FOM: nominalist, predicative, constructive physics

[Matthew Frank]

These are some comments and technical updates on sections D and E of
Jeffrey John Ketland's post of Jan. 31, concerning nominalist,
predicative, and constructive physics.  I may have responses to other
parts of this post later; I apologize in advance to Ketland if I have
misrepresented his views.

Section D, on Field's nominalist program:  "How do you do the nominalizing
for General Relativity?"  As Field's nominalized classical mechanics used
a synthetic Euclidean geometry, so a nominalized general relativity should
use a synthetic differential geometry.  Many have despaired of this, not
knowing that there was a synthetic differential geometry to draw on in the
works of Herbert Busemann.  I investigated this recently and have shown
how to formulate much of general relativity in such a framework; see my
paper "General Relativity Without Derivative Operators" on my web page.  
I believe that it would be possible to construct a nomininalized general
relativity as successful as Field's nominalized mechanics (though this
would require some significant technical work); however, I don't have
enough sense of or sympathy with nominalism to take on this task by
myself.

Section E, on Feferman's predicativist program:  I feel, with Feferman
(but not Ketland), that all the theorems of mathematical physics can
probably be recovered predicatively and in ACA_0.  In fact I find
Feferman's program so successful that I do not see interesting research
projects in the area.  (There is an open question about whether
predicative mathematics can handle non-separable Hilbert spaces well, but
it doesn't seem to make sense to pursue that until someone finds a really
good example of a theorem in mathematical physics that uses them -- and
that may well require some new physics.)

Section E, on the constructivist program and Hellman's critique of it:
In 1993, Hellman claimed that constructive mathematics was not enough for
physics because two central theorems in the mathematical foundations of
quantum mechanics were not constructively provable, and this motivated a
lot of research.  By now we have constructive proofs of both theorems.

1)  Hellman proved that a (coordinatized) version of Gleason's theorem
could not be constructively proved; Fred Richman and Douglas Bridges gave
a constructive proof of a more common (and coordinate-free) version in
last year's Journal of Functional Analysis.  This, along with
philosophical commentary by Richman, is on his web page at
http://www.math.fau.edu/~richman; I found a similar proof indepedently,
and my version is outlined on my web page.

2) Hellman argued from results of Pour-El and Richards (mentioned by
Ketland) that there could be no constructive treatment of unbounded
operators.  Bridges has discussed Hellman's arguments extensively in
articles in the Journal of Philosophical Logic, and I agree with his
criticisms.  In any case, Feng Ye has given a treatment of unbounded
operators (in his Princeton PhD thesis and a forthcoming JSL article), in
particular with proofs of the spectral theorem and of the self-adjointness
of quantum-mechanical Hamiltonians.  It is very impressive work.

3) Hellman's newer arguments about constructive treatments of general
relativity are a different case.  The singularity theorems as they stand
may not be constructively provable, though this says little about whether
there are appropriate constructively provable reformulations.  In the end,
I agree with Ketland that constructivizing these theorems is not an
interesting project, but for different reasons:  he because space time is
"right out there" and classical reasoning about it is therefore permitted;
I because the theorems are so inelegant that constructiving them would be
unlikely to be illuminating.  This is part of my general views about
constructive mathematics; they will appear on my web site, and possibly on
fom, soon.

In case all of that was too dry or critical:  I find these areas of
research very exciting.  The treatment of mathematical physics in
constructive mathematics is a live subject of research, going well beyond
the topics mentioned above.  More generally I find the various restrictive
programs surprisingly powerful and sometimes surprisingly enlightening.

--Matt Frank
http://www.math.uchicago.edu/~mfrank