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 <email@example.com> 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