Hyperspace is the Place

Writing about music, the saying goes, is like dancing about architecture. The implication is that even trying is futile and likely to make the person doing so look absurd.

The line has been attributed to various musicians over the years – wrongly, as it happens, though understandably, given how little of what they do while playing can be communicated in words to people who don’t know their way around an instrument. I don’t know if mathematicians have an equivalent proverb, but the same principle applies. Even more strictly, perhaps, since most nonmathematicians can’t even tell when things go out of tune. And in many of the higher realms, math drifts far from any meaning that could ever be expressed outside whatever latticework of symbols has been improvised for the occasion. (Kind of like if Sun Ra went back to performing on his home planet.)

Against all odds, however, there is good writing about music – as well as The Best Writing on Mathematics 2012, the third anthology that Mircea Pitici has edited for Princeton University Press in as many years. He is working toward a Ph.D. in mathematical education at Cornell University, and teaches math and writing there and at Ithaca College. A majority of the pieces come from journals such as Science, Nature, The Bulletin of the American Mathematical Society and The South African Journal of Philosophy, or from volumes of scholarly papers. But among the outliers is an article from The Huffington Post, as well as a chapter reprinted from an anthology called Dating Philosophy for Everyone: Flirting With the Big Ideas (Wiley-Blackwell, 2011)

There’s also a paper from the proceedings of the Fifth International Meeting of Origami Science, Mathematics, and Education, which opens with the sentence: “The field of origami tessellations has seen explosive growth over the past 20 years.”

Chances are you did not know that. It came as news to me, anyway, and I cannot claim to have followed every step of the presentation, which concerns the algorithms for creating fantastically intricate designs (resembling woven cloth) out of single flat, uncut sheet of paper.

The author, Robert J. Lang, is a retired specialist in lasers and optoelectronics; his standards of numeracy are a few miles above the national average, even if the math he’s using is anything but stratospheric. But Lang is also an internationally exhibited origami artist. The images of his work accompanying the article offer more than proof of what his formulas and diagrams can produce; they are elegant in a way that hints at the satisfaction the math itself must have yielded as he worked it out.

Other papers make similar connections between mathematics and photography, dance, and (of course) music. But one of the themes turning up in various selections throughout the book is the specificity of what could be called mathematical pleasure itself, which can’t really be compared to other kinds of aesthetic experience.

In his essay “Why is There Philosophy of Mathematics at All?" Ian Hacking -- retired from a university professorship at the University of Toronto – considers the hold that math has had on the imagination and arguments of (some) philosophers. Not all have been susceptible, of course. Among humans, “a high degree of linguistic competence is [almost] universally acquired early in life,” the ability “for even modestly creative uses of mathematics is not widely shared among humans, despite our forcing the basics on the young.” And as with the general population, so among philosophers.

But those who have thought carefully about mathematics (e.g., Plato and Husserl) or even contributed to its development (Descartes and Leibniz) share something that Hacking describes this way:

“[T]hey have experienced mathematics and found it passing strange. The mathematics that they have encountered has felt different from other experiences of observing, learning, discovering, or simply ‘finding out.’ This difference is partly because the gold standard for finding out in mathematics is demonstrative proof. Not, of course, any old proof, for the most boring things can be proven in the most boring ways. I mean proofs that deploy ideas in unexpected ways, proofs that can be understood, proofs that embody ideas that are pregnant with further developments…. Most people do not respond to mathematics with such experiences or feelings; they really have no idea what is moving those philosophers.”

Beyond the pleasure of proof (“Eureka!”) lies unfathomable mystery – of at least a couple of varieties. One is the problem addressed in “Is Mathematics  Discovered or Invented?” by Timothy Gowers, a professor of mathematics at Cambridge University.  Be careful how you answer that question, since the nature of reality is at stake: “If mathematics is discovered, then it would appear that there is something out there that mathematicians are discovering, which in turn would appear to lend support to a Platonist conception of mathematics….”

Or to put it another way and leave Plato out of it for a moment: If “there is something out there that mathematicians are discovering,” then just exactly where is “out there”? Answering “the universe” is dodging the question. We might naively think of arithmetic or even some parts of geometry as some kind of generalization from observed phenomena, But nobody has empirical knowledge of a seven-dimensional hypersphere. So how – or again, perhaps more pertinently, where, in what part of reality – does the hypersphere exist, such that mathematicians have access to it?

A neurobiological argument could be made that the higher mathematical concepts exist in certain cognitive modules found in the brain. (And not in everyone’s, suggests Hacking’s essay.) If so, it would make sense to say that such concepts are created. But if so, the mystery only deepens. Scientists have repeatedly found the tools for understanding the physical universe in extremely complex and exotic forms of mathematics developed by pure mathematicians who not only have no interest in finding a practical application for their work, but feel a bit sullied when one is found.

Translating math’s hieroglyphics into English prose is difficult but – as the two dozen pieces reprinted in Best Writing show – not always completely impossible. Mircea Pitici, the editor, pulls together work at various levels of complexity and from authors who pursue their subjects from a number of angles: historical or biographical narrative, philosophical speculation both professional and amateur, journalistic commentary on the state of math education and its discontents.

And the arrangement of the material is, like the selection, intelligent and even artful. Certain figures (the 19-century mathematicians Augustus de Morgan and William Hamilton) and questions (including those about math as experience and mystery) weave in and out of the volume -- making it more unified than “best of” annuals tend to be.

That said, I am dubious about there being a Best Writing ’13 given the dire implications of certain discoveries (or whatever) by Mayan numerologists. This will be the last Intellectual Affairs column for 2012, if not for all time. I’d prefer to think that, centuries ago, someone forgot to carry a digit, in which case the column will return in the new year. And if not, happy holidays even so.