I met Irving Segal in 1982 shortly after I came to MIT in order to get my Ph.D. in mathematics. As a slouching, scruffy grad student who preferred to be barefoot whenever possible, I was somewhat intimidated by his appearance. He was always impeccably dressed in a suit, he wore a goatee shaved short in a no-nonsense sort of way, and he made up for his lack of height by an erect posture and commanding manner. However, I decided to work with him because of all the pure mathematics faculty he seemed the most passionate about physics — not just as a source of math problems, but as an end in itself.

I wanted to work on quantum gravity, but at MIT everyone interested in this subject was working on superstrings, for which I had little taste. Segal himself found Einstein's equations too ill-behaved to bother trying to quantize them. The lack of a conserved energy, the tendency for solutions to develop singularities — all these qualities convinced him that general relativity was fatally flawed. My arguments failed to convince him, so I wound up working on one of his specialities, the mathematical foundations of quantum field theory.

I learned a lot and successfully completed a thesis, but I didn't have
much success proving really interesting theorems. Later, as a
postdoc, I decided that quantum field theory was too hard for me, so I
worked with Segal and Zhengfang Zhou on classical field theory —
i.e., nonlinear wave equations, another of Segal's specialties. The
three of us wrote some papers together and also coauthored a
book, *Introduction to Algebraic and Constructive Quantum Field
Theory*, summarizing Segal's work on quantum fields. Thus I spent
about 6 years in close contact with him and came to know him rather
well.

We would typically discuss mathematics in his office, taking turns scribbling equations on the blackboard. He had a devastating way of expressing doubt when my reasoning failed to convince him. Without saying a word, he would gradually raise his eyebrows higher and higher as I spoke. As they slowly climbed up his forehead, it became ever more difficult to keep up the momentum of my reasoning. When I finally lost the thread of what I was saying, he would interrupt and point out my error as he saw it. Being stubborn, I would not always accept these criticisms. As he was even more stubborn, our discussions sometimes became quite heated. Zhengfang Zhou served as a calming influence when he was around.

Segal's office was a cozy, lived-in place, cluttered with decades of accumulated papers. He had a couch where sometimes he would take short naps. He also made coffee in his office, refusing to touch the stuff served in the math department lounge. He took coffee very seriously, grinding the beans in his office, using only distilled water, and heating it to a precisely optimized temperature. (He claimed to have done a study to determine this optimal temperature.) He often let me work on his computer while he worked at his desk or typewriter. Sometimes when he wanted to prove a theorem he made a great show of setting a kitchen timer, allowing himself no more than 30 minutes to get the job done. This was but one of many ways he emphasized the importance of a businesslike attitude. When I passed my thesis defense, the first thing he said was "Good, now we can get back to work." He never slacked off; he often came to the office on weekends, and his retirement seemed not to slow him down in the least.

Everyone who knows Segal will recall his inability to do things any way other than his own. He was never one to accept something merely because others did. At various times I recall him making scathing criticisms of all the scientific disciplines in which he engaged. He was appalled by the lack of mathematical rigor shown by most theoretical physicists. He compared astronomers who put a lot of work into studying individual stars or galaxies to stamp collectors, and he repeatedly told me that explaining modern statistical methods to cosmologists was like trying to explain Western medicine to a witch doctor. On the other hand, he roundly criticized statisticians and probability theorists for their obsession with measure spaces as opposed to algebras of observables. He especially ridiculed mathematical purists who were unmotivated by applications, comparing them to tailors so engrossed in their art that they would as gladly make a shirt with three arms as with two. He often mocked "Bourbachic", and regarded a lot of mathematical formalism as the equivalent of "gold-plating the carburetor", saying "it may look nice, but it doesn't run any better."

People who failed to understand the essentially prickly nature of
Segal's relationship to the world would sometimes misinterpret his
actions. For example, he recently wrote a review of Alain Connes'
Noncommutative Geometry for the Bulletin of the A.M.S.. While largely
positive, at least by Segal's standards, the review contained a number
of serious criticisms. For example, he expressed disappointment that
Connes, with all his mastery of analysis, still treated quantum field
theory the way most particle physicists do, using perturbative
Lagrangian methods, rather than the more rigorous framework of
algebraic quantum field theory pioneered by Segal and others. Some
mathematicians were so upset by these criticisms that they organized a
special series of articles lauding Connes' work in the *A.M.S. Notices*
as a kind of gesture of apology. What they perhaps failed to
understand was that merely by writing the review, Segal was saying that
Connes' work was of the highest caliber! Indeed, the only other articles
I recall him writing about other people's work concerned von Neumann and
Wiener. Perhaps a joke Segal used to tell will clarify my point here.
When asked about the University of Chicago, where he worked before
coming to M.I.T., he would say in a deprecatory tone, "Oh, it's not very
good — just the best that there is."

In his earlier years Segal's tendency towards radicalism served him well. Long before most, he recognized that the algebra of observables is more fundamental in quantum physics than any particular representation of this algebra on a Hilbert space. His work on C*-algebras, infinite-dimensional integration theory, nonlinear semigroups, constructive quantum field theory — they all have a boldness typical of him. In his later years, unfortunately, I think his willingness to ignore the conventional wisdom led him astray. He developed an alternative to the big-bang cosmology in which redshifts were to be explained, not by the expansion of the universe, but by an effect of conformal geometry. According to him, his theory predicted a quadratic redshift-distance relation instead of the usual linear one. He spent a lot of time statistically analyzing redshift-brightness data for quasars and galaxies and wrote papers claiming they supported his theory. Most astronomers disagreed, and he became quite a pariah in the astronomy community.

I cannot judge his analysis of the data, but I thought long and hard about his derivation of the quadratic redshift-distance law from his theory, and it never seemed right to me. At first I hoped I was making a mistake, so I tried to get him to explain this derivation. His explanation did not convince me. Later I tried to explain what I thought was his error. He became quite angry. When I realized we would never see eye-to-eye on this subject, I tried to avoid it. But this was very difficult, since cosmology became ever more the center of his work as the years went by. Our relationship became even more strained when I finally started working on my real interest: quantum gravity. Since he didn't really believe in quantum gravity, and I didn't believe in his cosmology, and neither of us was very good at small talk, it was hard to know what to say. I'm sad to say that I eventually wound up avoiding him.

Despite this, I remain very fond of Segal, because he had a real passion for understanding the universe. He did not believe in God and was suspicious of all forms of organized religion. The quest for perfection which some express through religion, he expressed through mathematical physics. He could never take it lightly!

© 1999 John Baez

baez@math.removethis.ucr.andthis.edu