Rob Schneiderman is the author of my favorite article in Princeton University Press’s The Best Writing on Mathematics 2012, edited by Mircea Pitici, which I’ve talked about before in this blog. Schneiderman’s article, “Can One Hear the Sound of a Theorem?”, was the first piece of writing I ever read that finally provided me with some sort of answer to the question, which I’d been struggling with for a while, of what exactly was this vaunted link between music and mathematics I’d heard so much about. Schneiderman’s answer: music and math are both self-contained systems of expression, ones that require no references to the outside world to do what they do. Other comparisons are fluff and bluster.
And really, I’d say, Schneiderman (who apparently knows my great-uncle) is best in his fluff-and-bluster mode. Here’s one choice quote:
The problem is that mathematical content comes in the form of proven statements about well-defined structures, and attempts at “explaining” musical phenomena usually involve structures that are not well defined, with conclusions justified by carefully chosen examples and multitudes of counterexamples ignored. And any logical development of well-defined structure is inevitably based on dubious or pedantic musical principles, so that the resulting conclusions can say precious little about what is important in music. (p.97)
Over and over again, Schneiderman demonstrates that many of the links often drawn between music and math, taken in their entireties as disciplines, are superficial, tenuous, or worthy of extreme skepticism: they fail to get at the heart of what’s so beautiful and gripping in at least one, most likely both, of music and mathematics. Take this excerpt, too:
… after mentioning musical affinities of Galileo, Euclid, Euler, and Kepler, the author [of the book Emblems of Mind, Edward Rothstein] includes Schoenberg, Xenakis, and Cage among a short list of examples that seem to point back from music to mathematics. Even most mathematicians with an affinity for these composers would… surely recognize that this juxtaposition is way out of balance. This comparison leads to such contradictions as claiming the existence of “a systematic logic that guides musical systems” but then admitting later that the great musical compositions “create their own form of necessity, the binding coming not from logic but from the unfolding of ideas…” (p. 106)
The juxtaposition, of course, is way out of balance because the group of twelve-tone composers that includes Arnold Schoenberg, Iannis Xenakis and John Cage can in no way be said to stand in for all of music. And this, indeed, is Schneiderman’s point: that perhaps mathematics can form a basis for certain pieces or indeed compositional styles, but that that basis is merely one choice to be made within the universe of music, and has no bearing on the nature of that universe itself. (As for music forming the basis for mathematics, contrariwise — forget it.) Properly looked at, too, this point of view is liberating. Why indeed must a piece of music unfold according to an internal logic as tight, dare I say foreordained, as mathematics?
Yet Schneiderman implicates even the august Princeton Companion to Mathematics as falling victim to such false comparisons (p. 107). What’s going on here? A desperation, even among mathematicians who should know better, to imbue mathematics with just a drop of the intuitive beauty we all know can arise from a fragment of music? … Or something more sinister: a desire among the quantitative explainers of the world to own music as part of their own field, even if its internal richness is reduced thereby? A notion of mathematics as the golden road to absolute knowledge, outside of which further paradigms are unnecessary…?
To such attitudes, Schneiderman offers a throwdown. Music, as he sees it, exists in a realm apart, and that realm’s dialogue with math’s goes both ways. It’s a good thing, too: my love of math and my love of music combine to result in that much more love.