete structure, unitary in character and harmonious in form. We might define intuition as that power of the mind by which we build the great theories and fit phenomena into a plan designed along the lines of unifying principles. To be more exact, the mind creates a world of its own. This world is conditioned by what we call the outside world, but in many respects we are free to make it what we please, just as the architect is free to create his building although his material limits him. However, we endeavor to create this world with the maximum simplicity, mainly because simplicity implies harmony, that is, beauty. We are not satisfied with what William James called the "blooming confusion of consciousness" but we construct a replica of this consciousness which is simpler. Of two ways we can construct the replica, we choose the simpler. Thus we choose Euclidean geometry instead of Lobatchevskian, on account of its simplicity, although either might be applied to the world of phenomena. We choose to say the earth rotates on its axis, for that makes astronomy possible. This replica must have a plan, a design, a symmetry, a coherence. Intuition is the perception of this idealized structure. It is akin to the dream of the artist, or the vision of the prophet. Indeed the eminent literary critic, Émile Faguet, calls Poincaré a poet. Was it not Sylvester and Kronecker who said that mathematics was essentially poetry! That was as far as they ever got in defining it. In his address on "Analysis and Physics," Poincaré says:

Mathematics has a triple end. It must furnish an instrument for the study of nature. But that is not all, it has a philosophic end; and, I dare to say it, an esthetic end . . . these two ends [physical and esthetic] are inseparable, and the best way to attain the one is to keep the other in view.

The mathematician does not build in stone, nor paint on canvas, nor construct a symphony, though his harmonies are in and through all these; his medium is more ethereal; but is his creation therefore the less beautiful?

Since the intuition is necessary, the first problem of education becomes the conservation and development of this power. Poincaré points out that in mathematics, for example, we should not begin with general definitions and laws, nor with rigorous logic in the proofs of the theorems. Thus he recommends that in the special mathematics of the secondary school and in the first year of the Ecole Polytechnique, there should not be introduced the notion of functions with no derivatives. At most we should content ourselves with saying "there are such, but we are not concerned with them now." When integrals are first spoken of, they should be defined as areas, and the rigorous definition should be given later, after the student has found many integrals. He says:[10]

The chief end of mathematical instruction is to develop certain powers of the mind, and among these the intuition is not the least precious. By it the mathematical world comes in contact with the real world, and even if pure mathematics could do without it, it would always be necessary to turn to it to bridge the gulf between symbol and reality. The practician will always need it, and for one mathematician there are a hundred practicians. However, for the mathematician himself the power is necessary, for while we demonstrate by logic, we create by intuition; and we have more to do than to criticize others' theorems, we must invent new ones, this art, intuition teaches us.

We turn finally to the research student. How is he to bring the intuition to bear on his problem effectively? If creative work is to be hoped for only through this agency, how do we set it to work? This question Poincaré answers in his analysis of his creation of the fuchsian functions. He holds that the intuition does its work unconsciously. We can not use the term "subconsciously," for he had a repugnance to the doctrine of the superiority of the subliminal self. He points out that our unconscious activity forms large numbers of mental combinations, as an architect, we will say, makes many trial sketches, and of these combinations some are brought into consciousness. These are selected, he believes, by their appeal to the sentiment of beauty, the intellectual esthetic sense of the fitness of things, the unity of ideas, just as the architect from his haphazard sketches selects the right one finally by its appeal to his sense of beauty. Poincaré admits that this explanation of the facts is a hypothesis, but he finds many things to confirm it. One is the fact that the theorems thus suggested in mathematical creation are not always true, yet their elegance, if they were true, has opened the door of consciousness to them. It was Sylvester who used to declare:

Gentlemen, I am certain my conclusion is correct. I will wager a hundred pounds to one on it; yes, I will wager my life on it.

But it often turned out the next day that it was not true. However, it led eventually to things that were true. The direct conclusion from Poincaré's hypothesis would be that we must conserve and develop the esthetic sense of our field, whether mathematics, physics, chemistry, or what not. And we may well pause to consider whether the young investigator should not include some course in design in his work, in painting, architecture, music, poetry or sculpture. Courses in the appreciation of art, rather than the criticism of art, might also be very serviceable indirectly. The constructive philosophers, like Plato or Bergson, might furnish valuable indirect training. Reading that leads to an appreciation of the beauty and sublimity of the universe is of the same value. In any case whatever would intensify the esthetic sensitiveness would be worth while.

When the intuition does not favor us, the golden butterfly fails to emerge from its chrysalis, what is to be done? Here is his answer for whom time did not count, taken from one of his most recent papers.[11] There is a note of pathos in it as well as a hint of premonition. He presents some incomplete results of a new and very important theorem in geometrical transformation, which he is convinced is true, yet the proof of it encounters great difficulties. Every particular case he has been able to settle is favorable to the theorem. After explaining why he is publishing an incomplete paper for the first time, he says:

It would seem that in this situation I should abstain from all publication be long as I have not solved the problem, but after fruitless efforts for many months it seems to me wisest to let the whole problem ripen during several years. That would indeed be well, were I sure of some day being able to take it up again, but at my age I can not go bail for this. On the other hand, the importance of the subject is great. . . and the totality of results so far obtained is too considerable for me to resign myself to definitively allowing them to become unfruitful. I may hope that the mathematicians who interest themselves in the problem and who will be more fortunate than I without doubt will find some means to resolve it.

Again, Poincaré points out that these flashes from below the horizon of consciousness must be preceded by periods of prolonged attentive work. It is like setting Pegasus to plowing corn, but this conscious effort is necessary. This discouraging wandering over the hills and rocks, examining the promising paths and the fragments that point to a nearby mine, day after day, is indispensable to success. It is the weary search over the face of the mountain and the driving of many fruitless drifts that eventually lead the prospector to his mine of gold. On this kind of drudgery Poincaré spent two periods of two hours each daily. The unconscious action of his mind did the rest of his work.

Neither does the discovery of the mine develop it. After the unconscious power has led us to our eldorado, it has done all it can. The deductions, the demonstrations, the applications, must be carried out at the expense of prolonged effort again. The intuition can not do this kind of work. Its region is the nebulous part of thought where the mental ions unite, dissolve, and whirl away,—or we may say that it is found where the breakers surge against the shores of the unknown. But in the consciousness, the stable, the crystallized, the permanent combinations are formed; the new world is organized, surveyed, mapped, and the frontier is widened. Here everything proceeds under hard supervision.

Finally, the research student, the investigator, must have a burning love for the search for truth, as well as for the truth itself. And when in his somber moods he asks, what does it signify in the end? he finds the answer at the close of Poincaré's "Value of Science." He expresses the significance of science in these clear terms:

Civilizations are measured only by their science and their art. Some persons are surprised at the formula: science for science's sake; yet it is quite as good as life for life's sake, if life is only misery; and even as happiness for happiness' sake, if one does not place all pleasures on the same level, if one does not admit that the end of civilization is to furnish more alcohol to people who like to drink.

Every action must have an aim. We have to suffer, we have to work, we have to pay for our seat at the show, but it is in order that we may see, or at least that others may sometimes see.

What is not thought is nought; since we can think only thoughts, and every word we use in talking about things stands for a thought, to assert there is anything else than thought is a senseless affirmation.

Meanwhile—a strange contradiction for those who believe in time—geologic history teaches that life is only an episode between two eternities of death; and even in this episode conscious thought has endured and will endure but a moment. Thought is but a flash in the midst of a long night.

Yet this flash indeed is everything.

  1. For a biographical sketch of Poincaré see Revue des deux Mondes, 1912, September 15. Also the second edition of Lebon's book on Poincaré has appeared.
  2. Revue du Mois, Vol. 7 (1912), p. 133.
  3. C. R., 143 (1906), p. 996.
  4. Scientia, 12 (1912), 159-171 (posthumous).
  5. See Jour, de Physique, 1912 (5), 2; pp. 5-34; 347-360.
  6. Amer. Jour. Math., Vol. 12.
  7. "Scientific Papers," Vol. 4, p. 519.
  8. Scientia, 12 (1912), pp. 1-11.
  9. Amer. Jour. Math., Vol. 12.
  10. L'Enseignement Math., 1899, p. 157.
  11. Rend. Circ. Mat. Palermo, 33 (1912), p. 375.