[FOM] When is it appropriate to treat isomorphism as identity?
hendrik at topoi.pooq.com
Wed May 20 14:34:05 EDT 2009
On Wed, May 20, 2009 at 01:19:10AM +0200, Andrej Bauer wrote:
> Dear Monroe,
> 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.
When I went through my undergraduate studies in math&physics, several of
the physics students couldn't make sense of the dx's and dy's. It
seemed like witchcraft. But when the math professors provided
the epsilon-delta definitions, it made sense to them, and they could
accept dx's and dy's as a shorthand way of talking about epsilons and
Later they heard the differential-geometry way of defining diffentials
as approximating linear functions. They were fine with that, too.
> 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.
True. I was completely turned off computability theory because on the
first day the prof presented us with the nonconstructive proof that
there exists an algorithm for determining whether there are seven
consecutive sevens in the decimal expansion of pi. He said it was a
sign that our intuition was no good at understanding this subject. I
took it as an indication that the subject was using the wrong logic.
> > 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.
It gets in the way.
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