[FOM] Incompleteness and Physics

[joeshipman@aol.com]

The relationship of Godel Incompleteness to Physics is much more direct 
and interesting than the comments I have seen so far recognize.

On the one hand, Godel Incompleteness means that the physical 
predictions of certain theories may not be algorithmically calculable 
even when they are mathematically DEFINABLE.  For example,  Hartle and 
Hawking's quantum gravity model involves summations over homeomorphism 
classes of simplicial 4-manifolds, and the algorithmic unsolvability of 
this homeomorphism problem means no algorithm would be apparent even if 
the relevant dimensionless parameters were computable real numbers (and 
if they were not computable but were measurable, the experimental 
predictions would not be computable relative to those parameters). We 
don't have a well-defined enough "Theory of Everything" yet to know 
whether it will have the same type of incompleteness as Hartle and 
Hawking's model, but it's certainly conceivable, in a way that it is 
not for the differential-equation-based and obviously computable 
theories that other commenters apparently have in mind.

On the other hand, Godel Incompleteness also implies an influence of 
physics on mathematics -- as Godel pointed out, we might come to 
believe in new mathematical axioms because they made physics work. This 
will be the case if, according to some mathematized physical theory, 
there exists a definable but noncomputable real number that is 
measurable within the theory -- ZFC-proofs could only settle the value 
of finitely many bits of such a number, possibly fewer than can be 
measured.  (The measurements of a dimensionless physical quantity like 
the ratio of two particle masses or half-lives might be subject to 
statistical uncertainty but one could still attain probabilistic 
confidence about its value unobtainable from ZFC-proof.)

-- JS