Hermann Weyl at the Princeton Bicentennial

Hermann Weyl gave an address at the Final Dinner of the 'Problems of Mathematics Conference', held on 17-19 December 1946, one of fifteen conferences which were part of the Princeton Bicentennial. We give below a version of that address delivered on Thursday 19 December 1946 chaired by Howard Percy Robertson. There had been nine sessions at the 'Problems of Mathematics Conference':
(1) Algebra, chaired by Emil Artin;
(2) Algebraic Geometry, chaired by Solomon Lefschetz;
(3) Differential geometry, chaired by Oswald Veblen;
(4) Mathematical Logic, chaired by Alonzo Church;
(5) Topology, chaired by Albert Tucker;
(6) New Fields, chaired by John von Neumann;
(7) Mathematical Probability, chaired by Samuel Wilks;
(8) Analysis, chaired by Salomon Bochner;
(9) Analysis in the Large, chaired by Marston Morse.

Hermann Weyl's Address.

During the past three days the various speakers of this conference have discussed the actual state and current problems of our science in all its various branches, and have tried to prolong the lines beyond the present into the future. Will you now lend me your ear for a brief spell in which I shall evoke the past and let my memory roam over same of the outstanding mathematical events of my lifetime? I have reached the age where one likes "to the sessions of sweet silent thought, to summon up remembrance of things past," and is prone to believe that some lesson for the future may be drawn from comparing past with present. A deceptive belief, I hasten to add; what history can do to us is, as Jacob Burckhardt once said, not to make us more clever for the next time, but wiser for all time. I can give a little more life to my review if you will permit me to intersperse it with remarks about the influence of the mathematical events on my own work. The selection will be very subjective anyway. The balance will be weighted in favour of what happened in Central Europe, since I spent most of my productive years in Germany and Switzerland, and also in favour of youth: by the perspective that always has given rise to the myth of the good old times, my younger years seem to me to have been more crowded than the later with important events.

I wonder whether the organisers of this conference, when they assigned to me the task of talking to you about mathematics in general after the battle is over, had in mind the opening passage of G H Hardy's charming little book A Mathematician's Apology: "It is a melancholy experience for a professional mathematician to find himself writing about mathematics. The function of a mathematician is to do something and not to talk about what he and other mathematicians have done." "I write about mathematics," he continues a little later, "because like any other mathematician who has passed sixty I have no longer the freshness of mind, the energy or the patience to carry on effectively with my proper job." If I view the situation in which I find myself tonight a little less melancholically, it is not because I disagree with Hardy in that "mathematics is a young man's game," but because I do not quite share his scorn "of the man who makes for the man who explains." It seems to me that in mathematics, as in all intellectual endeavours, both things are essential: the deed, the actual construction, on the one side; the reflection on what it means on the other. Creative construction unguarded by reflection is in danger of losing its way, while unbridled reflection is in danger of losing its substance.

It was my good fortune that I took my first steps in mathematical research under the eyes of a master who combined both sides, mathematical creation and philosophical reflection, to an unusual degree, David Hilbert, who was then at the height of his productive power. So it happened that the first outstanding mathematical event of my life was the development of the theory of integral equations by Hilbert. Ivar Fredholm's great discovery lay before my time. What could have been more natural than the idea that a finite set of linear equations describing the motion of a discrete set of mass points gives way to a linear integral equation when one passes to the limit of the continuum? And yet science had to travel a long and tortuous road from Daniel Bernoulli's analysis of the vibrating string in 1730, before this general idea was conceived. But slow travel has its compensations: things become more concrete. Ideas ripen only in conjunction with the development of the concrete problems which they are destined to illumine; and that is good so. Fredholm treated the integral limit of linear equations, Hilbert that of a quadratic form. Its transformation onto principal axes led to a general theory of eigenvalues and eigenfunctions. Bernoulli's heuristic procedure of passing from a finite number of points to a continuum was converted into a mathematical proof. But shortly afterwards Erhard Schmidt gave a beautiful direct proof based on the same ideas as Daniel Bernoulli's and Karl Gräffe's method for the computation of the roots of an algebraic equation and Hermann Amandus Schwarz's construction of the gravest eigentone of a membrane. Hilbert did not yet use the axiomatic formulation in terms of what we now call a Hilbert space, though he exploited to the full the equivalence between the space of square integrable functions and of square summable sequences. By following the growing sequence step by step he established a general theory of bounded operators with their line and band spectra. Soon more direct proofs were found for his results in this wider field too by Frigyes Riesz and others. Their further development, two decades later, under the impact of quantum mechanics, is a familiar story.

One of the most interesting applications of integral equations made by Hilbert himself is to the solution of Bernhard Riemann's problem: Given a finite number of singular points in the complex plane, determine n analytic functions that by analytic continuation around each of these points undergo given linear substitutions. Again the simplifier followed on Hilbert's heels, Josip Plemelj, who analysed the problem as fully as the special case of algebraic functions on a Riemann surface with n sheets had been analysed long before. Important as the contributions of these simplifiers were, Erhard Schmidt, Frigyes Riesz, Josip Plemelj, I think the true moral of such happenings is this: the discoverer who first breaks through often does rough work, and yet to him belongs the highest honour. Only after the lock is broken, one can study it at leisure and construct a key that opens it more smoothly. From a purely logical standpoint there is no reason why the streamlined methods should not be invented straight away; but in the face of a profound problem man is seldom that clever.

The first discovery of my own of some consequence is closely connected with Hilbert's theory of eigenvalues. At a Göttingen conference on statistical thermodynamics in which Hendrik Antoon Lorentz, Arnold Sommerfeld and others participated, the physicists emphasised the need for a proof of the physically plausible fact that the asymptotic distribution of the eigenvalues of a membrane of given area, or of an elastic body or of radiation in a Hohlraum [blackbody] of given volume, is independent of the shape of the area or volume; a mathematically satisfactory foundation for statistical thermodynamics seemed to depend on this theorem. Characterising the successive eigenvalues in a non-recursive manner by a minimax principle, I succeeded in proving precisely that theorem. I still remember vividly the night in my Göttingen Studentenbude when this idea came to me, seemingly without effort, and how greatly surprised I was that it actually worked. In the meantime the kerosene lamp on my desk had started to smoke and the soot came down in flakes from the ceiling.

A number of years later, in Zurich, a pupil of mine, Fritz Peter (1899-1949), and I applied integral equations to the construction of a complete set of inequivalent irreducible representations of a compact Lie group. From this work I should like to draw another lesson. The task was to carry known results over from finite to continuous groups. Theodor Molien's old paper on algebras, published in 1893, served us as a springboard; his treatment is ingenious, but clumsy according to modern standards. Had I known more about the algebraically more polished investigations of Georg Frobenius, Issai Schur, Joseph H M Wedderburn, etc., the going might have been harder, not easier. An awkward method sometimes contains the seeds for important generalisations, wanting in the smoother varieties. Our work had just been completed when Harald Bohr visited Zurich and gave an inspiring talk on almost periodic functions. It did not require much imagination to see the connection between his basic completeness relation for almost periodic functions and ours for the representations of groups.

More recently, integral equations have furnished the tool for proving the main proposition in Hodge's beautiful theory of harmonic integrals.

The theory of integral equations has modified the face of analysis to an appreciable degree, and in view of this fact and of the impulse it gave to my own research I count the emergence of this theory with its far-flung applications as one of the outstanding, if not as the outstanding, mathematical event of my lifetime.

There have been others. Between 1907 and 1910 Paul Koebe was a dominant figure in Göttingen. The uniformisation theorem in all its various forms, which he and Poincaré first proved at that time, occupies a central position in the theory of analytic functions. Koebe never tired of composing new variations on this theme. In a limited field he had great powers of intuition; he was a constructive geometer of the first water. Felix Klein and Henri Poincaré, more than twenty years before, had tried to prove the theorem for the special case of algebraic functions by the so-called continuity method. Later the Riemann problem had been attacked by the same procedure, with equally unsatisfactory results. Abandoning this unwieldy instrument, Koebe and Poincaré now reached their goal by combining H A Schwarz's idea of the universal covering surface with simple estimates of the Harnack type for harmonic functions.

In my early Göttingen years I met Brouwer, I mean Egbertus Brouwer, the topologist and intuitionist. It was at this time (1911) that he published the series of his fundamental papers in Mathematische Annalen, vols. 70 and 71, proving, by means of his method of simplicial approximation, the invariance of dimension, the basic theorems about fixed points and about the degree of a topological mapping. I consider this series the second great impulse for the development of present-day topology after the first that originated from Poincaré's famous six memoirs in 1895-1904. Of all mathematicians I have met, Brouwer more than anybody else with the exception of Hilbert, impressed me as a man of genius. The imprint of his topological ideas is clearly visible in my book on Riemann surfaces (1913). Indeed, he and Koebe were its godfathers, - a strange couple when I now think of them, Koebe the rustic, and Brouwer the mystic. Koebe at that time used to define the notion of Riemann surface by a peculiar gesture of his hands; when I lectured on the subject I felt the need for a more dignified definition. I used the idea of cohomology for establishing the invariance of genus. Topology was in an innocent stage, then. Symptomatic for this early stage is also the fact that when Oswald Veblen and I, independently of each other, skinned Poincaré's Analysis Situs to a purely combinatorial skeleton, we did not dare to publish our investigations for several years. Topology, then a little mountain stream, has now widened into a broad rolling river. Many tributaries have flowed into it. Listing them here and now would mean to carry owls to Athens. The river has flowed far beyond my ken. But if everything is told, I still consider Brouwer's brilliant start in 1911 as the outstanding topological event of my life.

The next exciting event that comes to my mind is the solution of Waring's problem, presenting all integers as sums of a limited number of kth powers of integers, k being a given exponent. I was present when Hilbert in the first session of his mathematical seminar after Hermann Minkowski's death outlined his proof. More intimate is my connection with the great work of G H Hardy and J E Littlewood in this field, and that of Partitio Numerorum in general. I got involved in it because my investigations on equidistribution modulo 1 provided a link the lack of which had held the work up for a considerable time. The simplest case of equidistribution modulo 1, that of the multiples of an irrational number, had come my way in 1910 when I studied a very special question in heat conduction. But only when three years later Felix Bernstein told me about the application Piers Bohl had made of it to the astronomical problem of mean motion did I start to look into the matter in earnest. When I reported on my investigation in the Göttingen Mathematical Club, Carl Runge ridiculed my little drawings that illustrated equidistribution of points over a square by a mere dozen of points, while Harald Bohr overwhelmed me with his knowledge of literature from Leopold Kronecker on, of which I had been completely unaware. I had laid plans to proceed from linear functions to polynomials of higher degree, but shelved them, discouraged by these criticisms. There were distractions of another kind. After my marriage and transfer to Zurich, a big paper by Hardy and Littlewood on this subject appeared in the Acta Mathematica. I skipped through it in the reading room of the Technische Hochschule, walked out to the terrace in front of the building where one has a marvellous view over town and lake, sat down on a bench, and in a few minutes carried out the plan conceived many months before in Göttingen, finding to my satisfaction that it clicked. I could thus prove some of Hardy and Littlewood's conjectures. Of the bearing of my results on Waring's problem I had no idea, neither then nor when I published the result in more detail after I had returned from the war in 1916. Great progress in this line of research was later made by Ivan Vinogradov, who by more refined methods obtained estimates of unexpected accuracy.

I shall now mention by name only some major achievements of our science that in their time deeply impressed me. Personal relationships were often a contributing cause for my attention. I am a passive nature and have always been happier to learn than to think for myself. Thus passed before my eyes: Harald Bohr's theory of almost periodic functions (which was connected for me with integral equations and mean motion). The Bieberbach-Frobenius theory of the crystallographic groups in n dimensions. Hecke's analytic continuation and functional equation of the zeta function of an arbitrary algebraic field. The development of class field theory by Philipp Furtwängler, Teiji Takagi, Helmut Hasse, Emil Artin, Claude Chevalley, etc. (The greatest step had been made by Hilbert before I came to Göttingen as a student.) Élie Cartan's thesis on Lie groups had been written before the turn of the century, and the main body of Frobenius's work on representations of groups is not much later. But Cartan's infinitesimal determination of the irreducible representations of all semi-simple groups and Issai Schur's integral approach to the orthogonal group belong to the period under review and have influenced me profoundly. I witnessed the rise of non-commutative algebra and of the axiomatic viewpoint in algebra to its present position, under the aegis of Schur, Wedderburn, Emmy Noether, Artin and others. In 1930 Carl Siegel's memoir in the Berliner Abhandlungen that for the first time developed a systematic method for transcendency proofs made a big splash. Let this be enough, though many more titles are on my lips; even in Homer's epic the ship's catalogue makes dull reading.

Albert Einstein's "Grundlage der allgemeinen Relativitätstheorie" [Foundation of General Relativity Theory], published in 1916, announced a truly epochal event, the reverberations of which extended far beyond the confines of mathematics. It also made an epoch in my own scientific life. In 1916 I had been discharged from the German army and returned to my job in Switzerland. My mathematical mind was as blank as any veteran's and I did not know what to do. I began to study algebraic surfaces; but before I had gotten far, Einstein's memoir came into my hands and set me afire. You all know the influence general relativity had on the development of infinitesimal geometry. The idea for my unified field theory of gravitation and electromagnetism based on the principle of gauge invariance arose from a conversation with Willy Scherrer (1894-1979), then a young student of mathematics. I had explained to him that vectors when carried around by parallel displacement may return to their starting point in changed direction. And he asked "Also with changed length?" Of course I gave him the orthodox answer at that moment, but in my bosom gnawed the doubt. Gustav Mie's field conception of matter provided the ferment. Once more Pythagoras's and Kepler's music of the spheres seemed to descend from heaven. Over a paper of mine on the uniqueness of the Pythagorean metric I wrote as a motto Kepler's words: "Credo spatioso numen in orbe." [I believe in the geometrical order of the cosmos.] But I was not the only one who believed himself in all earnest to be on the road to the universal law of nature.

I have wasted much time and effort on physical and philosophical speculations, but I do not regret it. I guess I needed them as a kind of intellectual mediation between the luminous ether of mathematics and the dark depths of human existence. While, according to Kierkegaard, religion speaks of "what concerns me unconditionally," pure mathematics may be said to speak of what is of no concern whatever to man. It is a tragic and strange fact, a superb malice of the Creator, that man's mind is so immensely better suited for handling what is irrelevant than what is relevant to him. I do not share the scorn of many creative scientists and artists toward the reflecting philosopher. Good craftsmanship and efficiency are great virtues, but they are not everything. In all intellectual endeavours both things are essential: the deed, the actual construction, on the one side; the reflection on what it means, on the other. Creative construction is always in danger of losing its way, reflection in danger of losing its substance.

In the intervals between the brain tortures of mathematical problems we must seek somehow to regain contact with the world as a whole. The probing of the foundations of mathematics during the last decades seems to favour a realistic conception of mathematics: its ultimate justification lies in its being a part of the theoretical construction of the one real world.

As for my unified field theory, the further development of physics has shown that I was right in assuming a principle of gauge invariance standing in the same relation to the conservation of electric charge as the invariance with respect to coordinate transformations stands to the conservation of energy and momentum. I was wrong in assuming that it connects the electromagnetic potentials with the gravitational

g_{ij}

; it rather connects them with the Schrödinger-Dirac

\psi

s of electronic waves. But that could come to light only after the new quantum theory had been born. The birth of quantum mechanics is without doubt the most consequential physical event of the twentieth century; in view of its repercussions in mathematics, e.g., in the theory of operators, it is also an outstanding mathematical event. This birth is another of the great mathematical events that touched my own life. From close quarters I watched Erwin Schrödinger, who was my neighbour in Zurich, wrestle with his conception of wave mechanics. For myself, quantum mechanics became intimately interwoven with groups.

Last but not least, I have seen our ideas about the foundations of mathematics undergo a profound change in my lifetime. Bertrand Russell, Egbertus Brouwer, David Hilbert, Kurt Gödel. I grew up a stern Cantorian dogmatist. Of Russell I had hardly heard when I broke away from Cantor's paradise; trained in a classical gymnasium, I could read Greek but not English. During a short vacation spent with Brouwer, I fell under the spell of his personality and ideas and became an apostle of his intuitionism. Then followed Hilbert's heroic attempt, through a consistent formalisation "die Grundlagenfragen einfürallemal aus der Welt zu schaffen" [to answer the fundamental questions of the world once and for all], and then Gödel's great discoveries. Move and countermove. No final solution is in sight.

I am surprised how little this serious crisis has disturbed the mind of the average mathematician. What makes him so confident? How does he know that he builds on solid rock and does not merely pile clouds on clouds? In an example like that of solving an algebraic equation one can see that the demands of an intuitionistic or constructive approach coincide completely with the demands of the computer whose task it is actually to compute the roots - with an accuracy that can be increased indefinitely when the coefficients become known with ever greater accuracy. When I preached intuitionism Edmund Landau once said: "If Weyl really believes what he says, he ought to quit his job." I did not draw such drastic consequences, but the intuitionistic attitude influenced me to the extent that I directed my interest to fields I considered comparatively safe. Of course we cannot suspend all work and just wait, hands in lap, until the underground difficulties have been settled for good. What we can do, however, is to proceed with caution and put more emphasis on explicit construction. It is not a matter of black and white, but of grades. In his oration in honour of Lejeune Dirichlet, Hermann Minkowski spoke of the true Dirichlet principle, to face problems with a minimum of blind calculation, a maximum of seeing thought. I find the present state of mathematics, that has arisen by going full steam ahead under this slogan, so alarming that I propose another principle: Whenever you can settle a question by explicit construction, be not satisfied with purely existential arguments. One can certainly not dispense with such general notions as, e.g., that of a Riemann surface. But I prefer to look upon the general propositions about them and their alleged proofs, not as statements of facts but rather as instructions for procedure in broad outlines. I would not apply the theorems mechanically to a special case but would, following the instructions, go through all the steps of proof in concreto, and while checking them, make them as direct, economic, and constructive as possible. I recommend this attitude as an antidote to our present indulgence in boundless abstraction.

From recording, I have inadvertently passed to criticising. I hope I have not spoiled thereby the impression my account was intended to convey to you: It has been good to be a mathematician, and it has been a rich harvest that our science has brought in, during the last forty-odd years. I am happy that a few of its ears grew on my acre. Today we cannot help being beset by doubts in our scale of values. Yet I confess: I believe that mathematics, together with language and music, is the chief creative reaction of man to his universe, deeply founded in his nature. I still feel the deepest satisfaction before such marvels and sublime structures of ideas as, say, the algebraic theory of numbers. The blending of constructive and axiomatic procedures seems to me one of the most characteristic and attractive features of present-day mathematics. I would not miss the axiomatic component. And yet I wonder whether we have not overplayed the game and in the perpetual tension between the concrete and the abstract have not leaned too heavily on the latter side. The mathematician is in greater danger than the physicist of following the line of least resistance; for he is less controlled by reality. A helpful guide in judging mathematical production is the distinction that George Pólya and Gábor Szego made in the preface of their famous Aufgaben und Lehrsätze [Tasks and theorems] between cheap and valuable generalisation, generalisation by dilution and by condensation. Take this situation. On a certain level of generality A which I call the ground level, you have certain theorems that have been proved and certain unsolved problems P of recognised interest. Suppose you discover a generalisation of one of these theorems and thereby rise to a higher level of generality A'. Write it up, but lock it away in a drawer - unless or until it serves to solve one of the problems P on the ground level. At a conference in Bern in 1931 I said: "Before one can generalise, formalise, and axiomatise, there must be a mathematical substance. I am afraid that the mathematical substance in the formalisation of which we have exercised our powers in the last two decades shows signs of nearing exhaustion. Thus I foresee that the coming generation will have a hard lot in mathematics." That challenge, I am afraid, has only partially been met in the intervening fifteen years. There were plenty of encouraging signs in this conference. But the deeper one drives the spade the harder the digging gets; maybe it has become too hard for us unless we are not given some outside help, be it even by such devilish devices as high-speed computing machines. Should we thus seek a broader contact with reality and other fields of knowledge? That seems to be the trend among a considerable section of the younger set. It is to be welcomed. For even from a purely philosophical standpoint, the conception that mathematics is essentially a part of the theoretical construction of the one real world is in better accord with our probings of the foundations of mathematics than more idealistic views.

Mathematical papers are read by but a few people. How many more can enjoy a work of music! If the aesthetic side of mathematics is essential, our endeavours look somewhat futile. I think the main purpose of the day-by-day work in mathematics is to create the atmosphere in which the work of genius can thrive. When I compare the present average standard of mathematical papers with the past, I find that it has risen. I do not think that such a flood of mediocre papers as were written in the heyday of integral equations could now pass the gates of our journals. But the mass of little results that claim attention has made it harder to accomplish the exceptional; hence the net effect is a levelling one. What has definitely deteriorated is the art of writing along with the deterioration of modern languages in general. Writers of fifty years ago, fewer in number, tried harder to make their points clear to a wider group. Basic ideas, connections with other fields, formulated in a not too technical language, received more emphasis. Perhaps our predecessors were aided by a more solemn conviction of the importance of their scientific effort. A large percentage of our papers, if they are not outright monologues, exclude by their very style whoever does not belong to the narrow circle of people working along exactly the same lines. Research conversation between members of such a group should be distinguished from the communication of significant developments to a wider public. Not the first but the second is the proper purpose of publication. Maybe the first should be carried on in another way than by printed papers, say by classified or unclassified reports, and only communications of the second type should go into print.

Ever since Isaac Newton, in his modesty, spoke of playing with pebbles on the shore of a wide ocean, the attitude that we mathematicians play a nice game that ought not to be taken too seriously has enjoyed considerable popularity. In my opinion it is fundamentally unsound. Whatever analogies there are between the mental activities of a mathematician and a chess player, the problems of the former are serious in the sense that they are bound up with truth, truth about the world that is and truth about our existence in the world. That the game of mathematics and physics is a harmless game, one of Hardy's main points in defence of mathematics, nobody can claim any longer. Hardy distinguishes between the real mathematics, that of Carl Friedrich Gauss and Bernhard Riemann and Srinivasa Ramanujan, and the dull Hogben type of Mathematics for the Million [Lancelot Hogben (1895-1975) published Mathematics for the Million in 1936], and thinks that only the latter is useful or harmful as the case may be. I do not believe that his distinction is valid. Riemann is as respectable a mathematician in his papers on the hydrodynamics of shock waves as in his paper on the zeta function. If the progress of science is blamed for the impasse in which the world finds itself today, then the mathematician has to assume his share of responsibility.

We may well envy the nineteenth century for the feeling of certainty and the pathos with which it praised the sacrosanctity and supreme value of science and the mind's dispassionate quest for truth and light. We are addicted to mathematical research with no less fervour. But for us, alas! its meaning and value are questioned from the theoretical side by the critique of the foundations of mathematics, and from the practical-social side by the deadly menace of its misuse.

Crescet scientia, pereat mundus! [Let science grow, though the world perish!] The progress of science has led the world to a terrible impasse. Can we plead innocent before the tribunal of civilisation, and put all the blame on the wicked who misuse our knowledge? There is indeed much to be said in our defence: Without the gift of creative thought and its free exercise, human life would not be what it is; also this, that science is neutral towards good and evil. And yet, it is made by men for men and must not be isolated from man's total existence. Is this then our sin, that we let the unity of existence go to pieces? Or do we, in following the urge of intellectual curiosity that is deeply implanted in our nature, merely fulfil our destiny, like the silkworm?:

Verbiete du dem Seidenwurm zu spinnen,
Der sich dem Tode immer näher spinnt.

[Forbid a silkworm to weave. No use.
He will weave on, though weaving his own death.]

Some people think we can be saved by striking a better balance between the social and the natural sciences. With all respect for the social sciences, but this advice demands too much of them! Technological knowledge is such a dangerous tool in the hands of man, because of the second law of thermodynamics; it is much easier to blow up a building than to build it. What would really be needed to offset the menace of the progress of natural science is a development, not of social science, but of social behaviour and moral responsibility, of our whole attitude towards life. But alas! they change much more slowly than our knowledge. Here is the frightful dilemma that may spell our doom.

Thus the meaning and value of mathematical research is questioned, both from the theoretical side by the critique of the foundations of mathematics, and from the practical-social side by its dreadful implications. What to do is a question everyone of us must answer according to his own conscience. I can suggest no universal solution.

Last Updated September 2020