I did not really follow all of the talk on p-adic Galois groups yesterday, but that doesn’t bother me greatly. As the great genius John Von Neumann once said, “In mathematics you don’t understand things, you just get used to them.” I was already pretty used to the concept of a “group,” which is one of the most beautiful and powerful in higher mathematics. Crudely put, a group is a set of operations that hang together in a nice algebraic way. Group theory was invented back in the early 19th century by Evariste Galois, a young French political hothead who looked like a swarthier version of Leonardo DiCaprio and who died in a duel on the eve of his 21st birthday.
To a mathematician, nothing could be more real than a Galois group, or an endomorphism ring, or a noncommutative Banach space. Yet these entities can’t be found anywhere in the physical universe. Are they just elaborate fictions confined to the brains of the mathematical priesthood? Or are they really “out there,” existing in some higher realm as objectively as Mount Everest? In the middle of a lecture I was giving at MSRI some weeks ago, I asked the audience for a show of hands on this issue. About a third of the mathematicians present proudly admitted to being full-blooded Platonists, convinced that they were devoting their lives to the study of a reality that transcended the material world. Others privately told me that they felt this to be true in their gut, even though in their heads they knew it was metaphysical nonsense.
Though their vision might be grandiose, their tools of inquiry could not be more modest. The halls of MSRI are lined with chalkboards, which in the course of a typical day are covered over several times with runic figures and diagrams. Yellow legal pads are made abundantly available, as are wastepaper baskets. Free coffee is provided by the gallon. (“A mathematician is a machine for turning coffee into theorems,” someone once noted.) That is all it takes to conjure platonic entities into existence. What a contrast to the physicists just down the mountain at the Lawrence Livermore Labs, with their cycotrons and bevatrons and tons of other inelegant hardware. Of course, physicists arrogantly think that they’re the only ones dealing with reality. But, as G.H. Hardy once observed, imaginary universes are so much more beautiful than this stupidly constructed “real” one.
To get to MSRI, I catch a shuttle bus each day from the Berkeley campus. While it grinds its way up the steep and twisting (and surprisingly bumpy) road to the top of the mountain, I typically indulge in a bit of mathematical woolgathering.
This morning I have an epiphany. By using something in logic called the “compactness theorem,” I see a way to prove the consistency of the axiom of choice with the other axioms of set theory—a way far simpler than that discovered by Kurt Godel a half century ago. Thrilled by my seeming stroke of genius, I race through the front door of the institute and buttonhole a few mathematicians in the lobby. They seem faintly impressed by my reasoning, wondering only why no one had thought of it before. Then I run into the great Dutch number theorist Hendrik Lenstra, whom I also regale with my breakthrough. In the gentlest and most courtly way possible, Lenstra points out to me that the compactness theorem presupposes the axiom of choice (in a form known as Zorn’s Lemma, which incidentally used to be the name of a downtown jazz band in New York). In other words, my would-be proof moved in a tight little logical circle. My prospective glory disappeared like the breath off a razor blade.
Now I really feel like a “trivial being.”
