Applied Mathematics and Scrutability

Also via Leiter’s blog I was perusing the Philosopher’s Annual list of the ten best papers of 2008. The paper on Mill is very interesting and I have heard a lot about belief and alief lately but what really caught my attention is Penelope Maddy’s How Applied Mathematics Became Pure.

The whole paper is really very interesting and I would highly recommend that you read the whole thing but I want to quickly discuss one of the morals that she draws from the story she tells. She says,

This story has morals, it seems to me, about how mathematics functions both in application and in its pure pursuit. One clear moral for our understanding of mathematics in application is that we are not in fact uncovering the underlying mathematical structures realized in the world; rather, we are constructing abstract mathematical models and trying our best to make true assertions about the ways in which they do and do not correspond to the physical facts. There are rare cases where this correspondence is something like isomorphism – we have touched on elementary arithmetic and the simple combinatorics of beginning statistical mechanics, and there are probably others, like the use of finite group theory to describe simple symmetries – but most of the time, the correspondence is something more complex, and all too often, it is something we simply do not yet understand: we do not know the small-scale structure of space-time or the physical structures that underlie quantum mechanics. And even this leaves out the additional approximations and accommodations required to move from the initial mathematical model to actual predictions.

I wonder if this is right if it causes problems for the kinds of scrutability claims that David Chalmers wants to defend, and which for the most part I am highly sympathetic to (of course where we differ is over whether we need to include phenomenal truths in the base truths or not…I think probably not since they can be derived just as easily as other ordinary macroscopic truths).

The problem, it seems to me, is that if this is right (i.e. if at the limit we do not end up with a unified mathematical model of the world but rather patchwork models that apply only in various respects) then which mathematical model we apply or assumption we make will crucially depend on empirical knowledge (for instance knowing that the equations for a harmonic oscillator  are a good model of a molecule’s vibration only in the region of the minimum (see page 35)). Am I missing an easy response?

I’ll have to think about it later because now I’m off to Jared Blank’s cogsci talk

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