FOM: Re: Reuben Hersh: Mitteilungen der DMV (fwd)

[Peter White]

On Sat, 28 Feb 1998, Reuben Hersh writes

>	The difficulty with identifying all math with the part that
> has interpretations and applications in physics and engineering is
> that you then have no explanation for the big part of math that
> has no such interpretations or applications.  
> How about the undecidability of the axiom of 
> choice on the basis of the other Zermelo-Frankel axioms of set theory?  What 
> does that say to your goldfish?

As an engineer, I am using category theory, topology, higher order logic, and
abstract algebra in my day to day practice. I admit this is very unusual
as engineers go, but there are a few of us using such mathematics in
engineering applications. I do not use the axiom of choice, or Zorn's
Lemma directly, but these are used to help prove the theorems that
I do use. It seems to me that this gives some degree "reality" to
the axiom of choice, although I am not sure the axiom of choice was
in need of a vote of confidence from me.

The point to be made here is that it is surprising some of the mathematics
that gets used in engineering, not to mention physics. The mathematics of
Galois may have been of interest only to "pure" mathematicians in the
time of Galois, but these days you can hardly do communications engineering
without a Galois Field. Mathematics that is currently not applied to
any physics or engineering may well become stock in trade of some branch
of science or engineering in the future. I think uncertainty about
what can be applied blurs any distinction that can be made between
a part of mathematics that has an interpretation in physics and
engineering, and another part that has no such interpretation.

The axiom of choice may have a natural interpretation to a communications
engineer. Suppose I design a system that may have to cope with too many
inputs at a given time by choosing one and ignoring the others, and
suppose I want this system to operate non-stop (forever). Then part of what
this system is doing is choosing one element from a set of elements
over and over again. If the system operates forever, then it is
choosing a countably infinite sequence of elements from a
countably infinite sequence of sets. If the axiom of choice fails
for countably infinite sequences of sets, then this says that there
is some fundamental mathematical reason why this system will eventually
fail. I can accept that there are physical limits to the lifetime of 
a system, but a mathematical limitation would be harder to accept.

Regards,
Peter White, Motorola.