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Re: Using NLP to learn math
Posted:
Nov 25, 2001 4:04 PM
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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.
Yes, I really didn't say what I meant very carefully. It's a
good thing my article got cross-posted to sci.math, but it's
also a bit difficult speaking to two very different audiences
at once.
There's no doubt at all that there are certain parts of
geometry and topology where it's absolutely essential to
be able to see pictures of what's going on. Either one
has to be able to mentally visualize these pictures or
one has to be able to draw them on paper. (And I believe
that even the best geometers and topologists find the
need to actually draw things on paper much of the time.
It's just too complicated trying to keep track of a
complicated picture in one's mind.)
The main point I wanted to make in the paragraph quoted
above, in response to a previous comment by an NLP
person, is that to the extent that one is using an
image in doing geometry or topology, that image has to be
consciously available, because one needs to do very
deliberate manipulation of that image and that has to
be done on a conscious level.
There's also, however, no doubt that there are certain parts
of geometry and topology that go beyond things which one
can visualize or draw pictures of.
A large part of geometry and topology is done on a
formal level, i.e. by means of verbal and symbolic arguments.
For instance the use of homology groups and homotopy
groups and homological algebra and the like.
Unfortunately, in my article I used the phrase "geometric
topology," which for mathematicians is a technical term
and refers to that part of topology where visual reasoning
tend in fact to be the most prevalent. So much so
that one sometimes hears mathematicians in other disciplines
complain, "I'm not sure those people ever actually prove
anything; all they ever do is draw pictures."
> Your
>topological education seems rather atypical.
.....
>> In fact, when I took classes in geometric topology, I found
>> that my teachers, some very good topologists, very rarely
>> talked in terms of visualizations at all. Instead, they worked
>> on a purely formal (i.e. verbal and algebraic) level.
>
>What planet did you take these classes on?? Your profs must have been
>algebraists in disguise! Geometric topologists are a very visual bunch;
>to give you a source, the visual constructions in Dale Rolfsen's Knots
>and Links are *very* typical. I've met geometric topologists from half
>a dozen universities (not a large sample I'll admit) and they tend to
>draw really gory pictures that get them labelled as 'hand-wavers' by
>algebraist types.
>
>Also, if you read papers on geometric topology, you should find that
>these kinds of visualizations are done often.
You're completely right. Which makes me, um, wrong. Or at least
the way I stated things was, for a mathematical audience, quite
misleading.
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.)
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 really wanted to learn more algebraic topology, but when
people started talking about things like suspensions and
smash products, I found that I just couldn't visualize the
process. And when I heard lectures which explained these
topics, they didn't even try to explain the concepts
visually.
This was about 1968, and I think this was the time when
surgery was becoming all the rage. My friends who were
topology students at UCSD were taking seminars where surgery
seemed to be the primary tool. But by that point, I'd
become discouraged and decided that since I was never going
to become a topologist, it was just not worth the effort
to learn more.
But then I could also tell you about differential geometry.
The introductory differential geometry course I had was
taught out of the first chapter of Helgason, and contained
nothing whatsoever that any ordinary mortal could identify
as geometry. Not only do I not remember the professor
ever drawing any pictures on the board (except for the
very generic abstract circles that every mathematician of
every sort draws and which can represent anything one wants
them to), but it didn't seem like pictures were even
possible. I worked on on reading various books on the
subject and heard lots of talks by differential geometers,
and I kept thinking, "Why do they call this stuff geometry?"
What a relief it was when Spivak started publishing his
series of books which actually deigned to explain what
the intuitive content of the theorems was. But by
then I was no longer a student and had given up on the
subject, so I only read volume one of Spivak's series.
>This is standard! Many of the classic texts describe the same
>construction you wrote about (that I snipped for brevity). In fact,
>this construction is used *all the time*. Especially when you do knots
>or surgery on links, and countless more examples. I suppose certain
>algebraic topologists may not know this (although I doubt it), but
>certainly anybody teaching a geometric topology class should know this.
>And the drawings you made of the tesseract are done in analogous
>situations *all the time*.
Well, the tesseract representation is fairly obvious. And yet
I have sat in mathematics courses (undoubtedly not geometric
topology, though) and heard the professor say, "So if we go
to four dimensions, then we get a four-dimensional cube, which
is technically called a tesseract, and now we can't draw a picture
any more," and I'd be sitting there thinking, "Well I certainly
can!"
I don't know what course it was where a proof was presented that
the three-sphere is the union of two solid tori. Maybe it was
a general topology course. But I do remember that I had to
figure out the visualization for myself, and spent several days
doing it.
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.
And then eventually, more or less during the Eighties, after
I came to the University of Hawaii, my interests turned to
directions other than mathematics, because it started seeming
stupid to dedicate my life to something that I would never
be rewarded very well for when there were so many other things
that attracted me.
If I'd been able to stay at the University of Kansas, I
would probably have spent the rest of my life as a serious
mathematician. That was a place were my research was
definitely financially rewarded. But I got fired because
some faculty in non-science departments wanted to teach
the mathematics department a lesson. And then I came to the
University of Hawaii where everyone has always treated me
extremely well (often better than I deserve), but where the
bureaucratic structure does not provide any possibility of
rewarding faculty who do good work. During the Reagan era we
started going through a period of alarming inflation, and the
State of Hawaii grudgingly provided only very small across-the-board
salary increases, and I had a daughter in college, and at
the time it seemed obvious that within a few years it would
no longer be financially feasible for me to remain in the
academic world. (One possibility I seriously considered was
becoming a professional Russian translator. My Russian was
at that time almost good enough for that, and for several
years I was a member of the American Translators Association.)
Well, eventually things got better. But by then I'd already
reevaluated the direction of my life. Learned NLP, for one
thing, although from what I saw, it didn't seem realistic to
try and one a living that way.
--
Trying to understand learning by studying schooling
is rather like trying to understand sexuality by studying bordellos.
-- Mary Catherine Bateson, Peripheral Visions
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