FOM: physical computability

[Jeffrey Ketland]

Hello Stewart.

Joe and I (and some others) too are interested in this idea, that
uncomptable physical processes might occur in Nature (according to our "best
theory") and that by measuring the ouput, we get a physical reasons for
accepting mathematical facts independent of ZFC. It's not connected with
determinism (it might involve measuring the probabilistic statistics of some
quantum scattering experiment). And the issue of verifying the *theory* is
out of the question.

We accept the theory at face value. The theory says: there is a certain kind
of physical process P, and we can build a device M to generate this process,
and the output can be measured, and the output is an *uncomputable real*. We
then let it run - it produces a uncomputable ouput by this spooky physical
non-algorithmic process P. We humans (if what we can calculate is the same
as what some Turing machine can calculate) cannot "follow" this
computation - but it physically happens nonetheless, because that's how
Nature is built if the theory's right - and we can *measure* the result.

The theory states (in so many words) that the result is uncomputable (but
doesn't give a prescription for our computing it, a priori as it were).
Theories can *define* sets and quantities that are uncomputable (we can
define "theorem of PA" or "true sentence of lang of arithmetic" (in ACA
say), but Turing machines can't compute the characteristic function
involved).

Joe states (I've transcribed and removed the capitals):

>For any definable-but-not-recursive sequence a1, a2, a3, there is a finite
index N such that ZFC does not >decide the value of a_N; .... at some finite
time we will already have obtained a new mathematical fact >independent of
ZFC (but following from ZFC + T). This would demonstrate that mathematics is
not logically >prior to physics.

Sure - it's highly speculative, but it's a beautiful idea which I find very
attractive. It sheds quite a new light on Hilbert's optimism - his statement
"non ignorabimus" - that discovering new mathematical facts might involve
considerations from physical experiments.

Best wishes - Jeff

~~~~~~~~~~~ Jeffrey Ketland ~~~~~~~~~
Dept of Philosophy, University of Nottingham
Nottingham NG7 2RD United Kingdom
Tel: 0115 951 5843
Home: 0115 922 3978
E-mail: jeffrey.ketland at nottingham.ac.uk
Home: ketland at ketland.fsnet.co.uk
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~