FOM: Connections between mathematics, physics and FOM

[Jeffrey John Ketland]

Mark Steiner wrote:

> Discovery is very important; a recent paper by a mathematical physicist
> argues that for physics we don't need differentail equations over the
> reals at all, we could use difference equations over the rationals,
> given that we can represent all possible measurements by rational
> numbers.  But, the physicist goes on, we could never make discoveries
> without the mathematical form of the differential equations themselves.
 

My view is that kind of approach is fundamentally flawed, both 
technically and philosophically. The continuum model of spacetime 
is really fundamental to theoretical physics. One occasionally 
hears that people are trying to replace the manifold structure by 
something countable, or discrete or what have you. I think it won't 
work. There are many reasons, of which two are:

(i) One reason comes from quantum mechanics and the basic 
commutation relations for the self-adjoint operators Q and P which 
represent position and momentum. The Dirac commutation 
relations [Q, P] = ih have no countable representations and the 
spectra of these operators must be the whole real line (actually it 
can be shown that neither Q nor P have eigenfunctions in the 
Hilbert space L^2[R]). Roughly, you can't find two countable 
matrices A and B whose commutator is proprortional to the identity 
matrix. One can find a brief mention of this in Weyl's 1931 book 
"Group Theory and Quantum Mechanics". One cannot simply 
replace the real line by the ratios or something like that. That 
doesn't seem to work. (I read a recent (1999) paper by a young 
French author discussing some of these mathematical points on 
the Los Alamos preprint site, called "Mathematical Surprises in 
Quantum Mechanics". I'll find the details if anyone is interested).

(ii) The other reason is that serious progress in *real* theoretical 
physics has been obtained by trying to replace the basic 
differentiable manifold structure by a more complicated manifold 
structure. Eg, one moves to higher dimensions with 
compactification (as in Kaluza-Klein theories, supergravity and 
superstrings) or one considers various fibre bundle structures over 
the base manifold (Yang-Mills guage theories). Recent 
speculations, about "spacetime foam", make the structure even 
more complex, not less compex.

Replacing the reals by the ratios is primarily motivated by 
*epistemological* considerations, about what we (finite beings) can 
*experimentally measure* (ratios). But I'm very sceptical that this 
program of eliminating the reals from physics works though. (In 
fact, I think it is reactionary and anti-progressive).

Jeff Ketland


Dr Jeffrey Ketland
Department of Philosophy, C15 Trent Building
University of Nottingham, University Park,
Nottingham, NG7 2RD. UNITED KINGDOM.
Tel:    0115 951 5843
Fax:    0115 951 5840
E-mail: <Jeffrey.Ketland at nottingham.ac.uk>