[FOM] Re: The Myth of Hypercomputation

[Timothy Y. Chow]

Toby Ord wrote:

> I'm afraid that I see no reason to accept this thesis. I assume you are
> not simply saying that the computation must be verifiable by an unaided
> human (which would be a simple translation of the CTT), but allow
> non-human-verifiable processes, so long as they are 'finite'. Could you
> elaborate on some reasons for holding this?

I guess I believe something like the following.

1. If there are two competing physical theories and there is no
experimental test that we could perform, even in principle, to
discriminate between them, then the choice of one over the other
must be based on criteria that are less compelling than direct
experimental refutation.  That is, although there are many
criteria for accepting physical theories, experimental verification
is the "gold standard," and trumps other criteria.

2. The only experimental tests that we can perform, even in principle,
are "finite" ones, because human beings are finite and can only control
a finite amount of resources.

Let's consider your hypothetical hypercomputer for the halting problem.
Certainly if it were giving good answers then I would trust it at some
level.  I trust that ZFC is consistent, too (for different, though not
entirely unrelated, reasons).  What I am claiming is that my confidence
that ZFC is consistent is of a qualitatively different nature than my
confidence that, say, 34*27=918.  The latter admits a finite verification
and the former does not.  If your hypercomputer were to assure me that
ZFC is consistent, it might increase my confidence a little bit, but I
currently do not see how it could surmount the barrier I have posited.

Again, consider my argument that if your physical theory predicts that
your hypercomputer will correctly solve the halting problem, and then
the hypercomputer tells me that ZFC is consistent (which I still have
no way of finitely verifying), then you don't have a good answer to the
ZFC-skeptic who says, "Ha!  Since ZFC is inconsistent, your physical
theory has been experimentally disproved!"  In other words, when your
hypercomputer ventures into the realm of making assertions about things
that cannot be finitely verified, how do we know that it is yielding
new mathematical truths as opposed to falsifying your theory of how it
works?  There is no experimental test to discriminate between the two.

Intuitively, it seems to me that a finite amount of experimental data
can always be explained by two competing theories, one which says that
your hypercomputer works and one which says that it doesn't.  Therefore
I am never going to have a compelling (read: "experimental") reason to
assume that the hypercomputer works as advertised; I will have to rely
on, at best, silver standards.

Tim