[FOM] constructivism and physics

[Bas Spitters]

On Tuesday 07 February 2006 04:01, Steve Awodey wrote:
> Dear FOM'ers,
>
> Can someone please provide some references on connections between
> constructivism in mathematics and mathematical physics?  There must
> be some literature on this topic.

That depends on what you looking for.
1) In Bishop-style mathematics there is a large amount of papers showing how 
to develop mathematical physics (analysis) constructively. See the book by 
Bishop, and its successor by Bishop and Bridges. Recently, there has been 
some discussion (mostly between Hellman and Bridges) about whether it was 
possible to prove certain theorems constructively. Hellman stated that it is 
impossible to prove the Gleason theorem constructively and that is impossible 
to treat unbounded operators constructively.
The short conclusion can be found in the following papers: It is certainly 
possible.

A constructive proof of Gleason's theorem (Bridges and Richman)
Gleason's theorem has a constructive proof (by Richman)
on Richman's homepage:
http://www.math.fau.edu/Richman/HTML/DOCS.HTM

Feng Ye showed how to develop unbounded operators constructively:
Toward a constructive theory of unbounded linear operators.  J. Symbolic Logic  
65  (2000),  no. 1, 357--370.
(after reading this paper I showed that his choice of definitions is actually 
very natural: see the paper Located Operators)

Other theorems from math. phys. that I looked at personally:
Peter-Weyl theorem (with Thierry Coquand)
Almost periodic functions
Ergodic theorems
(All these papers can be found on my homepage
http://www.cs.ru.nl/~spitters/articles.html).
[Yes, I am plugging my own papers, but these are the ones that I know best.]

IMO all these theorems can be developed naturally in a constructive setting 
and I am not aware of any standing questions like Hellman's to the 
constructive community.

Note that I am not claiming that one *should* develop all mathematical physics 
constructively, just that it is not impossible to do so, as is sometimes 
claimed.
[Personally, I do like to develop my mathematics constructively and think that 
it often gives new insights, but that's a personal preference.]

2) Some mathematical physicists do seem to think that one should develop 
develop the subject observationally or phenomenologically. See for instance 
the work by von Neumann, Segal, Kolmogorov and many others. This seems to fit 
with the ideas in the constructive tradition of pointfree topology (aka 
locale theory). Although much work is needed here.

Recently, there seems to be some interest of mathematical physicists in topos 
theory:
(Here are some pointers which I would like to read carefully some day. I'd be 
grateful if someone could give me a better reading-list).
Smolin's book: Three Roads to Quantum Gravity
which seems to present to work of Markopoulou.
See also John Bell:
http://publish.uwo.ca/~jbell/Coverages.pdf
(I think John Bell is reading this list. I hope he will comment on this).
There is the work by Chris Isham applying topos theory to the interpretation 
of quantum mechanics. (His papers are in the arxiv)

John Beaz states: http://math.ucr.edu/home/baez/topos.html
" I'll warn you: despite Chris Isham's work applying topos theory to the 
interpretation of quantum mechanics, and Anders Kock and Bill Lawvere's work 
applying it to differential geometry and mechanics, topos theory hasn't 
really caught on among physicists yet. Thus, the main reason to learn about 
it is not to quickly solve some specific physics problems, but to broaden our 
horizons and break out of the box that traditional mathematics, based on set 
theory, imposes on our thinking."

I hope this is what you were looking for.

Bas