Schneiderman’s Throwdown

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.

Math Therapy

If mathematics is supposed to be the underlying structure and grain of the universe, but so many people have trouble resonating with it, then why not math therapy — to restore one’s personal harmony with math?  Sometimes I think that, at its best, the part of my job that’s actually teaching math should approach this ideal.  When you peel away the techniques of classroom management, when you peel away the signposts to important facts and theorems — what, ultimately, I mean, are you trying to pass on?

Even more so recently than ever before, I’ve come to feel that the search for interesting and rewarding mathematical topics leads in a plethora of unexpected directions.  I imagine that for many of these, even calculus might be mostly or entirely unnecessary.  Why shouldn’t it be one of the multitude of natural states available to a person to make, and then try to answer, interesting mathematical inquiries?  Why shouldn’t it be an integrated part, if one chooses, of personal, of social life?

Maybe I’ll write a story about a math therapist.

Mathematics, Interpreted Freely

I’ve been interested for a while in Princeton University Press’s annual Best Writing on Mathematics series, edited since 2010 by Mircea Pitici.  In his introduction to the 2015 edition, Pitici writes:

“Interpreting mathematics points toward protean qualities of mathematics not immediately obvious in doing mathematics per se.  An accepted mathematical result is merely the egalitarian premise from which each of us can part with the commonly shared view by interpreting it idiosyncratically, as we please or even as it suits us.  … Lack of reflection on the proper context of applied mathematical thinking perverted the humanities, the social sciences, and even the study and practice of law — to name just a few examples.” (p. xiv)

What a liberating view of mathematics and its relation to the world!  Pitici’s 2012 edition, too, includes an article by one Ian Hacking entitled, “Why Is There Philosophy of Mathematics at All?”.  Hacking argues for two main answers to his title question: one, that it is “astonishment that engenders philosophy of mathematics” (pp. 238-239) — astonishment, that is, that certain abstruse but awe-inspiring mathematical objects are even out there — combined with, two, an idea from “Mark Steiner… who asked more or less my title question, ‘Why is there philosophy of mathematics?’ He answered, in effect, application.” (p. 245)

And, for all those who’re worried that appeals to applied mathematics might sully the philosophical purity of math here — well, never fear, because Hacking distinguishes seven kinds of application, only a couple of which would normally be considered applied math (and the failure to distinguish between which, I imagine, could well have caused Pitici’s “perversions”).  They are: “Math Applied to Math” (about which Hacking states, “Why should there be so much ultimate connectedness behind so much apparent diversity?… This question needs a lot of philosophical work, right now” (p. 248)); “The Pythagorean Dream” (in which “mathematics of a deep and simple sort really is just the structure of reality” (p. 249)); “Mathematical Physics”; “Mission-Oriented Applied Math” (these last two being perhaps what most people think of as applied math); “Common or Garden”; “Unintended Social Uses” (i.e., to preserve an elitist hierarchy, which I find the least interesting inclusion on Hacking’s list); and finally, “Off the Wall”, for which Hacking quotes Wittgenstein: “Why should not the only application of the integral and differential calculus etc. not be for patterns on wallpaper?  Suppose they were invented just because people like a pattern of this kind?  This would be a perfectly good application.” (p. 251)

… The point that Pitici and Hacking are striving for, it seems to me, is that, yes, there is something unsettlingly alien about mathematics, something whose exact nature philosophers have disagreed about for eons, but that, in fact, this very alien quality is what creates (mostly unrecognized) freedoms in how we relate mathematics to the experiential world.  And, in my own experience teaching math with physical objects, certainly, I find a delicate tension between defaulting to instilling the conventional “what you see is an approximation of something mathematically ideal” notions, and something wilder.  The objects that we use to model mathematical ideas never seem to end up being as precise as we want them to be; in a mathematically governed universe, I mean, why would it take so much effort to create something in order to demonstrate mathematics?

de Gua’s Theorem

Last week I was first made aware of de Gua’s Theorem about triangle areas in 3-dimensional space.  It’s (I find) a supremely pretty theorem, easy to state, and not horrendous to prove either: I even attempted to demonstrate a proof I had come up with to some precalculus students this past week.  (Anyone who’s interested in a proof should let me know!) So why had I never heard of it before?  This is the kind of theorem for which I expect several gorgeous demonstrations….

A philosophical poem to kick things off!

There is always more to say….

Quid Est Veritas?

And after the magic’s been sifted through like a dream
curtain: what, again, is truth?
Father Time’s funky sibling clad in the hippest
shorts, tap-dancing ‘long a comet’s beard — you can’t catch
me!  O, it’s a long, long weekend by the mathematicians’ sea-
shore cave, and they’re ingenious with their pebbles
and sand-reckonings; let our Platonic Deity tenderly hand them
up to (well…) some heaven or other.  But Heaven holds no, holds
no enlightenment channel, and truth is Nirvana: Nirvana, the god
who vanishes.

Clouds are not spheres, quoth B. Mandelbrot, mountains are not cones.
So what the hell is a sphere or a cone in the first place?  The Greek
geometers were mystics, and so when they moved the earth,
it, obeying, brought forth factorial enslavements — while
two refugees, sick of time, fled to the Surreal as if
abstractions might dissolve their poison….

There is always more to say.  There
is always more to mean.  Awake
from a snuffly sleep and let the constituents of
the Art, which is your life, settle.  R. Fripp’s Lizard of
Kundalini energy dissolved into Discipline thereafter.  Mandelbrot,
at fractalism’s invention, had outlived the prescribed thoughtspan
of his kind by a decade or more.  What is truth?
It is the man who is before you.