Query for mathematicians

Does the proven unsolvability of Turing's halting problem mean that, though we know that either the axiom of choice or the axiom of determinacy is the case, we cannot know which is the case?

Not at all. The halting problem concerns Turing machines, and says that there is no Turing machine that can detect whether or not a given program will halt (because if the program won't halt, the Turing machine will just keep running, waiting for it to halt, and this will never end).

Now Turing machines are important because they are a model of computation, and the Church-Turing thesis says that all models of computation are "equivalent", in the sense that they permit precisely the same computations. So the unsolvability of the halting problem is a result for all models of computation.

What import does this have for the knowability of mutually incompatible axioms like the two you're describing. None, unless you believe both that:

(1) the problem of which of these two axioms is true is a problem that a Turing machine, or other model of computation, cannot decide.

(2) our minds are computers. More precisely, that *our* ability to decide between the two of them is a computation ability, and that our computation ability implements a model of computation meeting the Church-Turing thesis.

I see no good reason to believe (1) or (2).

Would be interested to hear more about (2). I read Penrose a long time ago, didn't find him that convincing, iirc his argument went something like "oooOOOOoooo Godel oooOOOOoooo quantum physics oooOOOOOOoooooo".

what good reason is there not to believe (2)?

I don't have a knock-down argument against (2), but it's not as though the only option remaining is to accept (2). Especially for a claim as massive as (2), shouldn't the default stance be to wait for good evidence before accepting it?

And I don't know what the evidence at this point is supposed to be. AI hasn't panned out. My own views at this time are like Hubert Dreyfus' in What Computers Can't Do. You can get a good summary of those views in the interview here.

I think a main reason people accept (2) is ideological: it comes from a desire to see nature as mechanized. But just because you want to believe nature is mechanized doesn't mean that it is.

Speaking for myself it's not a desire, it's just that, despite my strong dualist tendencies, I don't know what it means, for something to be non-computable. The arguments against computationalism are all negative - "consciousness|qualia|intuition|expertise|whatever seems to be [not even is, just seems to be] non-computable, therefore computationalism is false". But no-one knows, if it's not computation, what it is instead.

But I kinda promised myself I'd stop thinking about this, 'cause it's always just the same old merry-go-round.

huh I have now heard two references to Hubert Dreyfus in two days, having never heard of him before.

This is not my area, but my inclination for (2) is to say that it's different because the mind isn't a closed system in the way that a computer program is? Both in terms of the immediate and contingent access to extraneous information in the mind, and through ongoing sensory perception. And also in that the mind's computational processes can change to fit when working through a problem. Excuse me if I'm missing the point entirely.

Yeah, I don't know exactly what cognition is, if not computation, but I don't want to conclude from that that cognition is computation. We just don't understand cognition very well at this point.

xpost, interested in what you mean by "AI hasn't panned out". Do you mean, we don't have robots you can have a chat with yet?

Re. AI: sure, that would be a sign of passing. I'm talking about what's called "strong AI", a machine that wasn't just good at some specialized task like playing chess, but was more flexible. We've nothing like that on the horizon, even.

It'll creep up on us, and I'm sure it won't appear to be "human being" intelligent for a long time.

Then again, do you know I'm not a bot?

We now have flesh-eating robots, what more do you want?
(xpost)

Yeah, I don't know exactly what cognition is, if not computation, but I don't want to conclude from that that cognition is computation. We just don't understand cognition very well at this point.

otm.

I like analogy John Searle came up with which was something like "the planets are not calculating differential equations as they go through their orbits. They are not in that line of business."