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

[Timothy Y. Chow]

Andrej Bauer <andrej.bauer at andrej.com> wrote:
> I asked several physicists at my department whether they do anything in 
> their classroom or research by using the standard epsilon-delta 
> technique from analysis. The answer seems to be negative. They always 
> argue informally using infinitesimals. Which makes me wonder why we (the 
> math teachers) teach physics majors all those epsilons and deltas. They 
> don't need them. They can and do get along perfectly well with dx's and 
> dy's. So why don't we teach them dx's and dy's instead? I am pretty 
> certain physicists don't particularly care what logic comes with the 
> infinitesimals, as long as it works for them.

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.

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.

Another questionable presupposition is that it is the job of 
mathematicians to teach physicists how to reason like a physicist.  Once I 
put it this starkly, the absurdity should be clear.  It should be the 
*physicists* who teach physics students how to reason like a physicist.  
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.  Mathematicians should teach students how to reason like a 
mathematician.  It may surprise you to hear this, but sometimes physicists 
can benefit from familiarity with mathematical ways of thinking.  If the 
mathematicians don't try to teach mathematical ways of thinking, then who 
will?

Tim Chow