Category Archives: motivational stuff

Fourier Transformation in Data Science

via Fourier Transformation in Data Science

Fourier Transform in AI

via Fourier Transform in AI

an outlier

via an outlier

John Conway, Simons Foundation, Science Lives, Mathematics, Mathematicians

via John Conway, Simons Foundation, Science Lives, Mathematics, Mathematicians

because the human brain is plastic

via because the human brain is plastic

Way to go : MIT’s PRIMES — 2020 !!!

The Greatest Auction Ever Held

Reference: A Beautiful Mind by Sylvia Nasar.


Washington, D.C., December 1994:

On the afternoon of December 5, 1994, John Nash was riding in a taxi headed to Newark Airport on his way to Stockholm, where he would, in a few days time, receive from the King of Sweden the gold medal engraved with the portrait of Alfred Nobel. At around the same time, a few hundred miles in the south, in downtown Washington, D.C., Vice President Al Gore was announcing with great fanfare the opening of the “greatest auction ever.”

There was, as The New York Times would later report, no fast-talking auctioneer, no banging gavel, no Old Masters. On the auction block was thin air — airwaves that could be used for the new wireless gadgets like telephones, pagers, faxes — worth billions and billions of dollars, enough licenses for every major American city to have three competing cellular phone services. In the secret war rooms and building booths were CEOs of the world’s biggest communications conglomerates — and unlikely group of blue sky economic theoreticians who were advising them.When the auction finally closed the following March, the winning bids totaled more than $7 billion making it the biggest sale in American history of public assets and one of the most successful (and lucrative) applications of economic theory to public policy ever. Michael Rothschild, dean of Princeton’s Woodrow Wilson School, later called it “a demonstration that people thinking hard about a problem can make the world work better…a triumph of pure thought.”

The juxtaposition of Gore and Nash, the high-tech auction and the medieval pomp of the Nobel ceremony, was hardly an accident. The FCC auction was designed by young economists who were using tools created by John Nash, John Harsanyi and Reinhard Selten. Their ideas were specifically designed for analyzing rivalry and cooperation among a small number of rational players with a mix of conflicting and similar interests: people, governments, and corporations — and even animal species.

The prize itself was a long overdue acknowledgement by the Nobel committee that a sea change in economics, one that had been underway for more than a decade, had taken place. As a discipline, economics had long been dominated by Adam Smith’s brilliant metophor of the Invisible Hand. Smith’s concept of perfect competition envisions so many buyers and sellers that no single buyer or seller has to worry about the reactions of others. It is a powerful idea, one that predicted how free-market economies would evolve and gave policy-makers a guide for encouraging growth and dividing the economic pie fairly. But in the world of mega-mergers, big government, massive foreign direct investment, and whole-sale privatization, where the game is played by a handful of players, each taking into account the others’ actions, each pursuing his own best strategies, game theory has come to the fore.

After decades of resistance —- Paul Samuelson used to joke about “the swamp of n-person game theory” —- a younger generation of theorists began using game theory in areas from trade to industrial organization to public finance in the late 1970’s and early 1980’s. Game theory opened up “terrain for systematic thinking that was previously closed.” Indeed, as game theory and information economics have become increasingly entwined, markets traditionally seen as fitting the purely competitive mold have increasingly been studied using game theory assumptions. The latest generation of texts used in top graduate schools today all recast the basic theories of the firm and the consumer, the foundation of economics, in terms of strategic games. “Concepts, terminology and models from game theory have come to dominate many areas of economics,” said Avinash Dixit, an economist at Princeton who uses game theory in work on international trade and is the author of Thinking Strategically. “At last we are seeing the realization of the true potential of the revolution launched by von Neumann and Morgenstern.” And because most economic applications of game theory use the Nash equilibrium concept, “Nash is the point of departure.”

The revolution has gone far beyond research journals, experimental laboratories at Caltech and the University of Pittsburgh, and classrooms of elite business schools and universities. The current generation of economic policy-makers — including Lawrence Summers, undersecretary of the treasury, Joseph Stiglitz, chairman of the Council of Economic Advisers, and Vice-President Al Gore — are steeped in the stuff, which they say, is useful for thinking about everything from budget proposals to Federal Reserve policy to pollution cleanups.

The most dramatic use of game theory is by governments from Australia to Mexico to sell scarce public resources to buyers best able to develop them. The radio spectrum, T-bills, oil leases, timber, and pollution rights are now sold in auctions designed by game theorists — with far greater success than that of earlier policies.

Economists like Nobel Laureate Ronald Coase have advocated the use of auctions by government since the 1950’s. Auctions have long been used in markets where sellers of unusual items — from vintage wines to movie rights — have no idea what bidders are willing to pay. Their basic purpose is to make bidders reveal how much they value the item. But the arguments of Coase and others were stated in abstract, entirely theoretical terms, and little thought was given to how such auctions would actually be conducted. Congress remained skeptical.

Before 1994, Washington simply gave away licenses for free. Until 1982, it had been up to regulators to decide which companies deserved the licenses. Needless to say, the process was dominated by political pressures, outrageously expensive paperwork, and long delays. The pace of licensing lagged hopelessly behind market shifts and new technologies. After 1982, Washington awarded licenses using lotteries, with the winners free to resell licenses. Although the reform did speed up the granting of licenses, the process was still hugely inefficient — and unfair. Bidders with no intention of operating an actual telephone business spent millions to get into the game for the purpose of reaping a windfall. Further, although telephone companies were forced to pay the costs of obtaining licenses, Washington (and taxpayers) did not get the benefits of any revenues. There had to be a better way.

A young generation of game theorists, including Paul Milgrom, John Roberts, and Robert Wilson at the Stanford B-school, came up with that better way. Their chief contribution consisted of recognizing, as Milgrom said, that “the mere design of some auction was not enough…Getting the auction design right was also critically important.” In particular, they concluded that the most obvious auction designs —- auctioning licenses one by one in sequence using simultaneous sealed bids — was the way least likely to succeed in getting licenses into the hands of corporations that could use them best —- Washington’s stated objective. Game theorists treat an auction like a game with rules and try to evaluate how a given set of rules, taken together, is apt to affect the bidders’ behaviour. They take stock of the options the rules allow, the payoffs to the bidders associated with the options, and bidders expectations about their competitors’ likely choices.

Why did these economists conclude that traditional auction formats would not work? Mainly because the value of each individual license to a user depends — as is the case with a Rembrandt or a Picasso — on what other licenses the user is able to obtain. Some licenses are perfect substitutes for one another. That would be the case for similar spectrum bands to provide a given service. But others are complements. That would be the case for licenses to provide paging services in different parts of the country.

“To permit the efficient license assignment, an auction must allow bidders to consider various packages of licenses, combining complements and switching among substitutes during the course of the auction. Designing an auction to allow this is quite difficult,” writes Paul Milgrom., one of the economists who designed the FCC auction of which Gore was speaking.

A second source of complexity, Milgrom says, is that the purpose of the licenses is to create businesses for new services with unknown technology and unknown consumer demand. Since bidders’ opinions are bound to be wildly divergent, it is possible that license assignment would depend more on bidders’ optimism than on their ability to create a desired service. Ideally, an auction design can minimize that problem.

As Congress and the FCC inched closer to the notion of auctioning off spectrum rights, Australia and New Zealand both conducted spectrum auctions. That they proved to be costly flops and political disasters illustrated that the devil really was in the details. In New Zealand, the government ran a so-called second price auction, and newspapers were full of stories about winners who paid far below their bids. In once case, the high bid was NZ$7 million, the second bid was NZ$5000, and the winner paid the lower price. In another, an Otago University student bid NZ$1 for a television license in a small city. Nobody else bid, so he got it for one dollar. The government expected the cellular licenses to fetch NZ$240 million. The actual revenue was NZ$36 million, one-seventh of the advance estimate. In Australia, a botched auction, in which parvenu bidders pulled the wool over the government’s eyes, delayed the introduction of pay television by almost a year.

The FCC’s chief economist was an advocate of auctions, but no game theorists were involved in the first stage of the FCC auction design. The theorists’s phones started ringing only by accident after the FCC issued a tentative proposal for an auction format with dozens of footnotes to the theoretical literature on auctions. That was how Milgrom and his colleague Robert Wilson, leading auction theorists, got into the game. Milgrom and Wilson proposed that the FCC adopt a simultaneous, multiple round auction. In a simultaneous auction, a bunch of licenses are sold at the same time. Multiple rounds means that, after the first round of bidding prices are announced, and bidders have a chance to withdraw or raise one another’s bids. This is repeated round after round until the auction is over. The chief advantage of this format is that it allows bidders to take account of interdependencies among licenses. Just as sequential, closed-bid auctions let sellers discover what bidders are willing to pay for individual items, the simultaneous, ascending-bid auction lets them discover the market value of different groupings of items.

This early proposal —- which the FCC eventually adopted — did not cover seemingly small but critical details. Should there be deposits? Minimum bid increments? Time limits? Should the bidding system be wholly computerized or executed by hand? And so forth. Milgrom, Roberts and another game theorist, Preston McAfee, an adviser to AirTouch, provided proposals on these issues. The FCC hired another game theorist, John McMillan, of the University of Caliofornia, San Diego, to help evaluate the effect of every proposed rule. According to Milgrom, “Game theory played a central role in the analysis of the rules. Ideas of Nash equilibrium, rationalizability, backward induction, and incomplete information, though rarely named explicitly, were the real basis of daily decisions about the details of the auction process.

By late spring 1995, Washington had raised more than USD 10 billion from spectrum auction. The press and the politicians were ecstatic. Corporate bidders were largely able to protect themselves from predatory bidding and were able to assemble an economically sensible set of licenses. It was, as John McMillan said, ” a triumph for game theory.”


PS: a triumph of pure mathematical thought !:-)


Nalin Pithwa






Einstein told Banesh Hoffmann, “I am slow…!”

Picked up from Reader’s Digest: Indian Edition: March 2020 : just for the joy of sharing and learning with my students. 

January 1968: The Unforgettable Albert Einstein: A professor remembers his encounters with Albert Einstein, and pays a glowing tribute to the man’s genius and his many accomplishments.

By Banesh Hoffmann:


It was one of the greatest scientists the world has ever known, yet if I had to convey the essence of Albert Einstein in a single word, I would choose ‘simplicity’. Perhaps, an anecdote will help. Once, caught in a downpour, he took off his hat and held it under his coat. Asked why, he explained, with admirable logic, that the rain would damage the hat, but his hair would be none the worse for its wetting. This knack for going instinctively to the heart of the matter was the secret of his major scientific discoveries — this and his extraordinary feeling for beauty.

I first met Albert Einstein in 1935, at the famous Institute for Advanced Study in Princeton, New Jersey. Einstein had been among the first to be invited to the Institute, and was offered carte blanche as to salary. To the director’s dismay, Einstein asked for an impossible sum. It was far too small !! The director had to plead with him to accept a larger salary.

I was awe in of Einstein, and hesitated before approaching him about some ideas I had been working on. My hesitation proved unwarranted. When I finally knocked on his door, a gentle voice said, “Come” — with a rising inflection that made the single word both a welcome and a question. I entered his office and found him seated on a table, calculating and smoking his pipe. Dressed in ill-fitting clothes, his hair characteristically awry, he smiled a warm welcome. His utter naturalness at once set me at ease.

As  I began to explain my ideas, he asked me to write the equations on blackboard so that he could see how they developed. Then, came the staggering — and altogether endearing —request: “Please go slowly. I do not understand things quickly.” This from Einstein ! He said it gently, and I laughed. From then on, all vestiges of fear were gone. 

BURST OF GENIUS Einstein was born in 1879 in the German city of Ulm. He had been no infant prodigy; indeed, he was so late in learning to speak that his parents feared he was a dullard. In school, though his teachers saw no special talent in him, the signs were already there. He taught himself calculus, for example, and he told me that his teachers seemed a little afraid of him because he asked questions that they could not answer. At the age of 16, he asked himself whether a light wave would seem stationary if one ran abreast of it. It seems an innocent question, but this shows Einstein going to the heart of a problem. From it there would arise, 10 years later, his theory of relativity.

Einstein failed his entrance examinations at the Swiss Federal Polytechnic School in Zurich, but was admitted a year later. There he went beyond his regular work to study the masterworks of physics on his own. Rejected when he applied for academic positions, he ultimately found work, in 1902, as a patent examiner in Berne, and there, in 1905, his genius burst into fabulous flower.

Among the extraordinary things he produced in that memorable year were his theory of relativity, with its famous offshoot E = mc^{2} (energy equals mass times the speed of light squared), and his quantum theory of light. These two theories were not only revolutionary but seemingly self-contradictory as well: the former was intimately linked to the theory that light consists of waves, while the latter said that it consists of somehow of particles. Yet this unknown young man boldy proposed both at once — and he was right in both cases, though how he could possibly have been is far too complex a story to tell here.


Collaborating with Einstein was an unforgettable experience. In 1937, the Polish physicist Leopold Infeld and I asked if we could work with him. He was pleased with the proposal, since he had an idea about gravitation waiting to be worked out in detail. Thus, we got to know not merely the man and the friend, but also the professional.

The intensity and depth of his concentration were fantastic. When battling a recalcitrant problem, he worried it as an animal worries its prey. Often, when we found ourselves up against a seemingly insuperable difficulty, he would stand up, put his pipe on the table, and say in his quaint English, “I will a little tink” (he could not pronounce “th”). Then, he would pace up and down, twirling a lock of his long greying hair around his forefinger.

A dreamy, faraway yet inward look would come over his face. There was no appearance of concentration, no furrowing of his brow — only a placid inner communion. The minutes would pass, and then suddenly Einstein would stop pacing as his face relaxed into a gentle smile. He has found the solution to the problem. Sometimes it was so simple that Infeld and I could have kicked ourselves for not having thought of it. But the magic had been performed invisibly in the depths of Einstein’s mind, by a process we could not fathom.

When his wife died, he was deeply shaken, but insisted that now more than ever was the time to be working hard. I vividly remember going to his house to work with him during that sad time. His face was haggard and grief-lined but he put forth a great effort to concentrate. Seeking to help him, I steered the discussion away from routine matters into more difficult theoretical problems, and Einstein gradually became absorbed in the discussion. We kept at it for some two hours, and at the end his eyes were no longer sad. As I left, he thanked me with moving sincerity, but the words he found sounded almost incongruous. “It was a fun,” he said. He had a moment of surcease from grief, and these groping words expressed a deep emotion.


Although Einstein felt no need for religious ritual and belonged to no formal religious group, he was the most deeply religious man I have known. He once said to me, “ideas come from God,” and one could hear the capital ‘G’ in the reverence with which he pronounced the word. On the marble fireplace in the mathematics building at Princeton University is carved, in the original German, what one might call his scientific credo: God is subtle, but He is not malicious.” By this Einstein meant that scientists could expect to find their task difficult, but not hopeless. The Universe was a Universe of law, and God was not confusing with deliberate paradoxes and contradictions.

Einstein was an accomplished amateur musician. We used at play duets; he at the violin, I at the piano. One day he surprised me by saying that Mozart was the greatest composer of all. Beethoven, he said, “created” his music but the music of Mozart was of such purity and beauty that one felt he merely “found” it — that it had always existed as part of the inner beauty of the Universe, waiting to be revealed.

It was this very Mozartian simplicity that most characterized Einstein’s methods. His 1905 theory of relativity, for example, was built on two simple assumptions. One is the so-called principle of relativity, which means, roughly speaking, that we cannot tell whether we are at rest or moving smoothly. The other assumption is that the speed of light is the same, no matter what the speed of the object that produces it. You can see how reasonable this is if you think of agitating a stick in a lake to create waves. Whether you wiggle the stick from a stationary pier. or from a rushing speedboat, the waves once generated are on their own, and their speed has nothing to do with that of the stick.

Each of these assumptions, by itself, was so plausible as to seem primitively obvious. But. together they were in such violent conflict that a lesser man would have dropped one or the other and fled in panic. Einstein daringly kept both — and by doing so he revolutionized physics. For he demonstrated that they could after all, exist peacefully side by side, provided we give up cherished beliefs about the nature of time.

Science is like a house of cards, with concepts like time and space at the lowest level. Tampering with time brought most of the house tumbling down, and it was this made Einstein’s work so important —- and so controversial. At a conference in Princeton in honour of his 70th birthday, one of the speakers, a Nobel prize winner, tried to convey the magical quality of Einstein’s achievement. Words failed him, and with a shrug of helplessness he pointed to his wrist-watch, and said in tones of awed amazement, “It all came from this.” His very ineloquence made this the most eloquent tribute I have heard to Einstein’s genius.


Although fame had little effect on Einstein as a person, he could not escape it; he was, of course, instantly recognizable. One autumn Saturday, I was walking with him in Princeton discussing some technical matters. Parents and almuni were streaming excitedly toward the stadium, their minds on the coming football game. As they approached us, they paused in sudden recognition, and a momentary air of solemnity came over them as if they had been reminded of a world far removed from the thrills of football. Yet Einstein seemed totally unaware of the effect he was having on them, and went on with the discussion as though they were not there.

We think of Einstein as one concerned only with the deepest aspects of science. But he saw scientific principles in every day things to which most of us would give barely a second thought.He once asked me if I had ever wondered why a man’s feet will sink into either dry or completely submerged sand, while sand that is merely damp provides a firm surface. When I could not answer, he offered a simple explanation. It depends, he pointed out, on surface tension, the elastic-skin effect of a liquid surface. This is what holds a drop together, or causes two small raindrops on a window pane to pull into one big drop the moment their surfaces touch.

When sand is damp, Einstein explained, there are tiny amounts of water between the grains. The surface tensions of these tiny amounts of water pull all the grains together, and friction then makes them hard to budge. When the sand is dry, there is obviously no water between grains. If the sand is fully immersed, there is water between grains, but there is no water surface between them to pull them together. This is not as important as relativity; yet as his youthful question, about running abreast of a light wave showed, there is no telling what seeming trifle will lead an Einstein to a major discovery. And, the puzzle of the sand gives us an inkling of the power and elegance of Einstein’s mind.


Einstein’s work, performed with pencil and paper, seemed remote from the turmoil of everyday life. But his ideas were so revolutionary that they caused violent controversy and irrational anger. Indeed, in order to be able to award him a belated Nobel Prize, the selection committee had to avoid mentioning relativity, and pretend that his prize was primarily due to his work on quantum theory. Political events upset the serenity of his life even more. When the Nazis came to power in Germany, his theories were officially declared false because they had been formulated by a Jew. His property was confiscated, and it is said that a price was put on his head.

When scientists in the United States fearful that the Nazis might develop an atomic bomb, sought to alert American authorities to that danger, they were scarcely heeded. In desperation, they drafted a letter, which Einstein signed and sent directly to President Roosevelt. It was this act that led to the fateful decision to go all-out on the production of an atomic bomb — an endeavour in which Einstein took no active part. When he heard of the agony and destruction that his E=mc^{2} had wrought, he was dismayed beyond measure and from then on there was a look of ineffable sadness in his eyes.

There was something elusively whimsical about Einstein. It is illustrated by my favourite anecdote about him. In his first year in Princeton, on Christmas Eve, so the story goes, some children sang carols outside his house. Having finished, they knocked on his door and explained that they were collecting money to buy Christmas presents. Einstein listened then said, “Wait a moment.” He put on his scarf and overcoat, and took his violin from its case. Then, joining the children, he accompanied their singing of “Silent Night” on his violin.

How shall I sum up what it meant to have known Einstein and his work? Like the Nobel prize winner who pointed helplessly at his watch, I can find no adequate words. It was akin to the revelation of the great art that lets one see what was formerly hidden. And, when for example, I walk on the sand of a lonely beach, I am reminded of his ceaseless search for cosmic simplicity —- and the scene takes on a deeper, sadder beauty.



Nalin Pithwa.

PS: Thinking takes time, practice, perseverance and solitude. The reward of an intellectual discovery, mathematical or other, is far richer and complete than instant gratification. :-))

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


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.

Another little portrait of Hermann Weyl

(I simply loved the following introduction of Hermann Weyl; I am sharing it verbatim; as described by Peter Pesic in the book Mind and Nature by Hermann Weyl, Princeton University Press; available in Amazon India)

“It’s a crying shame that Weyl is leaving Zurich. He is a great master”. Thus, Albert Einstein described Hermann Weyl (1885-1955) who remains a legendary figure, “one of the greatest mathematicians of the first half of the twentieth century…No other mathematician could claim to have initiated more of the theories that are now being explored,” as (Sir) Michael Atiyah (had) put it. Weyl deserves far wider renown not only for his importance in mathematics and physics but also because of his deep philosophical concerns and thoughtful writing. To that end, this anthology gathers together some of Weyl’s most important general writings, especially those that have become unavailable, have not previously been translated into English, or were unpublished. Together, they form a portrait of a complex and fascinating man, poetic and insightful, whose “vision has stood the test of time>”

This vision has deeply affected contemporary physics, though Weyl always considered himself a mathematician, not a physicist. The present volume emphasizes his treatment of philosophy and physics, but another complete anthology could be made of Weyl’s general writings oriented more directly toward mathematics. Here, I have chosen those writings that most accessibly show how Weyl synthesized philosophy, physics and mathematics.

Weyl’s philosophical reflections began in early youth. He recollects vividly the worn copy of a book about Immaneul Kant’s Critique of Pure Reason he found in the family attic and read avidly at age fifteen. “Kant’s teaching on the “ideality of space and time” immediately took powerful hold of me, with a jolt I was awakened from my ‘dogmatic slumber,’ and the mind of the body found the world being questioned in radical fashion.” At the same time, he was drinking deep of great mathematical works. As “a country lad of eighteen,” he fell under the spell of David Hilbert, whom he memorably described as a Pied Piper “seducing so many rats to follow him into the deep river of mathematics”; the following summer found Weyl poring over Hilbert’s Report on the Theory of Algebraic Numbers during the “happiest months of my life.” As these stories reveal, Weyl was a very serious man; Princeton students called him “holy Hermann” among themselves, mocking a kind of earnestness probably more common in Hilbert’s Gottingen. There, under brilliant teachers, who also included Felix Klein and Hermann Minkowski, Weyl completed his mathematical apprenticeship. Forty years later, at the Princeton Bicentennial in 1946, Weyl gave a personal overview of this period and of the first discoveries that led him to find a place of distinction at Hilbert’s side. This address. never before published, may be a good place to begin if you want to encounter the man and hear directly what struck him most. Do not worry if you find the mathematical references unfamiliar; Weyl’s tone and angle of vision express his feelings about the mathematics (and mathematicians) he cared for in unique and evocative ways; he describes “Koebe the rustic and Brouwer the mystic” and the “peculiar gesture of his hands” Koebe used to define Riemann surfaces, for which Weyl sought “a more diginfied definition.”

In this address, Weyl also vividly recollects how Einstein’s theory of general relativity affected him after the physical and spiritual desolation he experienced during the Great War. “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. Both Weyl and Einstein had lived in Zurich and taught at its ETH (Eidgenoschule Technishce Hochschule) during the very period Einstein was struggling to find his generalized theory, for which he needed mathematical help. This was golden period for both men, who valued the freer spirit they found in Switzerland, compared to German, Einstein adopted Swiss citizenship, completed his formal education in his new country, and then worked at its patent office. Among the first to realize  the full import of Einstein’s work, especially its new, more general, phase, Weyl gave lectures on it at the ETH in 1917, published in his eloquent book Space Time and Matter (1918). Not only the first (and perhaps the greatest) extended account of Einstein’s general relativity, Weyl’s book was immensely influential because of its profound sense of perspective, great expository clarity, and indications of directions to carry Einstein’s work further. Einstein himself praised the book as a “symphonic masterpiece.”

As the first edition of Space Time Matter went to press, Weyl reconstructed Einstein’s ideas from his own mathematical perspective and came upon a new and intriguing possibility, which Einstein immediately called ” a first class stroke of genius”. Weyl describes this new idea in his essay “Electricity and Gravitation” (1921), much later recollecting some interesting personal details in his Princeton address. There, Weyl recalls explaining to a student, Willy Scherer, “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 {no} at that moment, but in my bosom gnawed the doubt.” To be sure. Weyl wrote this remembrance thirty years later, which thus may or may not be a perfectly faithful record of the events; nevertheless, it represents Weyl’s own self-understanding of the course of his thinking, even if long after the fact. Though Weyl does not mention it, this conversation was surrounded by a complex web of relationships: Weyl’s wife Helene was deeply involved with Willy’s brother Paul, while Weyl himself was the lover of Erwin Schrodinger’s wife, Amy. These personal details are significant here because Weyl himself was sensitive to the erotic aspects of scientific creativity in others, as we will see in his commentary on Schrodinger, suggesting that Weyl’s own life and were similarly intertwined.

In Space Time Matter, Weyl used the implications of parallel transport of vectors to illuminate the inner structure of the theory Einstein had originally phrased in purely metric terms, meaning the measurement of distances between points, on the model of the Pythagorean theorem. Weyl questioned the implicit assumption that behind this metrical structure is a fixed, given distance scale, or “gauge.” What if the direction as well as the length of meter sticks (and also the standard second given by clocks) were to vary at different places in space-time, just as railway gauge varies from country to country? Perhaps, Weyl’s concept began with this kind of homely observation about the “gauge relativity” in the technology of rail travel, well-known to travelers in those days, who often had to change trains at frontiers between nations having different, incompatible railway gauges. By considering this new kind of relativity, Weyl stepped even beyond the general coordinate transformations Einstein allowed in his general theory so as to incorporate what Weyl called relativity of magnitude.

In what Weyl called an “affinely connected space,” a vector could be displaced parallel to itself, at least to an infintely nearby point. As he realized after talking to Willy Scherrer, in such a space a vector transported around a closed curve might return to its starting point with changed direction as well as length (which he called “non-integrability,” as measured mathematically by the “affine connection.”) Peculiar as this changed length might seem, Weyl was struck by the mathematical generality of this possibility, which he explored in what he called his “purely infinitesimal geometry,” which emphasized infinitesimal displacement as the foundation in terms of which any finite-displacement needed to be understood. As he emphasized the centrality of the infinite in mathematics, Weyl also placed the infinitesimal before the finite.

Weyl also realized that his generalized theory gave him what seemed a natural way to incorporate electromagnetism into the structure of space-time, a goal that had eluded Einstein, whose theory treated electromagnetism along with matter as mass-energy sources that caused space-time curvature but remained separate from space-time itself. Here Weyl used the literal “gauging” of distances as the basis of a mathematical analogy; his reinterpretation of these equations led naturally to a gauge field he could then apply to electromagnetism, from which Maxwell’s equations now emerged as intrinsic to the structure of space-time. Though Einstein at first hailed this ‘stroke of genius’, soon he found what he considered a devastating objection: because of the non-integrability of Weyl’s gauge field, atoms would not produce the constant, universal spectral lines we actually observe: atoms of hydrogen on Earth give the same spectrum as hydrogen observed telescopically in distant stars. Weyl’s 1918 paper announcing his new theory appeared with an unusual postscript by Einstein, detailing his objection, along with Weyl’s reply that the actual behaviour of atoms in turbulent fields, not to speak of measuring rods and clocks, was not yet fully understood. Weyl noted that his theory used light signals as a fundamental standard, rather than relying on supposedly rigid measuring rods and idealized clocks, whose atomic structures was in some complex state of accommodation to ambient fields.

In fact, the atomic scale was the arena in which quantum theory was then emerging. Here began the curious migration of Weyl’s idea from literatly regauging length and time to describing some other realm beyond space-time. Theordor Kaluza (1922) and Oskar Klein (1926) proposed a generalization of general relativity using a fifth dimension to accommodate electromagnetism. In their theory, Weyl’s gauge factor turns into a phase factor, just as the relative phase of traveling waves depends on the varying dispersive properties of the medium they traverse. If so, Weyl’s gauge would no longer be immediately observable (as Einstein’s objection asserted) because the gauge affects only the phase, not the observable frequency of atomic spectra.

At first, Weyl speculated that his 1918 theory gave support to the radical possibility that “matter” is only a form of curved, empty space (a view John Wheeler championed forty years later). Here Weyl doubltess remembered the radical opinions of Michael Faraday and James Clerk Maxwell, who went so far as to consider so-called matter to be a nexus of immaterial lines of force.

Weyl then weighed these mathematical speculations against the complexities of physical experience. Though he still believed in his fundamental insight that gauge invariance was crucial, by 1922 Weyl realized that it needed to be re-considered in light of the emergent quantum theory. Already in 1922, Schrodinger pointed out that Weyl;s idea could lead to a new way to understand quantization. In 1927, Fritz London argued that gravitational scale factor implied by Weyl’s 1918 theory, which Einstein had argued was unphysical, actually makes sense as the complex phase factor essential to quantum theory.

As Schrodinger struggled to formulate his wave equation, at many points he relied on Weyl for mathematical help. In their liberated circles, Weyl remained a valued friend and colleague even while being Anny Schrodinger’s lover. From that intimate vantage point, Weyl observed that Erwin “did his great work during a late erotic outburst in his life,” an intense love affair simultaneously with Schrodinger’s struggle to find a quantum wave equation. But, then as Weyl inscribed his 1933 Christmas gift to Anny and Erwin (a set of erotic illustrations to Shakespeare’s Venus and Adonis), “the sea has bounds but deep desire has none.”

Weyl’s insight into the nature of quantum theory comes forward in a pair of letters he and Einstein wrote in 1922, here reprinted and translated for the first time, responding to a journalist’s question about the significance of the new physics. Einstein dismisses the question: for him in 1922, relativity theory changes nothing fundamental in out view of the world, and that is that. Weyl takes the question more seriously, finding a radically new insight not so much in relativity theory as in the emergent quantum theory, which Weyl already understood as asserting that “the entire physics of matter is statistical in nature,” showing how clearly he understood this decisive point several years before the formulation of the new quantum theory in 1923-26 by Max Born, Werner Heisenberg, Pascual Jordal, P.A.M. Dirac, and Schrodinger. In his final lines, Weyl also alludes to his view of matter as agent, in which he ascribed to matter an innate activity that may have helped him understand and accept the spontaneity and indeterminacy emerging in quantum theory. This view led Weyl to reconsider the significance of the concept of a field. As he wrote to Wolfgang Pauli in 1919, “field physics, I feel, really plays only the role of ‘world geometry’, in matter there resides still something different, (and) real, that cannot be grasped causally, but that perhaps should be thought of in the image of ‘independent decisions,’ and that we account for in physics by statistics.

In the years around 1920, Weyl continued to work out the consequences of this new approach. His conviction about the centrality of consciousness as intuition and activity deeply influenced his view of matter. As the ego drives the whole world known to consciousness, he argued that matter is analogous to the ego, the effects of which, despite the ego itself being non-spatial, originate via the body at a given point of the world continuum. Whatever the nature of this agents, which excites the field, might be — perhaps life and will — in physics we only look at the field effects caused by it. This took him in a direction very different from the vision of matter reduced to pure geometry he had entertained in 1918. Writing to Felix Klein in 1920, Weyl noted that “field physics no longer seems to me to be the key to reality; but, the field, the ether, is to me only a totally powerless transmitter by itself of the action, but matter rests beyond the field and is the reality that causes its states.” Weyl described his new view in 1923 using an even more striking image: “Reality does not move into space as into a right-angled tenement house along which all its changing play of forces glide past without leaving any trace; but, rather as the snail, matter itself builds and shapes this house of its own.” For Weyl now, fields were “totally powerless transmitters” that are not really existent or effectual in their own right, but only a way of talking about states of matter that are the locus of fundamental reality. Though Weyl still retained fields to communicate interactions, his emphasis that the reality of “matter rests beyond the field” may have influenced Richard Feynman and Wheeler two decades later in their own attempt to remove “fields” as independent beings. Weyl also raised the question of whether matter has some significant topological structure on the subatomic scale, as if topology were a kind of activation that brings static geometry to life, analogous to the activation ego infuses into its world. Such topological aspects of matter only emerged as an important frontier of contemporary investigation fifty years later. Looking back from 1955 at his original 1918 paper, Weyl noted that he “had no doubt” that the correct context of his vision of gauge theory was “not, as I believed in 1918, in the intertwining of electromagnetism and gravity” but in the “Schrodinger-Dirac potential \Psi of the electron-positron field…The strongest argument for my theory seems to be this that gauge-invariance corresponds to the conservation of electric charge in the same way that co-ordinate invariance corresponds to the conservation of energy and momentum,” the insight that Emmy Noether’s famous theorem put at the foundations of quantum field theory. Nor did Weyl himself stop working on his idea; in 1929 he published an important reformulating his idea in the language of what today are called gauge fields; these considerations also led him to consider fundamental physical symmetries long before the discovery of the violation of parity in the 1950’s. The “Weyl two-component neutrino field” remains a standard description of neutrinos, all of which are “left-handed” (spin always opposed to direction of motion), as all anti-neutrinos are “right-handed.” In 1954, (a year before Weyl’s death, but apparently not known to him), C. N. Yang, R. Mills, and others took the next steps in developing gauge fields, which ultimately became the crucial element in the modern “standard model” of particle physics that triumphed in the 1970’s, unifying strong, weak, and electromagnetic interactions in ways that realized Weyl’s distant hopes quite beyond his initial expectations.

In the years that Weyl continued to try to find a way to make his idea work, he and Einstein under went a curious exchange of positions. Originally, Einstein thought Weyl was not paying enough attention to physical measuring rods and clocks because Weyl used immaterial light beams to measure space-time. As Weyl recalled in a letter of 1952, “I thought to be able to answer his concrete objections, but in the end he said: “Well, Weyl let us stop this. For what I actually have against your theory is: ‘It is impossible to do physics like this (that is, in such a speculative fashion, without a guiding intuitive physical principle)!”‘ Today, we have probably changed our viewpoints in this respect. E. believes that in this domain the chasm between ideas and experience is so large, that only mathematical speculation (whose consequences, of course, have to be developed and confronted with facts) gives promise of success, while my confidence in pure speculation has diminished and a closer with quantum physical experience seems necessary, especially as in my view it is not sufficient to blend gravitation and electricity to one unity, but that the wave fields of the electron (and whatever there may still be of nonreducible elementary particles) must be included.

Ironically, Weyl the mathematician finally sided with the complex realities of physics, whereas Einstein the physicisit sought refuge in unified field theories that were essentially mathematical. Here is much food for thought about the philosophic reflections each must have undertaken in his respective soul-searching and that remain important now, faced with the possibilities and problems of string theory, loop quantum gravity, and other theoretical directions for which sufficient expeimental evidence may long remain unavailable.

Both here and through out his life, Weyl used philosophical reflection to guide his theoretical work, preferring “to approach a question through a deep analysis of the concepts it involves rather than by blind computations,” as Jean Dieudonne put it. Though others of his friends, such as Einstein and Schrodinger, shared his broad humanisitc education and philosophical bent, Weyl tended to go even further in this direction. As a young student in Gottingen, Weyl had studied with Edmund Husserl (who had been a mathematician before turning to philosophy), with whom Helene Weyl had also studied.

Weyl’s continuing interest in phenomenological philosophy marks many of his works, such as his 1927 essay on “Time Relations in the Cosmos, Proper Time, Lived Time, and Metaphysical Time,” here reprinted and translated for the first time. The essay’s title indicates its scope, beginning with his interpretation of the four-dimensional space-time Hermann Minkowski introduced in 1908, which Weyl then connects with human time consciousness (also a deep interest of Husserl’s). Weyl treats a world point not merely as a mathematical abstraction but as situating a “point-eye,” a living symbol of consciousness peering along its world line. Counterintuitively, that point-eye associates the objective with the relative, the subjective with the absolute.

Weyl used this striking image to carry forward a mathematical insight that had emerged earlier in the considerations about the nature of the continuum. During the early 1920’s, Weyl was deeply drawn to L.E.J. Brouwer’s advocacy of intuition as the critical touchstone of modern mathematics. Thus, Brouwer rejected Cantor’s transfinite numbers as not intuitable, despite Hilbert’s claim that “no one will drive us from paradise which Cantor created for us.” Hilbert argued that mathematics should be considered purely formal, a great game in which terms like “points” or “lines” could be replaced with arbitrary words like ‘beer mugs” or ‘tables” or with pure symbols, so long as the axiomatic relationships between the respective terms do not change. Was this, then, the “deep river of mathematics” into which Weyl thought this Pied Piper had lured him and so many other clever young rats?

By the mid-1920’s, Weyl was no longer an advocate of Brouwer’s views (though still reaffirming his own 1913 work on the continuum). In his magisterial Philosophy of Mathematics and Natural sciences (written in 1927 but extensively revised in 1949), Weyl noted that “mathematics with Brouwer gains its highest intuitive clarify…It cannot be denied, however, that in advancing to higher and more general theories the inapplicability of the simple laws of logic eventually results in an almost unbearable awkwardness. And the mathematician watches with pain the larger part of his towering edifice which he believed to be built of concrete blocks dissolve into a mist before his eyes.

Even so, Weyl remained convinced that we should not consider a continuum (such as the real numbers between 0 and 1) as an actually completed and infinite set but only as capable of endless subdivision. This understanding of the “potential infinite” recalls Aristotle’s critique of the ‘actual infinite. In his 1927 essay on “Time Relations,” Weyl applied this view to time as a continuum. Because an infinitely small point could be generated from a finite interval only through actually completing an infinite process of shrinkage, Weyl applies the same argument to the presumption that the present instant is a “point in time.” He concludes that “there is no pointlike Now and also no exact earlier and later.” Weyl’s arguments about the continuum have the further implication that the past is never completely determined, any more than a finite, continuous interval is ever exhaustively filled; both are potentially infinite because always further divisible. If so, the past is not fixed or unchangeable and continues to change, a luxuriant, ever-proliferating tangle of “world tubes,” as Weyl called them, “open into the future and again and again a fragment of it is lived through.” This intriguing idea is psychologically plausible: A person’s past seems to keep changing and ramifying as life unfolds, the past today seems different than it didi yesterday. As a character in Fauklner put it, “the past is never dead. It’s not even past.”

In Weyl’s view, a field is intrinsically continuous, endlessly subdividable, and hence, an abyss in which we never come to an ultimate point where a decision can be made. To be or not to be? Conversely pointlike, discrete matter is a locus of decisive spontaneity because it is not predictable through continuous field laws, only observable statistically. As Weyl wrote to Pauli in 1919, “I am firmly convinced that statistics is in principle something independent from causality, the ‘law’; because it is in general absurd to imagine a continuum as something like a finished being.” Because of this independence, Weyl continued in 1920:-

the future will act on and upon the present and it will determine the present more and more precisely; the past is not finished. Thus, the fixed pressure of natural causality disappears and there remains, irrespective of the validity of the natural laws, a space for autonomous and causally absolutely independent decisions. lI consider the elementary quanta of matter to be the place of these decisions.

“Lived time,” in Weyl’s interpretation, keeps evoked the past into furhter life, even as it calls the future into being. Weyl’s deep thoughts may still reap the further exploration, they have not received so far.

Weyl also contributed notably to the application of general relativity to cosmology. He found new solutions to Einstein’s equation and already in 1923 calculated a value for the radius of the universe of roughly one billion light years, six years before Edwin Hubble’s systematic measurements provided what became regarded as conclusive evidence that our galaxy is one among many. Weyl also reached a seminal insight, derived from both his matheamtical and his philosophical considerations, that the topology of the universe is “the first question in all speculations on the world as a whole.” This prescient insight was taken up only in the 1970’s and remains today at the forefront of cosmology, still unsolved and as important as Weyl thought. He also noted that relativistic cosmology indeed “left the door open for possibilities of every kind.” The mysteries of dark energy and dark matter remind us of how much still lies beyond that door. Then too, we still face the questions Weyl raised regarding the strange recurrence throughout cosmology of the “larger numbers” like 10^{20} and 10^{40} (seemingly as ratios between cosmic and atomic scales), later rediscovered by Dirac.

Other of Weyl’s ideas long ago entered and transformed the mainstream of physics, characteristically bridging the mathematical and physical through the philosophical. He considered his greatest mathematical work the classification of the semisimple groups of continuous symmetries (Lie groups), which he later surveyed in The Classical Groups: Their Invariants and Representations (1938). In the introduction to this first book, he wrote in English, Weyl noted that “the gods have imposed upon my writing the yoke of a foreign tongue that was not sung at my cradle.” But even in his adopted tongue he does not hesitate to criticize hte “too thorough technicalization of mathematical research” In America that has led to a ” mode of writing which must give the reader the impression of being shut up in a brightly illuminated cell where every detail sticks out with the same dazzling clarity, but without relief. I prefer the open landscape under a clear sky with its depth of perspective, where the wealth of sharply defined nearby details gradually fades away toward the horizon.” Such writing exemplifies Weyl’s uniquely eloquent style.

Soon after quantum theory had first been formulated, Weyl used his deep mathematical perspective to shape The Theory of Groups and Quantum Mechanics (1928). It is hard to overstate the importance of his marriage of the mathematical theory of symmetry to quantum theory, which has proved ever more fruitful, with no end in sight. At first, as eminent and hard-headed physicist as John Slater resisted the “group-pest” as if it were a plague of abstractness. But Weyl, along with Eugene Wigner, prevalied because the use of group theory gave access to the symmetries essential for formulating all kinds of physical theories, from crystal lattices to multiplets of fundamental particles. It was this depth and generality that moved Julian Schwinger to ‘read and to re-read that book, each time progressing a little farther, but I cannot say that I ever —- not even to this day — fully mastered it.”  Thus, Schwinger considered Weyl “one of my gods,” note merely an outstanding teacher, because “the ways of gods are mysterious, inscrutable, and beyond the comprehension of ordinary mortals.” This from someone regarded as god-like by many physicists because of his own inscrutable powers. Weyl’s insights about the fundamental mathematical symmetries led Schwinget and others decades later to formulate the TCP theorem, which expresses the fundamental indenticality between particles and anti-particles under the combined symmetries of time reversal (T), charge conjugation (reversal of the sign of charge, C) and parity (mirror) reversal (P), In one of his most powerful interventions in physics, Weyl used such symmetry principles to argue that Dirac’s newly proposed (and as yet unobserved) “holes” (anti-electrons) could not be (as Dirac had suggested) protons, which are almost two thousand times heavier than electrons. Weyl showed mathematically that anti-electrons had to have the same mass as electrons, though having opposite charge; this was later confirmed by cosmic ray observations. Weyl’s purely mathematical argument struck Dirac, who drew from this experience his often cited principle that “it is more important to have beauty in one’s equations than to have them fit experiment,” a principle that continues to be an important touchstone for many physicists. Even though Weyl’s mathematics moved Dirac to this radical declaration, Weyl’s own turn away from mathematical speculation about physics raises the question whether in the end to prefer beautiful mathematics to the troubling complexities of experience.

Whether a “god” or no, Weyl seemed to feel that the philosophical enterprise cannot remain on the godlike plane but really requires the occasions of human conversation. The two largest works in this anthology contain the rich harvest of Weyl’s long standing interest in expressing his ideas to a broader audience, both began as lecture series, thus doubly public, both spoken and written. To use the apt phrase of his son Michael, The Open World (1932) contains “Hermann’s dialogues with God” because here the mathematician confronts his ultimate concerns. These do not fall into the traditional religious traditions but are much closer in spirit to Spinoza’s rational analysis of what he called “God or nature,” so important for Einstein as well. As Spinoza considered the concept of infinity fundamental to the nature of God, Weyl defines “God as the completed infinite.” In Weyl’s conception, God is not merely a mathematician but is mathematics itself because “mathematics is the science of the infinite,” engaged in the paradoxical enterprise of seeking “the symbolic comprehension of the infinite with human, that is finite, means.” In the end, Weyl concludes that this God ‘cannot and will not be comprehended” by the human mind, even though “mind is freedom within the limitations of existence, it is open toward the infinite.” Nevertheless, “neither can God penetrate into man by revelation, nor man penetrate to Him by mystical perception. The completed infinite we can only represent in symbols.” In Weyl’s praise of openness, this freedom of the human mind begins to seem even higher than the completed infinity essential to the meaning of God. Does not his argument imply that God, as actual infinite, can never be actually complete, just an infinite time will never have passed, however long one waits? And if God’s actuality will never come to pass, in what sense could or does or will God exist at all? Perhaps, God, like the continuum or the field, is an infinite abyss that needs completion by the decisive seed of matter, of human choice.

Weyl inscribes this paradox and its possibilities in his praise of the symbol, which includes the mathematical no less than the literary, artistic, poetic, thus bridging the presumed chasm between the “two cultures.” At every turn in his writing, we encounter a man of rich and broad culture, at home in many domains of human thought and feeling, sensitive to its symbols and capable of expressing himself beautifully. He moves so naturally from quoting the ancients and moderns to talking about space-time diagrams, thus showing us something of his innate turn of mind, his peculiar genius. His quotations and reflections are not mere illustrations but show the very process by which his thought lived and moved. His philosophical turn of mind helped him reach his own finest scientific and mathematical ideas. His self-deprecating disclaimer that he thus “wasted his time” might be read as irony directed to those who misunderstood him, the hardheaded who had no feeling for those exalted ideas and thought his philosophizing idle or merely decorative. Weyl gained perspective, insight, and altitude by thinking back along the never unfolding past and studying its great thinkers, whom he used to help him soar, like a bird feeling the air under its wings.

In contrast, Weyl’s lectures on Mind and Nature, published only two years later (1934), have a less exalted tone. The difference shows his sensitivity to the changing times. Though invited to return to Gottingen in 1918, he preferred to remain in Zurich, finally in 1930, he accepted the call to succeed Hilbert, but almost immediately regretted it. The Germany he returned to had become dangerous for him, his Jewish wife, and his children. Unlike some who were unable to confront those ugly realities, Weyl was capable of political clear-sightedness, by 1933 he was seeking to escape Germany. His depression and uncertainty in the face of those huge decisions shows another side of his humanity, as Richard Courant put it, “Weyl is actually in spite of his enormously broad talents an inwardly insecure person, for whom nothing is more difficult than to make a decision which will have consequences for his life, and who mentally is not capable of dealing with the weight of such decisions, but needs a strong support somewhere. That anxiety and inner insecurity gives Weyl’s reflections their existential force. As he himself struggled along his own world line through endlessly ramifying doubts, he came to value the spontaneity and decisiveness he saw in the material world.

Weyl’s American lectures marked the start of a new life, beginning with a visiting professorship at Princeton (1928-1929), where he revised his book on group theory and quantum mechanics in the course of introducing his insights to this new audience.  Where in 1930 Weyl’s Open World began with God, in 1933 his lectures on Mind and Nature start with human subjectivity and sense perception. Here, symbols help us confront a world that “does not exist in itself, but is merely encountered by us as an object in the correlative variance of subject and object.” For Weyl, mathematical and poetic symbols may disclose a path through the labyrinth of “mirror land,” a world that may seem ever more distorted, unreal on many fronts. Though Weyl discerns “an abyss which no realistic conception of the world can span” between the physical processes of the brain and the perceiving subject, he finds deep meaning in “the enigmatic two fold nature of the ego, namely that I am both: on the one hand a real individual which performs real psychical acts, the dark, striving human being that is cast out into the world and its individual fate, on the other hand light which beholds itself, intuitive vision, in whose consciousness that is pregnant with images and endows with meaning, the world opens up. Only in this ‘meeting’ of consciousness and being both exist, the world and I.”

Weyl treats relativity and quantum theory as the latest and most suggestive symbolic constructions we make to meet the world. The dynamic character of symbolism endures, even if the particular symbols change, “their truth refers to a connected system that can be confronted with experience only as a whole.” Like Einstein, Weyl emphasized that physical concepts as symbols “are constructions within a free realm of possibilities,” freely created by the human mind. “Indeed, space and time are nothing in themselves, but only certain order of the reality existing and happening in them.” As he noted in 1947, “it has now become clear that physics needs no such ultimate objective entities as space, time, matter, or ‘events’, or the like, for its constructions symbols without meaning handled according to certain rules are enough.” In Mind and Nature, Weyl notes that in nature itself, as (quantum) physics constructs it theoretically, the dualism of object and subject, of law and freedom, is already most distinctly predesigned.” As Niels Bohr put it, this dualism rests on “the old truth that we are both spectators and actors in the great drama of existence.”

After Weyl left Germany definitively for Princeton in 1933, he continued to reflect on these matters. In the remaining selections, one notes him retelling some of the same stories, quoting the same passages from great thinkers of the past, repeating an idea he had already said elsewhere. These repetitions posed a difficult problem, for the latest essays contain some interesting new points along with the old, Because of this, I decided to include these later essays, for Weyl’s repetitions also show him reconsidering. Reiterating a point in a new or larger context may open further dimensions. Then too, we as readers are given another chance to think about Weyl’s points and also see where he held to his earlier ideas and where he may have changed. For he was capable of changing his mind, more so that Einstein, whose native stubbornness may well have contributed to his unyielding resistance to quantum theory. As noted above, Weyl was far more able to entertain and even embrace quantum views, despite their strangeness, precisely because of his philosophical openness.

Weyl’s close reading of the past and his philosophical bent inspired his continued openness. In his hitherto unpublished essay “Man and the Foundations of Science” (written about 1949), Weyl describes an ocean traveler who distrusts the bottomless sea and therefore clings to the view of the disappearing coast as long as there is in sight no other coast toward which he moves. I shall now try to describe the journey on which the old coast has long since vanished below the horizon. There is no use in staring in that direction any longer.” He struggles to find a way to speak about “a new coast [that] seems dimly discernible, to which I can point by dim words only and may be it is merely a bank of fog that deceives me.” Here symbols might be all we have, for “it becomes evident that now the words ‘in reality’ must be put between quotation marks, we have a symbolic construction, but nothing which we could seriously pretend to be the true real world.” Yet even legerdemain with symbols cannot hide the critical problem of the continuum. “The sin committed by the set theoretic mathematician is his treatment of the field of possibilities open into infinity as if it were a completed whole all members of which are present and can be overlooked with one glance. For those whose eyes have been opened to the problem of infinity, the majority of his statements carry no meaning. If the true aim of the mathematician is to master the infinite by the finite means, he has attained it by fraud only — a gigantic fraud which, one must admit, works as beautifully as paper money.” By his reaffirmation of his critique of the actual infinite, we infer that Weyl continued to hold his radical views about “lived time,” especially that “we stand at that intersection of bondage and freedom which is the essence of man himself.”

Indeed, Weyl notes that he had put forward this relation between being and time years before Martin Heidegger’s famous book on that subject appeared. Weyl’s account of Heidegger is especially interesting because of the intersection between their concerns, no less than their deep divergences. Yet Weyl seemingly could not bring himself to give a full account of Heidegger or of his own reactions, partly based on philosophical antipathy, partly (one infers) from his profound distaste for Heidegger’s involvement with the Nazi regime. Though he does not speak of it, Weyl may well also have known of the way Heidegger abandoned their teacher, Husserl, in those dark days. Most of all, Weyl conveys his annoyance that Heidegger had botched important ideas that were important to Weyl himself and, in the process, that Heidegger lost sight of the future of science. “Taking up a crucial term, they both use, Weyl asserts that ‘no other ground is left for science to build on than this dark but very solid rock which I once more call the concrete Dasein of man in his world.” Weyl grounds this Dasein, man’s being-in-the-world, in ordinary language, which is “neither tarnished poetry nor a blurred substitute for mathematical symbolism; on the contrary, neither the one nor the other would and could exist without the nourishing stem of the language of everyday life, with all its complexity, obscurity, crudenss, and ambiguity.” By thus connecting mathematical and poetic symbolism as both growing from the soil of ordinary human language, Weyl implicitly rejects Heidegger’s turn away from modern mathematical science.

In his late essay “The Unity of Knowledge” (1954), Weyl reviews the ground and concludes that “the shield of Being is broken beyond repair,” but does not take this disunity in a tragic sense because “on the side of Knowing there may be unity. Indeed, mind in the fullness of its experience has unity. Who says ‘I’ points to it. Here he reaffirms his old conviction that human consciousness is not simply the product of other, more mechanical forces, but is itself the luminous centre constituting that reality through its “complex symbolic creations which this lumen built up in the history of mankind.” Even though “myth, religion, and alas! also philosophy” fall prey to “man’s infinite capacity for self-deception,” Weyl implicitly holds our greater hope for the symbolic creations of mathematics and science, though he admits that he is still struggling to find clarity.

The final essay in this anthology, “Insight and Reflection” (1955) is Weyl’s rich Spatless, the intense, sweet wine made from grapes long on the vine. This philosophical memoir discloses his inner world of reflection in ways his other, earlier essays did not reveal quite so directly, perhaps aware of the skepticism and irony that may have met them earlier on. WE are reminded of his “point-eye,” disclosing his thoughts and feelings while creeping up his own world line. Nearing its end, Weyl seems freer to say what he feels, perhaps no longer caring who might mock. He gives his fullest avowal yet of what Husserl meant to him, but does not hold back his own reservations; Husserl finally does not help with Weyl’s own deep question about “the relation between the one pure I of immanent consciousness and the particular lost human being which I find myself to be in a world full of people like me (for example, during the afternoon rush hour on Fifth Avenue in New York).” Weyl is intrigues by Fichte’s mystic strain, but in the end Fichte’s program (analyzing everything in terms of I and not=I) strikes him as “preposterous.” Weyl calls Meister Eckhart “the deepest of the Occidental mystics…a man of high responsibility and incomparably higher nobility than Fichte.” Eckhart’s soaring theological flight beyond God toward godhead stirred Weyl alongwith Eckhart’s fervent simplicity of tone. Throughout his account, Weyl interweaves his mathematical work, his periods of soberness after the soaring flights of philosophical imagination, though he presents them as different sides of what seems to his point-eye a unified experience. Near the end, he remembers with particular happiness his book Symmetry (1952) which so vividly unites the poetic, the artistic, the mathematical, and the philosophical, a book no reader of Weyl should miss. In quoting T. S. Eliot that “the world becomes stranger, the pattern more complicated,” we are aware of Weyl’s faithful openness to their strangeness, as well as the ever more complex and beautiful symmetries he discerned in it.

Weyl’s book on symmetry shows the fundamental continuity of themes throughout his life and work. Thinking back on the theory of relativity, Weyl describes it not (as many of his contemporaries had) as disturbing or revolutionary but really as “another aspect of symmetry” because “it is the inherent symmetry of the four-dimensional continuum of space and time that relativity deals with.” Yet as beautifully as he evokes and illustrates the world of symmetry, Weyl still emphasizes the fundamental difference between perfect symmetry and life, with its spontaneity and unpredictability. “If nature were all lawfulness then every phenomenon would share the full symmetry of the universal laws of nature as formulated by the theory of relativity. The mere fact that this is not so proves that contingency is an essential feature of the world.” Characteristically, Weyl recalls the scene in Thomas Mann’s Magic Mountain in which his hero, Hans Castorp, nearly perishes when he falls asleep with exhaustion and leaning against a barn dreams his deep dream of death and love. An hour before when Hans sets out on his unwarranted expedition on skis he enjoys the play of the flakes “and among these myriads of enchanting little stars,” so he philosophizes, “in their hidden splendour, too small for man’s naked eye to see, there was not one like unto another, an endless inventiveness governed the development and unthinkable differentiation of one and the same basic scheme, the equilateral, equiangular hexagon. Yet each in itself — this was the uncanny, the antiorganic, the life-denying character of them all — each of them was absolutely symmetrical, icily regular in form. They were too regular, as substance adapted to life never was to this degree — the living principle shuddered at this perfect precision, found it deathly, the very marrow of death === Hans Castrop felt he understood how the reason why the builders of antiquity purposely and secretly introduced minute variation from absolute symmetry in their columnar structures.”

Weyl’s own life and work no less sensitively traced out this interplay between symmetry and life, field and matter, mathematics and physics, reflection and action.

So rich and manifold are Weyl’s writings that I have tried to include everything I could while avoiding excessive repetitiveness.

Not long after making his epochal contributions to quantum theory, Dirac was invited to visit universities across the United States. When he arrived in Madison, Wisconsin, in 1929, a reporter from the local paper interviewed him and learned from Dirac’s laconic replies that his favourite thing in America was potatoes, his favourite sport Chinese chess. Then the reporter wanted to ask him something more:

“They tell me that you and Einstein are the only two real sure-enough high brows and the only ones who can understand each other. I won’t ask you if this is straight stuff for I know you are too modest to admit it. But I want to know this — Do you ever run across a fellow that even you can’t understand?

“Yes”, replied Dirac.

“This will make a great reading for the boys down at the office,” says I (reporter). “Do you mind releasing to me who he is?”

“Hermann Weyl,” says he (Dirac).

The interview came to a sudden end just then, for the doctor pulled out his watch and I dodged and jumped for the door. But he let loose a smile as we parted and I knew that all the time he had been talking to me he was solving some problem that no one else could touch.

But if that fellow Professor Weyl ever lectures in this town again I sure am going to take a try at understanding him. A fellow ought to test his intelligence once in a while.

So should we —- and here is Professor Weyl himself, in his own words.

Reference: Amazon India link: Mind and Nature (Hermann Weyl) — Peter Pesic.