Re: Penrose's reply to Chalmers


Daryl McCullough wrote:
|Well, I was just thinking that looking at the code of the robot,
|one could formalize the r.e. set T of all sentences of arithemetic
|that the robot could ever convince itself was true. Let's assume
|that that's at least the axioms of PA. Then let T* be
|the deductive closure of T.

It's sort of an interesting way of side-stepping the issue of
whether the sentences of arithmetic that the robot can convince
itself of is closed under deduction.

|Then we can formulate in the language
|of arithmetic a sentence G such that it is provable in PA that
|
| G <-> T* does not prove G
|
|Then, assuming certain closure properties about T*, we should be
|able to prove in T* that
|
| G <-> T* is consistent
|
|So if the robot believes that T* is consistent, then T* will
|be able to prove G, and presumably can prove that T* proves G,
|and so T* will be inconsistent.
|
|I guess you can defeat this argument if you weaken T* so that it
doesn't
|include PA, but such a robot wouldn't be a very good mathematician
|(and so couldn't actually be Torkel Franzen).

Believing itself to be consistent and believing its deductive
closure to be consistent are not exactly the same.

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. 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). 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.

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.

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. (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? 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. And, although it is surely realistic to assume that one
is
fallible in general, being infallible about a single question, while
sitting for a short period of time in a chair, where by infallible we
include the case of someone who just doesn't make any claim at all--
this kind of infallibility is definitely possible.

Refusing to apply modus ponens may seem wrong, but in other paradoxes
we
consent to suspending what otherwise would seem like reasonable rules.
We
already have seen the paradox concerning statements of the form:

(*) Brad doesn't believe (*).

where we can (for lots of people anyway) correctly deduce (*) from
basic
properties of Brad's world view, even though Brad can't. Because of the
special relationship Brad has to (*), he has to refuse to believe it,
much like the way the person inside my box has to refuse to disbelieve
the conjecture even though for anyone else with enough faith in them,
it's possible to be sure it's false.

Keith Ramsay

.

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