[FOM] Re: A Newtonian computing design

[Timothy Y. Chow]

Harvey Friedman wrote:
> This simple design is based on Newtonian inverse square law gravitation.
> The ideas can be adapted to other deterministic contexts. We also
> comment on how this could be adapted to probabilistic theories.

As usual, Harvey Friedman succeeds in formulating a specific and very
interesting proposal.  Here is how I look at this setup.  For further
concreteness, suppose that one of the Turing machines in question
searches for a contradiction from ZFC, or ZFC + "there exists a measurable
cardinal" if you prefer.  Let's split into two cases:

Case 1: The Newtonian experiment admits finite simulation.  That is, not
only the initial conditions and the detection of the collision lie within
our ability to measure and control, but the time evolution of the system
can be sufficiently accurately simulated by taking a finite number of time
steps, finite accuracies for the positions, velocities, and forces, and so
on.  In that case, what we have essentially done is to reduce the problem
of the consistency of ZFC to a finite computation.  (Assuming that the
proof of the equivalence of the halting of the Turing machine and the
Newtonian system took place within ZFC, this effectively shows that ZFC
is inconsistent.)  Although this is remarkable, it is not remarkable in
the sense that hypercomputation proponents want to claim.  The physical
system is not solving anything that an ordinary computer couldn't
(although maybe it's solving it much *faster*---I'll come back to this
point later).  Therefore the interesting case is:

Case 2: The Newtonian experiment resists finite simulation.  There could
be something subtle in the equations of classical mechanics that implies
that any fixed finite approximation to the system fails to correctly
predict the collision/non-collision outcome for all initial conditions
in the specified range.  In that case, my objection would be that "belief
in" the outcome of the physical system requires

  *believing that Newtonian mechanics remains valid in regions beyond
   what we can measure*.

For being finite ourselves, we can only measure the system a finite number
of times to some finite accuracy.  I do not see why we should believe in
Newtonian mechanics when it is extrapolated to untested limits.  Note that
Harvey's setup, if it is to fit in a normal-sized lab, involves distances
close to the Planck length.

I believe that the same sort of considerations should apply to almost any
kind of thought-experiment of this type, regardless of whether classical
or modern physics underlies the experiment.  Trying to "solve the
unsolvable" is probably going to involve extrapolating physics to
arbitrarily extreme limits, beyond what we can measure.  Even if we
had physics that was stable for 1000 years that we had tested over and
over again, the testing would take place within finitely measurable limits
and our confidence could not justifiably be extended to realms in which
we had no ability to confirm the theory experimentally.

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Let me switch gears slightly by making a more positive suggestion that
just occurred to me yesterday.  Instead of using the slogan "computing
the uncomputable," how about using

  *solving any decidable problem in constant time*

instead?  Some advantages of this are:

1. It seems to capture the spirit of many of the proposals for
hypercomputation that I have seen.  That is, if these proposals
work as they are supposed to, then they should also be able to
solve any decidable problem in constant time.

2. It avoids the recursive/non-recursive issue, which I have
argued is a complete red herring, leading to protracted and
irrelevant arguments about various Church-Turing theses.

3. It sidesteps entirely all the objections I have been raising,
by concentrating attention only on finitely verifiable computations.

4. It would still make trillions of dollars and be an incredible
scientific breakthrough.

5. It naturally directs attention towards the all-important practical
implementation issues (making sure that no assumptions of infinitely
powerful engineering ability are used) rather than towards arcane
questions about whether certain physical theories are "non-recursive"
in an unusable way.

One disadvantage might be that adopting this slogan does mean abandoning
the hope of (affirmatively) settling the consistency of ZFC by a finite
physical experiment, but as I argued before, I don't think this was really
in the cards in the first place.

The new slogan might be less sensational, which might be bad if the main
goal is to capture the attention of journalists and of funding sources.
But actually I'm not sure it's any less sensational.  I guess that's for
the public to decide.

Tim