The truth about mathematics.

I often read paraphrases of one or other of Gödel’s theorems that talk about true, unprovable statements. I’ve said before that I’m a formalist of sorts. Talk of undecidable statements in a system being true gives me headaches. And I’m an analyst so I work in ZFC. If I say “statement X is true” I’m telling you that there exists a proof of statement X in ZFC. If you ask me if I think the continuum hypothesis is true, I’ll explain to you that it’s known to be undecidable. If you tell me you know it’s undecidable but still want to know if I think it’s true, I’ll look at you as if you asked me what colour integrity is.

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Reasonable effectiveness

In which, without lapse into solipsism, your correspondent defends mathematical formalism in the face of deep connections between mathematics and the physical universe.

In response to a comment of mine at Ars Math the unapologetic John Armstrong challenges me thus:

So, Matt, you’re a formalist? You seem to have a similar underlying belief that mathematics is a formal system, and a product of the activities of human minds (brains).

Not to claim a Platonic position here, but I challenge you with the same response as I’d give to a hardcore formalist: how do you explain the “unreasonable effectiveness” of mathematics in the physical sciences? Why should the output of human brains have anything to do with physical law, and how is it that truly well-formed sciences are invariably expressed in mathematics? Escapes into radical solipsism will be discarded as the jokes they are.

My response is after the jump.