Re: Penrose's reply to Chalmers

Keith Ramsay says...

>In a previous discussion I came up with a thought-experiment meant
>to get around some of the qualms about unbounded time. Suppose that
>the laws of physics are deterministic and suppose you're put
>temporarily inside of a box able to measure your state and having
>a description of the laws of nature. Now suppose you get to play
>a little game where you answer mathematical questions for prize money.
>All you have to do is push either a "true" button or a "false" button.
>Incorrect answers lose you money.
>
>Under these conditions, one could in principle rig up the machine in
>such a way that it would pose you a conjecture that is true if and only
>if you press the "false" button.

I think many people would dispute this, if you restrict your
game to answering *mathematical* questions. If you allow non-mathematical
questions, then you can certainly do it:

True or false: The next button you push will be "false".

>You might think that the fact that it
>hands you a description of your own state might disturb you in such a
>way as to keep the conjecture it gives you from really being equivalent
>to you, after having been given it, pressing the "false" button. But by
>the same sort of trick as used in the Goedel incompleteness theorem,
>the conjecture can be contrived so as to be equivalent to your pressing the
>"false" button (in accord with the deterministic laws of physics).

Right, that assumes that the laws of physics are deterministic.

>In a different version of the thought-experiment, we can imagine your
>belief that the conjecture is true or false determining the reward
>directly, rather than your pushing the button, assuming it's possible
>for us to define what brain states constitute belief or disbelief in a
>conjecture you've been given.
>
>How should one cope with such a situation? In the first version of the
>thought-experiment, I would say you should consider the conjecture
>false, agree that if it's false and you press "false" you get money,
>but refuse to press "false" because it would render the conjecture
>true, and thus lose you money. A lot of people balk at this
>conclusion, especially the notion that what we do can somehow
>render mathematical conjectures true or false. It's essentially
>a version of the Newcomb paradox. But I don't see any more plausible
>conclusion to reach. Certainly the sense in which
>pressing the button "renders" the conjecture true is unusual, but from
>the point of view of the person inside the box, I think it's the
>relevant one.

This reminds me of an essay I once read by an intuitionist
(I don't remember his name) explaining why he rejected the
law of excluded middle. He assumes that you give a
mathematician the following rule for computing the decimal
expansion of a real number:

If by stage n, you have constructed a proof that the
number you are computing is irrational, then the nth
decimal place is 0.

Otherwise, the nth decimal place is equal to the nth
decimal place of pi.

The author claimed that in this scenario, it is wrong
to assert that the number being constructed is rational,
and it is also wrong to assert that it is irrational.

>How should one cope with the second type of box? It's going
>to be much more difficult, since one has less voluntary
>control over one's beliefs, but I think it's reasonable as
>well as consistent to decide to believe
>all the following:
>
> (a) I'm partially consistent, in the sense that I'm not going to
> arrive at a mistaken belief concerning this conjecture.
> (b) In any case where someone plays this game, when (a) holds for
> them, then the conjecture they are handed is false.
> (c) I believe (a) and (b) but not their consequence under
> modus ponens, because someone who applies modus ponens in this
> case arrives at a mistaken belief about the conjecture they are
> handed by the machine and loses money as well.

Well, that's sort of strange, because modus ponens basically
defines what it *means* for A -> B to be true. It's a little
like believing forall x Phi(x), but not believing Phi(0).

>Penrose would have me disbelieve that the laws of nature as
>documented in the machine are correct, but I don't see how
>such a conclusion could be mandated "just because". Note that
>I haven't assumed the laws of nature are computable, just
>that they're deterministic.

I guess you mean that the consequences of the laws are mathematically
definable, even if they aren't computable.

>With indeterminism, one could modify the thought experiment to say,
>if the probability of my believing or disbelieving the
>conjecture is >4/5, then we proceed as if I was going to.) Stuff
>all the exotic quantum gravity you like into the laws of nature,
>just so long as in the end the laws are sufficient to predict
>the behavior of a cubic meter's worth of ordinary matter for a
>limited period of time.

>Others would have me be unsure of myself. But here again, what about
>the scenario requires being unsure?

Well, I would think that being sure of your beliefs would mean
that your beliefs are deductively closed. I don't see how you
can be sure that A is true, and be sure that A implies B, without
being sure that B is true.

>If I'm convinced that I can't determine the truth or falsity of the
>conjecture within the conditions specified, I can in principle be
>sure that I'm not making an error by not attempting to answer it.

Okay, we have a sentence G such that we can establish the
following:

If your belief system is sound, then G is true.
If you believe G, then G is false.

Your resolution is to consciously avoid considering
whether you believe G or not. The implication

If you believe that you are sound, then you believe G.

can be avoided by simply *not* considering all the logical
consequences of your beliefs. You believe that you are
consistent, and you believe that if you are consistent,
then G follows, but you don't believe G.

I guess I agree that that resolution works, except for
the fact that, as I already said, it makes it unclear
what it *means* to believe A implies B, if you are not
going to conclude B whenever you believe A.

But what if we take the deductive closure of your beliefs?
Suppose that we have a device that is reading your brain
state, and computing the set of all statements that you
currently believe, and is also working out the deductive
closure of this set. You are given a sentence to consider,
and if that sentence is in the deductive closure of your
current beliefs, then a green light will flash.

The sentence given is

The green light will not flash.

Now, it would seem that if you believe that you are
sound, then the fact that the green light will not
flash is in the deductive closure of your beliefs.

>Refusing to apply modus ponens may seem wrong, but in
>other paradoxes we consent to suspending what otherwise
>would seem like reasonable rules.

Okay, but what if we build modus ponens into the mind-reading
device?

--
Daryl McCullough
Ithaca, NY

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