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

[Andrej Bauer]

On Wed, May 20, 2009 at 11:19 PM, Timothy Y. Chow <tchow at alum.mit.edu> wrote:
> There are so many incorrect presuppositions in the question "why do we
> teach physics majors all those epsilons and deltas?" that I hardly know
> where to begin.

I think there's a difference in education across countries here.

> The first answer is that we don't.  For starters, undergraduate-level
> textbooks that teach calculus from the point of view of nonstandard
> analysis do exist and have been used in various schools.  Even at schools
> where epsilons and deltas are taught, they are usually taught only to math
> majors.  "Service courses" typically dispense with epsilons and deltas, or
> at least downplay them severely.

In Slovenia (and I would bet in the countries of the former Eastern
block, too) physics majors are taught very serious mathematics, with
lots of epsilons and deltas, and continuity, and theorems about
existence of solutions of differential equations, and all those
things.

> And by and large they do a fine job of that.  They don't need math
> teachers to teach physics students how to reason informally with
> infinitesimals because they are perfectly capable of teaching that
> themselves.

What I was saying is that they need math teacher to teach physics
students how to reason FORMALLY with infinitesimals, so that they
don't have to do the informal hocus pocus. In this sense, we agree
that

> Mathematicians should teach students how to reason like a
> mathematician.

With kind regards,

Andrej Bauer