Developing a fitting awareness of the marvels of the human voice became a priority. Somehow, I was assigned for a week to an air base near Aix-en-Provence during the newly founded music festival, modeled after Salzburg’s, which featured opera. Next came a weeklong assignment near the newly revived Salzburg Music Festival, where I heard great performers. Yehudi Menuhin played Bach’s Chaconne for Violin for a score of music students in a very small and ornate room. Wilhelm Furtwängler led the Vienna Philharmonic in a Bach Brandenburg Concerto, conducting from the harpsichord and including his own (strange) cadenza. A young woman ran down the stairs and out of the building, whistling beautifully, and I recognized the famous soprano Irmgard Seefried. Below is a picture of me in Salzburg during that most pleasant and educational week—more than I could have ever imagined from the military. Later, I stopped in Vienna, went to the opera, and heard Carmen, with another great soprano, Hilde Gueden, in a minor role.
(Illustration Credit 11.1)
Opera became a passion. A true opera nut remembers the best performances to his dying day. I soon became a demanding expert on singing in general. One day the radio was broadcasting from Toulouse a stunning concert by the unknown soprano, Victoria de los Ángeles. The announcer mentioned that she was to sing the next day in Paris. I rushed to the Salle Gaveau, bought the cheapest ticket, and got a choice seat in the orchestra. Why? Only a dozen people attended—half of them recognizable artists. When she sang a few months later at the Théâtre des Champs-Élysées, it was jammed.
The ancient, elongated Salle du Conservatoire—a few blocks from home—never sold out. I often dropped in for concerts and “discovered” several other future greats. A young and skinny (!) flutist named Jean-Pierre Rampal played beautifully to a near-empty hall. I also heard the venerable George Enescu—the legendary teacher of Yehudi Menuhin—play in a hall so packed that I was seated on the side of the stage. Bent and arthritic, he held his instrument straight down. To accompany him might have been unfeasible, and indeed I recall no accompanist. He too played the Bach Chaconne (!!) to rapturous fans crying in unison. Of course.
In mid-twentieth-century Paris, the prevailing taste in music was not daring at all: Claude Debussy and Maurice Ravel (long dead)—not to mention Igor Stravinsky—were still widely viewed as wild modernists. This helps explain the virulence of the French musical avant-garde that was to be exemplified by my contemporary Pierre Boulez.
Since then, I have become attuned to more way-out current fare. I boast the composer Charles Wuorinen as a friend and was close to composer György Ligeti, who died in 2006. What brought the three of us together was a special development—the observation that music has a fractal aspect.
13
Life as a Grad Student and Philips Electronics Employee, 1950–52
IN 1950, I BECAME A NOT-SO-YOUNG mathematics student at the University of Paris in search of a good topic for a doctoral dissertation. Unlike today, Carva did not grant doctorates, so I went to the University of Paris—then at a low point in its long and often glorious history. Its requirements for the doctorate had not changed in years, and shortly afterward were made stricter. The course requirements were minimal, and I had fulfilled them with no sweat in 1947. French academia was about to be pushed from ancient anachronism and immobility into perpetual modernization. These exceptional conditions were freewheeling from any viewpoint, and it was a perfect fit for me.
A few one-semester courses taught by regular professors covered scattered specialized topics, and “conference cycles” (ten lectures or fewer) were given by all kinds of short-term visiting professors. There were limited openings for holders of the Ph.D., hence a fear of oversupply, a small number of candidates (called thésards), and no justification for investing in a proper graduate school.
Also, the Paris doctorate came in several flavors, because in the past different political pressures had called for different diplomas with anything but obvious titles. The Doctorat de l’Université de Paris sounded splendid, but requirements were left to the discretion of the faculty, and it had no legal value. It was tailored to foreign students who did poorly and might become hostile to France if sent back without some piece of paper. To be short-listed for a position, the only flavor that mattered was analogous to the German (and now also French) habilitation, the Doctorat d’État ès Sciences.
For the thesis, I was largely left to myself, a widespread and disorderly practice. Many thésards and professors bemoaned it, but for me, disorder was a godsend. Serious teachers and enlightened guides might have done far more harm than good. The orderly United States might have constricted me beyond reason.
Life-Altering Verbal Lashing from Szolem
From the time I quit the École Normale, Szolem had been increasingly bothered by my ways. One day, when I was twenty-eight and stopped by that lifesaving country house near Tulle, he lost his temper, and a polite conversation shifted abruptly to a ferocious verbal lashing, an old-fashioned “visit behind the woodshed.”
“You are like a student I had before the war, reading everything and always ready to discuss a new book or article. I told him that the next time I saw him in the library, I would suspend his scholarship and let him starve. He took it to heart and wrote a beautiful thesis in no time.… It’s a tragedy that he vanished during the war.
“Too many good students are nothing but well-trained monkeys; they know everything they are taught—and nothing more. If you continue to be of that breed, you will become—at best—a slavish scholar … like too many in our family. You can do better. If you want to amount to anything, hurry up and find out what you can do. Settle down—now!”
Szolem’s wife, Gladys, so sweet and almost always close by, repeated the same thoughts more kindly: “You must already have some kind of an idea for a thesis. Try to write it down and see.”
Oddly, this episode worked. It literally turned my worldview around—for a while anyway. I ceased to be a know-it-all intellectual dandy and plunged into a serious search for a doctoral dissertation topic.
Gladys convinced me to think of what I had at hand as a possible topic, and Szolem soured me on “well-trained monkeys.” Unlike Szolem, I enjoy intellectual fencing and occasionally showing off. Otherwise—like Szolem—I absolutely stopped having patience for their games.
I do not deny that plain old-fashioned scholarship is a source of enjoyment, including the hunt for old, musty books hidden on hard-to-reach library shelves. Szolem viewed having a quick memory as detrimental to creativity, but in my case it has been neither detrimental nor an empty distraction. Also, the Keplerian style of research that I came to practice happens to be powerfully assisted by flipping through reference books and forgotten texts. The goal is not to copy them passively into one’s memory but to link them to one another over high intellectual walls or across wide intellectual abysses. My memory has been a key asset—so far.
A Flawed Ph.D. Dissertation Well Ahead of Its Time
The “woodshed” episode made me listen to Szolem more than usual. For a thesis topic, he suggested a theory to which I have already alluded, one originated in the 1910s by the mathematicians Gaston Julia and Pierre Fatou and now called quadratic dynamics. I did my best but soon gave up—much to Szolem’s consternation—because the topic seemed hopelessly “stuck” and, perhaps, because I was a young rebel. Only after a deliberation that lasted thirty years did I feel up to facing quadratic dynamics, and I discovered something that became its most recognized icon—the Mandelbrot set.
Instead, I wrote a somewhat strange two-part dissertation for the Doctorat d’État ès Sciences, which was soon overtaken by far better work. But it largely determined the course of my life and—arguably—the work that led to changes in the course of several sciences.
The first part of the dissertation concerned George Kingsley Zipf’s universal power law distribution for words. The other part was an incursion into the foundation of an ancient area of physics: generalized statistical thermodynamics. One of my mo
dels of word frequencies relied on that second part in a very exotic form. Unfortunately, this mixture was dreadful academic politics. More important, my thoughts in physics were still very much in flux. In fact, they took many more years to become ready for publication. In 1952, this combination was viewed as wild and every onlooker warned me that it would in no way be perceived as natural. The stretch across the abyss between fields was too extreme. In addition, the first part presented a subject that did not yet exist, and my main goal was not to help linguistics become mathematical but to explain Zipf’s law.
Why the rush? At a meeting in London, I had been offered a postdoc at MIT. The desire to take off pushed me to cram everything I had at hand into the dissertation. Lacking any advice, my result was unfinished, and I presented it in a grossly incompetent style.
Any half-friendly professor would have rejected my topic. But I had no Ph.D. adviser. All that mattered was finding a well-disposed chairman powerful enough to place me on the short list of prescreened candidates for an academic job. Even if my choice had been less exotic, the selection of advisers was pitifully small because the science faculty at the University of Paris had very few professors. Every Ph.D. dissertation had to be printed, and the page facing the title page had to name all the professors, irrespective of discipline. An amazingly small list! Even worse, the few professors at Carva and the Collège de France were denied the right to supervise Ph.D.’s.
Having an actual adviser was a novelty at that time in Paris. Szolem had not had one; only after Szolem’s defense did Hadamard read his thesis and become his patron. Szolem told me of a case when he saw Hadamard livid that someone had asked him for a thesis topic and supervision. “Can you imagine that? If he has no topic of his own, he should not even think of a Ph.D.!”
A report had to be written, and a Ph.D. committee had to be selected. By default, the task fell to the sitting professor of probability theory and mathematical physics. That chair’s prestige had climaxed with the great Henri Poincaré. Paul Lévy had amply deserved and desperately wanted it, but the Sorbonne faculty first chose the miscast Maurice Fréchet (1878–1973), and then a scientific lightweight, savvy statistician named Georges Darmois (1888–1960), who was moonlighting as the manager of his wife’s iron foundry.
Darmois was by nature unfriendly, and we always talked standing in the corridor with many others waiting around for their turn. Taking on an additional Ph.D. student demanded little time and made him look good. He probably took it for granted that I would continue at Philips of Holland and therefore merely glanced at my thesis while on an airplane, having decided in advance—without telling me—that I would only make the meaningless long list of candidates for a job, not the desirable short list.
In any event, how should this dissertation be pigeonholed? The science faculty had no formal departments, and Darmois’s chair overlapped mathematics and physics. I could choose either, with inescapable consequences. But my plight interested nobody, least of all Szolem.
Luckily, I chanced to cross paths on the street with the physicist Alfred Kastler (1902–84), a close friend of Szolem and an exceptionally nice man, whom I had met when I was twelve. Later in life, after receiving the Nobel, he proclaimed that a lifelong collaborator deserved equal honor. Splitting the medal was not possible, but he gave Szolem half of the money. Later, he published a book with a French title that translates as Poems in German by a French European. Indeed, he was born in Alsace, and until he entered the École Normale on a special deal, spoke only German. During the war, he never yielded to Hitler, kept Szolem’s apartment safe by living there, and left as soon as Szolem came back. This man was attuned to nuance and comfortable with living between two cultures. The perfect man to consult. We stopped to chat. I sketched my thesis and described my quandary.
He sighed with foreboding and confirmed that one must not combine two very different topics—especially when neither would lead to a job. Thermodynamics was inactive and jobs no longer existed; quantitative linguistics did not yet exist. In physics, I would compete with a relative flood of strong dissertations on currently fashionable topics. Kastler realized that fate was on my side. Mathematics offered one big advantage. The level of abstraction of nearly all the new dissertations had become so extreme that Kastler and fellow physicists had seen enough. Some new openings were to be reserved for applied mathematics, and—miracle—my chances of landing a job might in fact be rather good.
This advice proved wise beyond any short-term consideration of bureaucratic politics. In order to accommodate my longings around 1950, and also my life’s accomplishments in physics or mathematics, the scope of either science must be given a very broad interpretation. It used to be that terms like “mathematics” were defined broadly, and physics actually began as a corner of mathematics. A hundred years ago, however, it separated from mathematics and from engineering. Only recently has physics expanded again, both in directions where it becomes hard to distinguish from mathematics and in directions (hard and soft materials) where it becomes hard to distinguish from engineering. For example, several Nobel Prizes have recently been granted in physics for work that in decades past might not have qualified. Back in 1950, when it had a big impact on my life, Kastler was right that my work belonged in broadly understood mathematics.
Darmois agreed, but decided that my committee’s chairman need not be a mathematician. His unexpected choice was Prince Louis de Broglie (1892–1987)—the official reasons being that de Broglie publicly praised interdisciplinary work and that my thesis would benefit from association with a broad-minded professor familiar with flying solo. Twenty-five years earlier, this aristocrat had been a key contributor to quantum theory.
The cover of every French dissertation of that day referred to a second thesis, with the universal title Propositions données par la Faculté, which was never published. This all-purpose boilerplate denoted a reading course requirement meant to amplify and balance the skimpy graduate course work. By unwritten tradition, the topic had to be very different from the main one. A heavily computational first (real) thesis could be balanced by assigning a philosophical issue.
My assignment was long and heavily computational: the then-recent Ph.D. dissertation of the mathematician Yvonne Choquet Bruhat (1923–) asked a key question: Do the equations of gravitation discovered by Albert Einstein have a solution and only one? Physicists found this question of no interest, but mathematicians found it very difficult and hence fascinating. Bruhat managed to prove that it was sufficient for the initial conditions to have well-behaved derivatives at least to the magic order of 7.
At my thesis defense, I was reporting on this proof elegantly enough when Darmois suddenly broke in. “Your presentation was excellent. But could you tell us more specifically why the topic of your second thesis is important?” The second thesis was a report on a very long recent article on gravitation—a fine but early and provisional technical stage in a long-range program that was bound to (and did) proceed much further. As I was fumbling for a suitably noncommittal response, Darmois smoothly took over. He turned ninety degrees to address face-to-face the committee chairman—who was none other than de Broglie. We were informed that something Darmois had written in the 1920s on relativity theory deserved mention.
At that point, it became limpidly clear why Darmois agreed to report on my thesis topics and suggested the topic he did for the second. He was campaigning for election to the Académie des Sciences as an astronomer—his early area of expertise. So he welcomed an opportunity to give an uninterrupted twenty-minute presentation to the academy’s perpetual secretary, de Broglie. This political maneuver soon succeeded.
At no time did I feel that any member of my Ph.D. committee gave serious thought to the content of my thesis. Since then, my experience with these committees has made me realize that mine faced an impossible task. An unwise and dreadfully rushed presentation did not help, and my excuse—a waiting postdoc in the United States—was weak. Even a flawless job would not h
ave affected the fundamental obstacle noted by Kastler: my key topic was far from any mainstream. At the time, even I did not know that my dissertation was but a seed from which a mighty tree was to rise.
Do I Regret This Messy Doctorate?
I do not regret my messy doctorate in the least. It was preferable to falling under the supervision of a maniac who would delay me until I had been shaped to fit his own agenda. On several occasions in my life, an element of freewheeling in the system proved a blessing.
The irony of my failing to be short-listed in 1952 is that it no longer mattered by 1956. Enrollments were exploding, jobs were opening everywhere in France, and every passed-over warm body languishing in a dark corner was short-listed. Darmois telephoned (!) to inform me that I was urgently needed; in fact, he allowed me the luxury of selecting Lille. I could live in Paris, with a commute of only two hours.
Recounting those long-past events never fails to both amuse and hurt. The choice of a Ph.D. label was one of many critical decisions I faced in my life with no precedent to help. In each instance, a wrong choice might have thrown my life orbit in a totally different and possibly very unfortunate direction. Furthermore, those critical choices had steadily increasing consequences. As a result, the great promise I had held at age twenty had largely dissipated. In between, I suffered many bruises and not a few indignities. But in hindsight, I was the only one to blame, or to praise, since my lowly dissertation turned out to be the germ of all I went on to accomplish.
The Fractalist Page 15