[FOM] Incompleteness and Physics
joeshipman@aol.com
joeshipman at aol.com
Sun Oct 19 10:35:20 EDT 2008
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
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