The Incompleteness of Gödel

In Slate, mathematician and novelist Jordan Ellenberg takes a solid whack at metaphysical interpretations of Gödel’s Incompleteness Theorem:

Any scientific result that can be approximated by an aphorism is ripe for misappropriation. The precise mathematical formulation that is Gödel’s theorem doesn’t really say “there are true things which cannot be proved” any more than Einstein’s theory means “everything is relative, dude, it just depends on your point of view.”

The article does somewhat understate the breadth of the theorem, which actually applies not just to arithmetic but to any formal system. Still, formal notions of undecidability and incompleteness have little relevance to our common-sense notions of evidence and proof.

I do find Gödel’s theorem valuable as synecdoche for the larger truth that any deductive system is based on unprovable assumptions. But as Ellenberg writes:

But what’s most startling about Gödel’s theorem, given its conceptual importance, is not how much it’s changed mathematics, but how little. … The mathematics we do today is very much like the mathematics we’d be doing if Gödel had never knocked out the possibility of axiomatic foundations.

And the same is true of non-formal reasoning as well.

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