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

[Andrej Bauer]

Dear Monroe,

On Tue, May 19, 2009 at 8:33 AM, Monroe Eskew <meskew at math.uci.edu> wrote:
>> Nevertheless, too much reliance onclassical set theory has its negative effects, such as:
>> - it makes topologists blind to the topological notion of overtness,which is just as important as compactness (Paul Taylor has said >moreabout this in a parallel post)
>
> Of course, a classical mathematician (set theorist or not) would find
> anti-classical notions to be either meaningless or trivial when
> interpreted from their viewpoint, and would thus not consider this
> blindness.

I think we're in agreement here. It's a bit like when a crazy person
does not recognize himself to be crazy. In the case of the topological
property of overtness, classical mathematicians are blind to it
_because_ they see it as a trivial property that all spaces satisfy.

> Almost all engineers and scientists are trained in classical rather
> than anti-classical mathematics, and all these examples can be
> adequately understood with classical logic (and standard analysis, and
> Robinson's nonstandard analysis).

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.

> I don't know what you mean.  Set theory does not contradict computer
> science.  As I said, set theory is designed to be inclusive of all
> classical mathematics.

Classical mathematics creates the wrong kind of mathematical intuition
and expectations for a computer scientist to have. He is much better
off knowing (also) constructive mathematics, because it fits more
naturally with the nature of computation.

Here is an example: a well educated computer scientist typically knows
that a polynomial (with real coefficients) has finitely many roots. He
therefore naturally expects that there is a thing called "the number
of distinct roots of a polynomial". Surely, such a simple number can
be computed, yes? No.

> People may study any aspect of logic they wish, and that they do.

Yes, but teaching computer scientists classical logic _only_ does not
exactly help them.

> There is a huge and active discipline called "theoretical computer science."

I hardly need to be reminded that there is such a discipline.

With kind regards,

Andrej Bauer