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?