> From: Leigh
>
> From: Rob Bass
> Sent: Friday, August 08, 1997 12:54 PM
> >I agree with Chris (I think) that several prominent claims made on behalf
> >of Gödel’s theorem are unwarranted, for example, that it vindicates
> >mathematical Platonism (Gödel himself seems to have thought that) or that
> >it proves the falsehood of determinism or mechanism (in philosophy of mind)
>
> I’m not sure what the parenthetical means. But excluding the extent to which
> it might change what I would otherwise take to be your meaning, it was
> Heisenberg and his fellow quantum physicists who proved the falsehood of
> determinism, yes?

The distinction is this. On most interpretations of quantum theory, physical determinism is ruled out for quantum-scale objects. But that doesn’t by itself mean that macro-scale objects aren’t so close to deterministic as to make no difference because the law of large numbers can overwhelm the underlying quantum “noise.” For lack of a better name, call that “quasi-determinism.” Down at the micro-level, there’s quantum randomness, but in aggregating to the macro-level, the randomness tends to cancel out so that large objects can be treated for practical purposes as if they just conformed to deterministic laws. And human brains, of course, are macro-scale objects.

Some people – the British philosopher, J.R. Lucas, is the best-known – following up on hints to be found in Gödel’s own writing, argued that Gödel’s theorem shows that even quasi-determinism can’t be true of human beings. Sketching, the argument is that any machine that could represent human intelligence can be modeled as a formal system. But there’s something that, by Gödel’s theorem, we know such a formal system can’t do, namely, it can’t produce its own Gödel-sentence as true. But human beings (human mathematicians, anyhow) can do what a particular, proposed formal system cannot. They can find the Gödel-sentence for that system. Ergo, that formal system can’t be an adequate model for human intelligence. Since this is a generalizable result – the argument can be applied to any formal system that may be proposed as a model for our intelligence – human intelligence can’t adequately be modeled by any formal system and therefore not by any machine.

To my mind, this line of argument has come in for devastating criticism. I’ll mention just one objection. It’s correct that if a mathematician can produce a true Gödel sentence for a formal system, then that system can’t be an adequate model of his intelligence. But Gödel didn’t provide a procedure for finding and identifying a true Gödel-sentence for an arbitrary formal system. Rather, he provided a proof that there was one for an arbitrary (consistent) formal system, even though we might not be able to find it. The Lucas-type argument does nothing to show that if a mathematician were presented with a formal system sufficiently rich and powerful to represent human intelligence that he would be able to find and identify its Gödel-sentence.

[….]

Rob