Monday, April 18, 2011

A way to for mathematics to apply to the real world

Here are two standard stories about how mathematics is useful for figuring out what's going on in the physical world.

Abstract structure isomorphism: Mathematical truth is to be understood in a Platonist way as concerning abstract entities, but these abstract entities have a structure isomorphic to structures found in the world.  This structure isomorphism allows one to transfer Platonic mathematical facts to physical facts.

Axiomatism ("if-thenism"): Mathematical truths should be understood in an if-then way.  A mathematical theorem says that when some structure satisfies certain axioms, something else follows.  As it happens, certain things in the world satisfy the axioms, and hence the theorems apply directly to things in the physical world.

I wanted to offer a third option which may or may not be original.

Inferential structure monomorphism: Sometimes we have meta-theorems that say that whenever you prove p from the axioms in A using the rules of inference in R, then there exists a proof of a proposition p* from the axioms in some other set A* using the rules of inference in R*.  A trivial example is given by notational variants.  Any proof in infix-notation first order logic can be transformed to a proof in Polish notation.  For a non-trivial example, for every a theorem p in real analysis that can be proved using intuitionistic logic from a certain set of axioms, there is a theorem p* about certain kinds of toposes.  So, the suggestion is that as it happens, to some families of inferentially connected mathematical propositions there correspond families of physical propositions.

Abstract structure isomorphism will entail inferential structure monomorphism.  But inferential structure monomorphism is more general.  For instance, it could be that while on the mathematical side we have a quantified claim, on the physical side, we have some other kind of claim.

What's also interesting about inferential structure monomorphism about it is that it's not mathematical truths that end up being applicable to the world, but only mathematical theorems, i.e., provable truths.  As a result, this view is neutral as to the correct philosophy of mathematics--it's compatible with Platonism and if-thenism, for instance.

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