[FOM] When is it appropriate to treat isomorphism as identity?

[Andrej Bauer]

On Fri, May 22, 2009 at 2:54 AM, Monroe Eskew <meskew at math.uci.edu> wrote:
> Your responses seem to come from a pragmatist perspective.  So I have
> a few questions.

I will admit to being a pragmatist to a degree, but not quite as
radical as you and Timothy Chow seem to see in me. In particular, I am
_not_ willing to sacrifice the mathematical method for a "whatever
works" approach. However, deciding which kind of mathematics
(constructive, classical, with large cardinals, with green cardinals
and ham) is to be used in natural sciences _is_ subjected to a
"whatever works" approach.

> 1) Why bother with foundations?  You even dismiss such concerns as too
> puritan in the discussion about approximation processes.

Can you be more explicit? I don't know what you're referring to. If
you are referring to my remark about the Markov principle, all I was
saying there was that Markov principle was the mathematical
formulation of ideas behind your discussion about how approximation
sequences eventually lead to good enough results, even if we don't
know when that will be. I was not dismissing the discussion as
irrelevant.

Markov principle (in one formulation) states: if a binary sequence is
not always zero then it contains a one. It is consistent with
classical logic, and is is very useful in constructive analysis
(although Bishop does without it). All I intended to say was that
Markov principle guarantees convergence of an approximation sequence
whose error bounds do not stay above a positive number.

>  If your goal
> is to just do whatever works "as a physicist," then it would seem
> unnecessary to have some formal foundation for infinitesimal numbers,
> or to reform the semantics of existential quantification.

I completely fail to see how you arrive at this conclusion. Timothy
Chow arrived at a similar one regarding teaching maths to physicists.

As a physicist (which I am not, just to make things clear), I would
want to use lots of cool mathematics built on proper mathematical
foundations (even physicists prefer to avoid logical fallacies), but I
would not particularly like it if someone forced on me mathematical
methods that I judge to be useless in my applications.

I suspect physicists would gladly sacrifice mathematics for anything
else that "worked better". I can't imagine what that would be, perhaps
something along Wolfram's "New Kind of Science" where proper
mathematical formulation and proofs are secondary to "experimental
mathematical discovery". Except I would say Wolfram is applying
methods of physics  to mathematics, not vice versa.

> 2) Aren't the epsilon-delta notions of limits practical?  I know that
> in experimental science, one wants to approximate and compute all the
> time, and it is also of interest to do so within a margin of error.
> This means a FINITE margin of error, since infinitesimal error is not
> available in the real world.

Did I ever say that epsilon-delta notions were not practical? In fact
some of my work concerns computaton with exact real numbers in which
epsilon-deltas abound. But again, just because epsilon-delta notions
are useful in practical computation, why does it follow that physcists
should use _only_ epsilon-deltas? They could use infinitesimals to do
some things (discuss smooth manifolds, vector fields and differential
equations) and epsilon-deltas for others (compute approximate
solutions of differential equations). It is the job of a mathematician
to explain how all this can be done consistently, with proper
foundations.

With kind regards,

Andrej