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.

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Gödel’s theorems on “In Our Time” by the BBC

Hat tip to Yemon Choi (via email) for this very interesting discussion of Gödel’s theorems on Radio 4. It features the aforementioned Marcus du Sautoy, John Barrow of Cambridge and Philip Welch of Bristol. It is pitched nicely with interest whether you are a professional mathematician or if you have no training in maths at all.

I was particularly glad to hear someone at least draw some attention to the philosophical trickiness of the usual popularization of the theorems which talks about “true statements that cannot be proved in the system”. This always erks me a bit, because it isn’t entirely clear that a statement in a formal system has a meaning that survives being taken out of the system (and if it does that is a pretty subtle thing for this level of discussion). Thus just hearing the warning (from Welch?) about the theorems and their proofs being basically syntactic rather than semantic was nice. (It was du Sautoy giving the usual “true statement” version with but I will forgive him since he does so much good work and most mathematicians seem to be happy with the truthiness).

Incidentally, I suspect my feelings of awkwardness towards the claim we can talk about “truth” outside of “proof in a given system” may be related to the sort of maths I work in. My “grand-supervisor” Garth Dales discusses here how those who work in abstract analysis (and in algebra) tend to view their work as essentially formal (although using “realistic pictures” to help us).