FOM: Application of reverse maths to theoretical physics

[Joe Shipman]

Ketland:
So an important question to ask is to see exactly what mathematical
axioms
are *needed* to prove important theorems required (already proved
informally) in theoretical physics. There are at least two examples that

have been recently discussed in the literature by the philosopher
Geoffrey
Hellman:
        [1] Gleason's Theorem: every state on a Hilbert space of dim > 2
is
regular.
        [2] Hawking-Penrose Singularity Theorem(s): certain spacetime
structures (M, g, T)    satisfying Einstein's field equations plus
further
conditions must contain an initial      singularity.
Hellman claims that there cannot be constructive proofs of these
theorems.
There are many other mathematical results that could be investigated
similarly. E.g., Weyl's Theorem that the spectra of the position Q and
momentum P operators on Hilbert space, which satisfy the commutation
relation [Q, P] = ih, must be the whole real line.

Very good question!  You might also consider the proof that
renormalization in Quantum Electrodynamics "works" (which depends on the
Heine-Borel theorem of analysis in the treatment in Bjorken and Drell's
"Relativistic Quantum Mechanics").
Unfortunately, the mathematical status of relativistic quantum mechanics
is really very bad -- "renormalization works" is the best anyone has
been able to do, but that only shows that you get a finite value for the
Feynman integral for a particular diagram, and there is NO proof that
the sum over all diagrams converges (and informal reasons for believing
that it diverges).  Hellman's two examples are taken from quantum
mechanics and general relativity, respectively, and it is the
combination of the two that gives trouble.

The deepest example of reverse mathematics applied to theoretical
physics is in my thesis, "Cardinal Conditions for Strong Fubini
Theorems" (Transactions of the AMS 10/90), where I showed that the
existence of "spin-1/2 functions" of the type postulated by the
physicists Itamar Pitowsky and Stanley Gudder was independent of ZFC.
In order to get around the EPR paradox and the Bell inequality, they had
used the Continuum Hypothesis to develop a "local hidden-variables"
theory where measuring noncommuting observables in different orders
corresponded to taking the iterated integral of a function of several
variables in different orders.  An essential feature of their theory was
that "Strong Fubini Theorems" fail so that the measurements can fail to
commute.  This allows one to avoid several philosophical and
methodological problems that arise in the orthodox treatment of quantum
mechanics or the standard alternatives to it; the cost is the
replacement of a "physical weirdness" (collapse of the wave function or
faster-than-light action at a distance) with a "mathematical weirdness"
akin to, but stronger than, the Banach-Tarski paradox.  I showed that
their use of CH (the foundational implications of which they ignored,
except to justify what they did on the grounds of consistency) or
something like it was NECESSARY, because it was consistent with ZFC that
iterated integrals (of bounded functions) are always equal whenever they
exist.

This shows a way in which mathematics is not logically prior to
physics.  Even though the theory of Pitowsky and Gudder was constructed
to be experimentally indistinguishable from standard quantum mechanics,
it is conceivable that some other theory that requires new mathematical
axioms may someday be put to the test; evidence for the theory will
represent physical evidence for mathematical statements that are
independent of our mathematical axioms, and "usefulness in physics" may
become a criterion for adopting new axioms, as Godel envisioned.  (Even
without an experimental test, those who prefer Pitowsky and Gudder's
local hidden-variables theory on philosophical grounds will have
extra-mathematical evidence for the existence of spin-1/2 functions and
hence, indirectly, for CH.)

Another way in which we could get at mathematical truth from physics is
if Church's thesis is false and we could set up an experiment to
calculate a definable but noncomputable real number; the value of any
digit beyond a certain finite level of precision would be a mathematical
fact independent of ZFC.

-- Joe Shipman