Date: Nov 26, 2001 1:00 AM
Author: Chan-Ho  Suh
Subject: Re: Using NLP to learn math





Lee Lady wrote:

> Everything here is more or less well known to mathematicians.
> The discussion is really for the benefit of NLP people, but
> is being cross-posted to sci.math primarily so that I can
> be corrected when I say something really stupid.
>
> In article <3C006FF1.83C3F64@math.ucla.edu>,
> Chan-Ho Suh  <csuh@math.ucla.edu> wrote:
> >lady@bogus.Hawaii.Edu wrote:
>         ....
> >> In order to be able to figure things out visually in geometry
> >> and topology, you have to be able to consciously manipulate
> >> your images.
> >
> >Exactly, which is why your comments below really confuse me.
>
> Yeah, well I was not educated as a topologist.  I just
> had the usual general topology stuff, and then a year's
> course in algebraic topology, half of which was taught
> by Guido Lehner, a geometric topologist who stated that
> he was teaching algebraic topology in order to learn the
> subject.  Lehner could draw beautiful pictures, but he
> was also capable of incredibly pedantic arguments proving
> things that were visually pretty much self-evident.  (Of course
> it's essential that students learn to write careful proofs,
> so this was not completely inappropriate in a beginning course.
> One learns to write proofs by starting out with things that
> are easy.)
>

I can't say how "self-evident" those things were, but one thing I can say is
that an important part of a topologist's education is learning what is obvious
and what is not.  I found personally that sometimes I would think something
was very obvious (visually) but the prof would go on and on for days
sometimes, proving that 'obvious' something.  Later I would learn it was not
so obvious and only my ignorance made it seem so.  Other times I would think
something was not obvious and be very annoyed when a topologist would just
brush details aside and comment, "It's pretty clear."  But later (in most
cases) I would find it was relatively trivial.  Sometimes I think topologists
can be a little careless, but for the most part I've found that the
hand-waving is quite rigorous after all.


>
> Then I sat in on a few seminars, one of which was devoted to
> the generalized Schoenflies theorem, and another of which
> was taught by Morton Brown at UCSD, where he was visiting
> for a year.  I was surprised by the extent to which Brown
> used algebraic topology, since (as I recall) he was one of
> Bing's students.
>

I hope at least in this Brown seminar you saw some great visualizing going
on.  Brown is probably the prototypical 1960s topologist.  I know Bing draws
lots of pictures in his papers, while Brown typically doesn't, but one of his
students, Marshall Cohen, has told me that he picked Brown as his advisor
because he thought Brown was a very precise, meticulous guy.  Then one day
Brown comes in to his geometric topology class and "started drawing these
huge, gory pictures."

> I've looked through a lot of books on topology, because when
> I was a student I used to like just hanging out in the math
> library and looking at various books on subjects that seemed
> interesting.  And I never came across a book that presented
> this sort of visualization.  Of course an incredible number
> of books on mathematics have been published since the Sixties.

This is really unfortunate.  Somehow you missed all the standard texts
(published in the 70s) which are very visual.  I don't think it was until the
70s or so until the kind of visual mathematics that were being done by the 60s
topologists was condensed and put into books.  Rourke&Sanderson's classic text
on PL topology, Rolfsen's Knots and Links, etc., weren't written until the
early 70s.  Having a good library helps of course.  There's a lot of this kind
of material in various article anthologies and conference books from the 60s,
but your library probably didn't have those.

What's really funny is that nowadays doing computer animations of topological
stuff is all the rage.  I think that's especially good since nowadays topology
students tend to be more of the algebraic sort than the geometric sort.