From: jbaez@riesz.mit.edu (John C. Baez)
Subject: Re: Applied inconsistency
Organization: MIT Department of Mathematics, Cambridge, MA
Date: Mon, 14 Sep 92 03:12:03 GMT

pratt@Sunburn.Stanford.EDU (Vaughan R. Pratt) writes:
In article <88808@netnews.upenn.edu> weemba@sagi.wistar.upenn.edu (Matthew P 
Wiener) writes:
[Physicists do cool stuff with the Riemann Hypothesis, RH, but then some
jerk shows it's false using induction up to some silly ordinal....] 
>>Do the
>>physicists abandon ship?  Of course not.  They let the mathematicians
>>figure it out, and will continue to assume RH.  In essence their TOE
>>is modeled inside Large_Fragment(PA)+RH

I can't imagine that happening unless this "theory of everything" they
had gotten using RH was experimentally verified and in great shape.  
In that case it would be rather interesting, and I can imagine
physicists finally getting interested in the foundations of math, and
some trying to correct their theory of everything to take the bad news
into account, while others write papers on how physics has shown that
induction up to big ordinals is nonsense...

Anyway, while this would make great science fiction, we can't really tell
what would happen in such a remote eventuality.

>Whoa, that's a remarkably strong claim.  George Mackey's understanding
>of physics seems to involve Borel sets, is he completely alone in
>that?  

No, there are a few more of us mathematical physicists out here who
use Borel sets.

>Anyway, if you want to understand Mackey you need to know about
>Borel sets.  Now these can boot you way way up above PA, even above
>ZC+V=L (Zermelo plus Choice plus Every-Set-is-Constructible).  Here's
>one instance.

>It is a theorem that a symmetric Borel relation (one whose graph is a
>Borel set in the plane) on the interval [0,1] either contains or is
>disjoint from some Borel function on [0,1].  (We're doing ordinary
>mathematics here, i.e. this is a theorem of ZFC.)

>While I have no idea where this particular theorem would come up in
>physics, I've seen quite enough facts I couldn't grasp appealed to in
>physics that I would not turn a hair if this fact that I can easily
>grasp entered into some physics argument.

Well, I can't possibly imagine how needing to know this for ALL Borel
sets would ever enter into mathematical physics.  In fact, even though
I have done some very abstract stuff under the heading of mathematical
physics, I have long noticed that lots of the generalities of analysis
only find application to honest physics, i.e., calculations predicting
measurements, through very particular cases.  For example, while the fact
that every Hilbert space has an orthonormal basis is probably
equivalent to the axiom of choice, and is used all the time in
mathematical physics, in reality all the Hilbert spaces you wind up
using in physics come equipped with obvious (not necessarily
canonical, but existent!) bases.  If ZF+C was suddenly discovered to
be inconsistent and people decided to jettison C, one could simply
define a Hilbert space as one for which a basis existed, and
everything would be fine, although it would be a pain in the butt.

In short, I believe mathematical physics, insofar as it is really
physics, to be fairly insensitive to the axioms of math one uses. 
Indeed, since when it comes down to it one really wants algorithms for
computing things, mathematical physics could even weather a coup
d'etat by a band of radical constructivists, though it would be most
unpleasant.  See e.g. Pour-El and Richards' book recently cited on
sci.physics.  (Yes, I know they're not constructivists... but their
book would become popular if one had to learn to live with
constructivists.)   

You all may be pleased to know that I figured out how to get around
one ugly use of the axiom of choice in my book on quantum field theory
with Segal and Zhou.  I am pro-choice, but favor doing things as
constructively as possible, as long as one doesn't need to break a
sweat.  

>The belief that somehow physics will never need to venture anywhere
>near where mathematicians have been venturing has no rational basis.
>There is no law saying physicists can't use any mathematics they can
>understand (though they should pass one against their using the other
>kind).

Here I agree completely.  

Newsgroups: sci.physics,sci.math