Is Mathematics Unreasonably Effective?

In yet another confirmation of Betteridge’s,

I argue that aesthetic virtues in mathematics are highly correlated with more general theoretical virtues—simplicity, generality, explanatory power, etc—of the sort that play an important role within scientific practice. (Waxman 2)

Cliffs Notes: it’s plausible that the so-called ‘unreasonable effectiveness of mathematics in the natural sciences’ — an observation first attributed in that form to Eugene Wigner — is less ‘unreasonable’ than it appears.

Why? Because while mathematics itself may be an a priori discipline — in the methodological sense, if not in the epistemic — the aesthetic sensibilities of mathematicians are formed in no such vacuum.

Evolution by natural selection was what generated humans and, with us, our ‘cultures’ and our ‘aesthetic preferences’. Aesthetics is closely entangled with the physical world. It’s also closely entangled with other intellectual ‘virtues’.

The explanatory challenge posed by the ‘applicability of mathematics’ to the natural sciences thus dissolves in the face of two observations:

  1. the judgments that mathematicians call ‘aesthetic’ are (often) barely separable from the identification of theoretical virtues — such as simplicity, unificatory power, parsimony, importance — which are themselves generated by connection to the demands of the real world; and,
  2. humans find ‘aesthetic’ virtue in things that are also correlated with being useful abstractions over the world.

Waxman gestures to some fruitful examples:

What should we call the variety of argument that is in general being prosecuted by Waxman in this paper? ‘Argument from correlation of culture to world’? ‘The naturalist’s sanity-check’?

In some sense, it’s an obvious argument. Little more than: “an external world exists and substantially optimises humans via their preferences, actually”. Yet it’s also one that seems systematically underrepresented in the broader spaces of philosophy-of-science and science & technology studies.

I think there’s a one-level-more-meta- insight, here, though.

Others have explored the so-called ‘unreasonable’ effectiveness. (The Quinean naturalism and ‘realism’ in Sorin Bangu’s The Applicability of Mathematics in Science: Indispensibility and Ontology seems relevant.) This by itself isn’t a massive contribution.

In ‘Is Mathematics Unreasonably Effective?’, what Waxman is doing is casting doubt on whether we ever ought to have taken ‘complete ineffectiveness’ as the prior. I see a similar point made gesturally by a literal banana in ‘Against Automaticity’, and by Adam Mastroianni in ‘I’m so sorry for psychology’s loss, whatever it is’.

When we see some new work which ‘dismantles’ or ‘disproves’ a long-standing finding P, it’s probably worth, like, taking a breath, man. Were we even justified in believing P in the first place? How much of this is new information? How much of this is “we were processing the information we had surprisingly poorly”?

Like, fuck, dude.

How strange and implausible-on-its-face was that P claim, anyway?