Is Mathematics Unreasonably Effective?
In yet another confirmation of Betteridgeâs,
I argue that aesthetic virtues in mathematics are highly correlated with more general theoretical virtuesâsimplicity, generality, explanatory power, etcâof the sort that play an important role within scientific practice. (Waxman 2)
Cliffs Notes: itâs plausible that the so-called âunreasonable effectiveness of mathematics in the natural sciencesâ â an observation first attributed in that form to Eugene Wigner â is less âunreasonableâ than it appears.
Why? Because while mathematics itself may be an a priori discipline â in the methodological sense, if not in the epistemic â the aesthetic sensibilities of mathematicians are formed in no such vacuum.
Evolution by natural selection was what generated humans and, with us, our âculturesâ and our âaesthetic preferencesâ. Aesthetics is closely entangled with the physical world. Itâs also closely entangled with other intellectual âvirtuesâ.
The explanatory challenge posed by the âapplicability of mathematicsâ to the natural sciences thus dissolves in the face of two observations:
- the judgments that mathematicians call âaestheticâ are (often) barely separable from the identification of theoretical virtues â such as simplicity, unificatory power, parsimony, importance â which are themselves generated by connection to the demands of the real world; and,
- humans find âaestheticâ virtue in things that are also correlated with being useful abstractions over the world.
Waxman gestures to some fruitful examples:
- Diracâs equation;
- Galois theory; and,
- the Kac-ErdoĚs theorem.
What should we call the variety of argument that is in general being prosecuted by Waxman in this paper? âArgument from correlation of culture to worldâ? âThe naturalistâs sanity-checkâ?
In some sense, itâs an obvious argument. Little more than: âan external world exists and substantially optimises humans via their preferences, actuallyâ. Yet itâs also one that seems systematically underrepresented in the broader spaces of philosophy-of-science and science & technology studies.
I think thereâs a one-level-more-meta- insight, here, though.
Others have explored the so-called âunreasonableâ effectiveness. (The Quinean naturalism and ârealismâ in Sorin Banguâs The Applicability of Mathematics in Science: Indispensibility and Ontology seems relevant.) This by itself isnât a massive contribution.
In âIs Mathematics Unreasonably Effective?â, what Waxman is doing is casting doubt on whether we ever ought to have taken âcomplete ineffectivenessâ as the prior. I see a similar point made gesturally by a literal banana in âAgainst Automaticityâ, and by Adam Mastroianni in âIâm so sorry for psychologyâs loss, whatever it isâ.
When we see some new work which âdismantlesâ or âdisprovesâ a long-standing finding P, itâs 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 âwe were processing the information we had surprisingly poorlyâ?
Like, fuck, dude.
How strange and implausible-on-its-face was that P claim, anyway?