What Gödel’s Incompleteness Thoerem means and doesn’t mean

Mathematicians often seem to get irritated by the invocation of Gödel’s Second Theorem as proof or evidence of the a priori limits of human knowledge. When Freeman Dyson did so in his review of Brian Green’s The Fabric of the Cosmos, Solomon Feferman weighed in to raise the mathematican’s objection.

Via Sean Carroll I came across a nice piece by Cosma Shalizi on the theorem and its abuses.

“There are two very common but fallacious conclusions people make from this [Gödel’s theorem], and an immense number of uncommon but equally fallacious errors I shan’t bother with. The first is that Gödel’s theorem imposes some some of profound limitation on knowledge, science, mathematics. Now, as to science, this ignores in the first place that Gödel’s theorem applies to deduction from axioms, a useful and important sort of reasoning, but one so far from being our only source of knowledge it’s not even funny. It’s not even a very common mode of reasoning in the sciences, though there are axiomatic formulations of some parts of physics. . . .

This brings us to the other, and possibly even more common fallacy, that Gödel’s theorem says artificial intelligence is impossible, or that machines cannot think. The argument, so far as there is one, usually runs as follows.”

Read on.