Devlin's Angle

July/August 2001

Witten at 50

Edward Witten -- the man who has often been described as the Isaac Newton of the modern age -- celebrates his fiftieth birthday on Saturday 26 August this year. I am sure that many mathematicians will want to join with me in wishing him the very best as he passes the half-century mark. Which, on the face of it, is a little odd, since Witten is not a mathematician but a physicist, a Professor in the School of Natural Sciences at the Institute for Advanced Study in Princeton, New Jersey. Why would mathematicians want to celebrate the birthday of a physicist -- other than the obvious fact that it's always good to have an excuse for a party? Can Witten really be compared to the great Isaac Newton, the seventeenth century genius who brought us not only powerful scientific theories of light and of gravity but calculus as well?

I believe the comparison is entirely apt, in several ways. First, though, a few words about Witten the man. He was born 26 August 1951 in Baltimore, Maryland. He studied at Brandeis University, where he received his BA in 1971. From Brandeis he went to Princeton, where he received an MA in 1974 and a Ph.D. in 1976. He was a postdoctoral fellow at Harvard for 1976-77, then a junior Fellow there for 1977-80. In 1980 he was appointed a Professor at Princeton University, from where he moved across town to the Institute for Advanced Study in 1987.

Now to the comparison of Witten with Newton. Although there is no doubt that Witten is a physicist, like Newton he is a powerful mathematician. Very much in the tradition of Newton, Witten's mathematics arises out of physics -- he does his mathematics in order to advance his -- and hence our -- understanding of the universe. The British mathematician Sir Michael Atiyah has written of Witten that:

"... his ability to interpret physical ideas in mathematical form is quite unique." [Michael Atiyah: On the work of Edward Witten, Proceedings of the International Congress of Mathematicians, Kyoto, 1990 (Tokyo, 1991), pp.31-35.]
Atiyah's words were written in 1990, on the occasion when the international mathematical community give Witten their most prestigious award, a Fields Medal, often described as the mathematicians' equivalent of a Nobel Prize. In addition to the Fields Medal, physicist Witten has been invited to give two major addresses at national meetings of the American Mathematical Society: he was AMS Colloquium Lecturer in 1987 and three years ago, in 1998, he gave the Gibbs Lecture.

Like Newton, the physics Witten does is deep, fundamental, and center stage. Both men set out to answer ultimate questions about the nature of the world we live in. In Witten's case, he works in the hot research areas of supersymmetry and string theory.

Just as questions in physics led Newton to develop some far reaching new mathematics that found many applications, often well outside of physics, so too Witten's mathematics has been of a depth and originality (and incidentally of a difficulty equaled by few mathematicians) that will surely find other applications. Witten has used infinite dimensional manifolds to study supersymmetric quantum mechanics. Among the results for which he was awarded a Fields Medal was his proof of the classic Morse inequalities, relating critical points to homology.

Witten's work in manifold theory brings up yet another comparison with Newton. Neither of them were concerned with finding mathematically correct proofs to support their arguments. Relying on their intuitions and their immense ability to juggle complicated mathematical formulas, they both left mathematicians reeling in their wake. It took over two hundred years for mathematicians to develop a mathematically sound theory to explain and support Newton's method of the infinitesimal calculus. Similarly, it might take decades -- maybe even centuries -- before mathematicians can catch up with Witten. Commenting on this state of affairs in a presentation at a Millennium Meeting at the University of California at Los Angeles, in August 2000, Witten said:

"Understanding natural science has been, historically, an important source of mathematical inspiration. So it is frustrating that, at the outset of the new century, the main framework used by physicists for describing the laws of nature is not accessible mathematically.'' [Edward Witten: Physical Law and the Quest for Mathematical Understanding.]
British mathematician Sir Michael Atiyah, wrote of Witten's work:
"... he has made a profound impact on contemporary mathematics. In his hands physics is once again providing a rich source of inspiration and insight in mathematics. Of course physical insight does not always lead to immediately rigorous mathematical proofs but it frequently leads one in the right direction, and technically correct proofs can then hopefully be found. This is the case with Witten's work. So far the insight has never let him down and rigorous proofs, of the standard we mathematicians rightly expect, have always been forthcoming." [Michael Atiyah: On the work of Edward Witten.]
This feature of Witten's work not only tells us that Witten is a remarkable physicist, it also says something about mathematics. For all that mathematics is a product of the human mind, the very logical rules that have to be satisfied for mathematical creations to be mathematics means that there is -- potentially -- a shortcut path to mathematical truth that avoids the long and painful "official" route of logically correct, step-by-step deductions we call proofs. For most mathematicians, myself included, the only way to convince ourselves that something is true in mathematics is to find a proof. A very small number of individuals, however, seem to be blessed with such deep and powerful insight that, guided by little else besides their intuitions and a sense of "what is right", they can cut through the logical thickets and discover the truth directly -- whatever that means. Newton did it with calculus. The great Swiss mathematician Leonhard Euler did much the same thing with infinite sums in the eighteenth century. Arguably the Indian mathematician Srinivasa Ramanujan did something similar with the arithmetical patterns of numbers he discovered. And now Witten is doing the same with infinite dimensional manifolds. On several occasions, Witten has made a discovery -- a physicist's discovery since it is technically not a mathematical discovery -- that mathematicians subsequently showed to be "correct" by the traditional means of formulating a rigorous proof. Given the complexity of the "insights" that Newton, Euler, Ramanujan, and Witten have made -- and the difficulty of the subsequent proofs -- this cannot be a case of making lucky guesses. So what is going on?

As a mathematician, when I work on a mathematical problem, my sense is very much one of discovering facts about some pre-existing (abstract) world "out there". If I try hard enough, and am lucky, I'll discover the right path that leads me to my goal, and I'll solve the problem. If I fail, sooner or later someone else will come along and find the path. Very likely I'll then see that it's the very path I was trying to find!

Nevertheless, for all that mathematical research feels like discovery, I firmly believe that mathematics does not exist outside of humans. It is something we, as a species, invent. (I don't see what else it could be.)

But mathematical invention is not like invention in music or literature. If Beethoven had not lived, we would never have heard the piece we call his Ninth Symphony. If Shakespeare had not lived, we'd never have seen Hamlet. But if, say, Newton had not lived, the world would have gotten calculus sooner or later, and it would have been exactly the same! Likewise, if Witten had not lived we'd have obtained his results eventually. (Although the wait would almost certainly have been much longer for Witten's work than it was for calculus.)

Since mathematical creativity is not tied exclusively to one particular individual, the patterns of mathematics must tell us something very deep and profound about the human brain and the way we interact with our environment. If you want, you can "reify" (objectify) the results of that interaction and think of it as an "outside (Platonic) world". But to my mind that's just playing with words. A much more honest way to think of it, I suggest, is that our mathematical creations arise from the world we live in and are constrained by our experience of that world. This means that, very occasionally, a person such as Ed Witten can come along whose understanding of the physical world is so good that he can bypass the normal methods of mathematical discovery and "see", directly, the results that the rest of us can, at best, stumble upon.

The result is that mathematicians are able to get occasional glimpses of the mathematics of tomorrow -- or possibly even the next century. For me, and I'm sure for many fellow mathematicians, that alone is good reason to say:

Happy fiftieth birthday, Ed Witten.


Devlin's Angle is updated at the beginning of each month.
Mathematician Keith Devlin ( devlin@csli.stanford.edu) is the Executive Director of the Center for the Study of Language and Information at Stanford University and "The Math Guy" on NPR's Weekend Edition. His latest book is The Math Gene: How Mathematical Thinking Evolved and Why Numbers Are Like Gossip, published by Basic Books.