Yau’s Voice, PopularizedDecember 14, 2010
Philip J. Davis
The Shape of Inner Space: String Theory and the Geometry of the Universe's Hidden Dimensions. By Shing-Tung Yau and Steve Nadis, Basic Books, New York, 2010, 377 pages, with an extensive glossary and notes, $30.00.
Shing-Tung Yau, a professor of mathematics at Harvard, is a polymath who has been honored with multiple awards, including the Fields Medal (1982). Steve Nadis is a science aficionado and writer who has made contributions to several dozen books. It would appear from a quick initial scan that the goal of this present collaboration is to provide a popularization of string theory. To ease the reader in, the authors have provided historical developments and biographical bits. We can read about quarks and quantum tunneling, singularities and supersymmetry. Regarding strings, it turns out that there are at least five varieties---which was news to me. There are constant references or puffs to Calabi–Yau manifolds. (The senior author is no shrinking violet.)
Actually, it is not clear who the intended readers of this book are. I know from experience that popularization is a tricky business. A writer who simplifies in pursuit of wide comprehension risks criticism from the mavens that this or that hasn't been stated correctly. Is it fatuous to think that a reader who needs an explanation of complex numbers a + ib on page 81 will be able to understand on page 231 what it means that four authors, whose joint identity is abbreviated KKLT, were able to create a stable space–time?
My level of understanding about such matters is pretty close to that of the man on the street. Popularizations of contemporary developments often fail to connect with me. What I was able to get from reading a bit here and a bit there was entrée into the mindset of the numerous physicists and mathematicians whose work is cited. Yet while I could sense the mindset, only the slightest whiff of understanding came through. Nonetheless, even a whiff of understanding is something positive. My experience in reading this book may be akin to that of a kibitzer in the presence of some moment of high creativity---perhaps of an onlooker in the atelier of Titian, watching how he painted the famous equestrian portrait of Charles V.
I skip now to the penultimate chapter, titled "Truth, Beauty and Mathematics," on which I feel rather more capable of venting some thoughts. The basic question is whether a theory of mathematical physics that describes the world with some accuracy must be beautiful or, conversely, whether a beautiful theory has a leg up in describing the world. Chapter 21 of my book Mathematics and Common Sense (2006) takes up this question, and I hope the reader will excuse me if I borrow just a bit from it.
People have been mulling over these questions for millennia. Pythagoras opined that "All is number." Plato noted that "God ever geometrizes" (quoted with approval by Yau and Nadis on page xvii). All well and good, but when is mathematical beauty present in geometry, or elsewhere? Is mathematics, by its essential nature and substance, automatically beautiful? Can there be no ugly mathematics? I've heard it asserted by purists that industrial math is ugly math. Listen to one response to the late Vladimir I. Arnold's remarks on this question:
"Very often we come across people who declare that some classes of mathematics is "ugly" or "boring". When I was a young student, such a class was non-existent---there was only mathematics that I could understand and that which I could not. That which I could not understand was of course more challenging and required greater effort. Of course, some mathematics becomes boring because we understand it well and do not see any challenges in pursuing it. However, a more common context in which people call some subject ugly is when they do not understand the point of the exercise. In this case it probably needs more study and a clearer perspective to see the worth of what one's colleagues are doing. Ugliness, like beauty is often in the eye of the beholder."
An adequate discussion of this question would thus have to deal with, among other things, ideas of beauty, of simplicity/complexity, of what is subjective and what is objective, of what is utility as opposed to pure aesthetics. If we believe, erroneously, that truth is beauty, as in the ode of John Keats, we may very well ask what truth is, as---according to Francis Bacon---"jesting Pilate" did, although he ran away without waiting for an answer.
Beauty = truth in mathematical physics? This has led to a great debate. Steven Weinberg wrote:
"I once heard Dirac say in a lecture which consisted largely of students that students of physics shouldn't worry too much about what the equations of physics mean, but only about the beauty of the equations. The faculty members present groaned at the prospect of all our students setting out to imitate Dirac."
Well, Dirac was a top-down thinker. He believed that he could describe the world by pure and beautiful mathematical ideas, culled, as it were, from pure cogitation. His genius (or perhaps his luck) was that he was able to derive physical consequences from his equations, rather than the reverse. Thus, he predicted the existence of antiparticles, subsequently discovered in detailed studies of cosmic rays.
On the other hand, as philosopher and historian of science James McAllister wrote,
"Whether beauty in theories is a reliable indicator of proximity to the truth depends ultimately on empirical facts about the world. I shall argue that the evidence so far is negative. . . . Aesthetic preferences . . . have played a conservative role in scientific revolutions."
To put it more strongly, revolutions in science alter the underlying aesthetic.
To conclude, what Yau appears to be asserting---and I agree heartily---is that it's a wonderful thing that the real world leads to beautiful mathematical theories, even though such theories may not describe the world. Models may come and models may go, but the mathematics is forever. Yay!
Philip J. Davis, professor emeritus of applied mathematics at Brown University, is an independent writer, scholar, and lecturer. He lives in Providence, Rhode Island, and can be reached at firstname.lastname@example.org.