FOM: Maths from physics: reply to Ketland

[Joe Shipman]

Yes, you have the right idea.  For example, one can conceive of an
extension of quantum electrodynamics in which one can derive that the
fine-structure constant alpha (currently inserted into QED as a free
dimensionless parameter) is a noncomputable real number like Chaitin's
"Omega" (halting probability on random input tape of a Universal Turing
Machine).  Then the "non-algorithmic machine" consists of a sequence of
experiments to measure alpha to increasingly greater precision.  The
uncertainty principle imposes no theoretical limit on the precision of
*dimensionless* measurable quantities like mass ratios or
probabilities.  (You may have to deal with statistical fluctuations, so
that e.g. you need  1,000,000,000,000 events rather than 1,000,000 to be
confident of the first 20 bits of a frequency, but in principle there is
no problem.)

In QED, experimentally measurable quantities are regarded as computable
(actually, computable RELATIVE to alpha), though it has not been proven
that the relevant summations (over Feynman diagrams) converge.  But
other theories don't necessarily have this property.  Fr example, some
theories of quantum gravity involve summations over homeomorphism
classes of 4-dimensional spacetime topologies; since 4-manifolds cannot
be classified (Markov), there is no apparent way to carry out the
computation of these (theoretically) experimentally accessible
quantities, though that doesn't yet prove they're noncomputable reals.
[QED is more tractable because Feynman diagrams are topologically
classifiable.]

-- Joe Shipman