FOM: predicativist, constructive physics -- reply to Frank and Ketland

[Joe Shipman]

Matt and Jeff:

I don't think Feferman's program has been all that successful.  The key
test is quantum field theories.  The theory of the strong interaction
has not even been precisely formulated, and the theory of the
electroweak interaction has not been precisely formulated in a
computable way, and even the best quantum field theory, pure quantum
electrodynamics, has only been precisely formulated and shown to be
computable under the dubious mathematical assumption that the power
series in the fine-structure constant whose coefficients are sums of
n-interaction Feynman integrals converges.  (Dubious because the number
of distinct diagrams grows factorially and will eventually overwhelm the
decay factor of 1/137).  See my paper "Aspects of Computability in
Physics".

It ought to be a much bigger scandal that the "Standard Model" is not
actually a model.  It ought to be provable, for example, that the
half-lives of various nuclei are at least DEFINABLE from the free
parameters of the theory (quark, lepton, and boson mass ratios,
fine-structure constant, a few others), even if computability is to much
to ask.  I have never seen a good argument from a physicist that these
are even determined in principle.  Only in QED do we have an algorithm,
but no proof it converges.

This is not a criticism of Feferman, he can't reduce to predicative math
something that isn't properly math yet, but it belies his claim that
mathematical physics needs only predicative mathematics, because the
physics that he reduced is just an idealized approximation of a deeper
theory we don't understand fully yet, and it's only natural that our
idealizations will be tractable, that's why we make idealizations in the
first place.  If he could formulate QED predicatively he would have a
better case.

I also do not think Feferman treated General Relativity, though I am
more optimistic that GR could be done in a predicative type theory than
that Quantum Field Theories could.

I would be very happy to have the previous comments proved wrong by
someone who understands the fundamental physical theories better than I
do; but nowhere that I have looked have I found an exposition of Quantum
Field Theories that is acceptably mathematically careful.

-- Joe Shipman