[FOM] Re: The Myth of Hypercomputation

[Timothy Y. Chow]

On Tue, 10 Feb 2004 JoeShipman at aol.com wrote:
> For such a result to be established, the number in question would have
> to be DEFINABLE but nonrecursive; then ZFC would only determine the
> value of finitely many places in the decimal expansion of the number,
> and more precise measurement would allow us to derive new mathematical
> truths from experiment, that could not be proven in ZFC.
>
> (Of course we would have to believe that the physical theory was correct
> and the experimental setup was appropriate, but that is not so different
> from the current situation where we accept computer-assisted proofs
> because we believe the theory behind the construction of the computers
> and believe that the algorithms have been appropriately implemented.)

What I was trying to capture in my two Theses was the intuition that these
two situations are radically different.  I'm not sure yet if the Theses
successfully capture this intuition, so let me try to argue directly.

Let me slightly rephrase your setup.  We have a physical theory that
predicts that if I perform a certain finite experiment, and the observed
result is "1," then ZFC is consistent, whereas if the observed result
is "0," then ZFC is inconsistent.  I do the experiment and observe a "1."

How could I come to believe such a physical theory to the point where I
accept the result of this experiment as settling the question of whether
ZFC is consistent?  Presumably I came to the theory by performing a bunch
of finite experiments and making some sense of them.  This set of finite
experiments cannot prove that the theory is correct, or that any
predictions the theory might make are correct.  So now, when I perform
the "ZFC experiment," what makes me so sure that I've now learned that
ZFC is consistent?  Perhaps ZFC is really inconsistent and I've just
experimentally disproved the theory by refuting one of its predictions.

The reason that the computer-assisted proofs are different is that they
can be finitely verified.  I do not need to extrapolate any theory to a
region that cannot be directly verified.  Intuitively, something that can
be finitely verified is different in quality from something that cannot.
I've tried to capture this difference in my "Theses," but maybe they still
need to be tweaked.

Tim