Monthly Archives: January 2020

Yet another little portrait of Hermann Weyl: as painted by John Archibald Wheeler

Reference: 

The Continuum, Hermann Weyl, translated by Stephen Pollard and Thomas Bole.

This is what famous physicist, John Archibald Wheeler writes as the foreword:

Hermann Weyl was-is-for many of us, and for me, a friend, a teacher, and a hero. A North German who became an enthusiastic American, he was a mathematical master figure to mathematicians, and to physicists a pioneer in quantum theory and relativity and discoverer of gauge theory. He lives for us today, and will live in time to come, in his great findings, his papers and books, and his human influence.

I last knew Weyl after I last knew him. Day after day in Zurich in late 1955 he had been answering letters of congratulations and good wishes received on his seventieth birthday, walking to the mailbox, posting them, and returning home. December eighth, thus making his way homeward, he collapsed on the sidewalk and murmuring, “Ellen, died. News of his unexpected death reached Princeton by the morning New York Times. Some days later our postman brought my wife and me Weyl’s warm note of thanks. I like to think he sent it in that last mailing.

I first knew Weyl before I first knew him. Picture a youth of nineteen seated in a Vermont hillside pasture, at his family’s summer place, with grazing cows around, studying Weyl’s great book, Theory of Groups and Quantum Mechanics, sentence by sentence, in the original German edition, day after day, week after week. That was one student’s introduction to quantum theory. And what an introduction it was! His style is that of a smiling figure on horseback, cutting a clean way through, on a beautiful path, with a swift bright sword.

Some years ago I was asked, like others, I am sure, to present to the Library of the American Philosophical Society the four books that had most influenced me. Theory of Groups and Quantum Mechanics was not last on my list. That book has, each time I read it, some great new message.

If I had to come up with a single word to characterize Hermann Weyl, the man, as I saw and knew him then and in the years to come, it would be that old fashioned word, so rarely heard in out day, “nobility.” I use it here not only in the dictionary sense of “showing qualities of high moral character, as courage, generosity, honour,” but also in the sense of showing exceptional vision. Weyl’s eloquence in pointing out the peaks of the past in the world of learning and his aptitude in discerning new peaks in newly developing fields of thought surely were part and parcel of his lifelong passion for everything that is high in nature and man.

Erect, bright-eyed, smiling Hermann Weyl I first saw in the flesh when 1937 brought me to Princeton. There I attended his lectures on the Elie Cartan calculus of differential forms and their application to electromagnetism — eloquent, simple, full of insights. Little did I dream that in thirty-five years I would be writing, in collaboration with Charles Misner and Kip Thorne, a book on gravitation, in which two chapters would be devoted to exactly that topic. At another time Weyl arranged to give a course at Princeton University on the history of mathematics. He explained to me one day that it was for him an absolute necessity to review, by lecturing, his subject of concern in all its length and breadth. Only so, he remarked, could he see the great lacunae, the places where deeper understanding is needed, where work should focus.

The man who ranged so far in his thought had mathematics as the firm backbone of his intellectual life. Distinguished as a physicist, as a philosopher, as a thinker, he was above all a great mathematician, serving as professor of mathematics from 1913 to 1930 at Zurich, from 1930 to 1933 at Gottingen, and at the Princeton Institute for Advanced Study from October 1933 to his retirement. What thinkers and currents of thought guided Weyl into his lifework: mathematics, philosophy, physics?

“As a schoolboy,” he recalls, “I came to know Kant’s doctrine of the ideal character of space and time, which at once moved me powerfully.” He was still torturing himself, he tells us, with Kant’s Schematismus der reinen Verstandesbegriffe when he arrived as a university student at Gottingen. That was one year before special relativity burst on the world. What a time to arrive, just after David Hilbert, world leader of mathematics, had published his Grundlagen der Geometrie, breaking with Kant’s predisposition for Euclidean geometry and taking up, in the great tradition of Karl Friedrich Gauss and Bernhard Riemann, the construction and properties of non-Euclidean geometries, and — having just published an important book on number theory Zahlericht — was giving absorbing lectures on that field of research. Philosophy! Mathematics! Physics! Each was sounding its stirring trumpet blast to an impressionable young man. Mathematics, being represented in Gottingen by its number one man, won the number one place in Weyl’s heart.

Weyl tells us the impression made upon him by Hilbert’s irresistible optimism, “his spiritual passion, his unshakable faith in the supreme value of science, and his firm confidence in the power of reason to find simple and clear answers to simple and clear questions.” No one who in his twenties had the privilege to listen to Weyl’s lectures can fail to turn around and apply to Weyl himself those very words. Neither can anyone who reads Weyl, and admires his style, fail to be reminded of Weyl’s own writing by what he says of the lucidity of Hilbert: “It is as if you are on a swift walk through a sunny open landscape; you look freely around, demarcation lines and connecting roads are pointed out to you before you must brace yourself to climb the hill; then the path goes straight up, no ambling around, no detours.”

Electrified by Leibnitz and Kant, and under the magnetic influence of Hilbert, Weyl leaped wholeheartedly, as he later put it, into “the deep river of mathematics.” That leap marked the starting point of his lifelong contributions to ever widening spheres of thought.

For the advancing army of physics, battling for many a decade with heat and sound, fields and particles, gravitation and spacetime geometry, the cavalry of mathematics, galloping out ahead, provided what it thought to be the rationale for the real number system. Encounter with the quantum has taught us, however, that we acquire our knowledge in bits; that the continuum is forever beyond our reach. Yet for daily work the concept of the continuum has been and will continue to be as indispensable for physics as it is for mathematics. In either field of endeavour, in any given enterprise, we can adopt the continuum and give up absolute logical rigour, or adopt rigour and give up the continuum, but we can’t pursue both approaches at the same time in the same application.

Adopt rigour or adopt the contiuum ? These ways of speaking should not be counted as contradictory, but as complementary. This complementarity between the continuum and logical rigour we accept and operate with today in the realm of mathematics. The hard-won power thus to assess correctly the continuum of the natural numbers grew out of titanic struggles in the realm of mathematical logic in which Hermann Weyl took a leading part. His guidance, his insights and his wisdom shine out afresh to the English-speaking world with the publication of the present volume. The level of synthesis achieved by now in mathematics is still far beyond our reach today in physics. Happily the courageous outpost-cavalry of mathematical logic prepares the way, not only for the main cavalry that is mathematics, but also for the army that is physics, and nowhere more critically so than in its assault on the problem of existence.

Hermann Weyl has not died. His great works speak prophecy to us in this century and will continue to speak wisdom in the coming century. If we seek a single word to stand for the life and work of Hermann Weyl, what better word can we find than passion? Passion to understand the secret of existence was his, passion for that clear, luminous beauty of conception which we associate with the Greeks, passionate attachment to the community of learning, and passionate belief in the unity of knowledge.

— John Archibald Wheeler, University of Texas, Austin.