fostering mathematical conversation

Expect the frequency of posting here to drop off somewhat after tomorrow.

This is because Wednesday is the first day of classes here. I originally began this blog in part because I didn't have much else to do with my time during the summer; now I will have classes and teaching and (at least in theory) research that I'm working on.

I will be taking courses in algebraic topology (homotopy theory, etc.; I'm not particularly looking forward to this one, but it's required for my program); logic and computability (using Enderton's text, A Mathematical Introduction to Logic); and probability inequalities and machine learning. (This last one is probably the one that will inspire the most posts over the course of the semester, if I had to guess.) I'm also TAing a multivariate calculus course.

The reason I mention the text for the logic course is that Enderton has produced an Author's commentary for his text, about which he says:

The purpose of these comments is to explain, section by section, what I am trying to do in the book. My hope is that the commentary will add a helpful perspective for the reader.

In addition being a place for helpful material, this website gives me the chance to post additional material. Users of the book might have varying views on this role of the website.

I welcome suggestions for how the commentary can be made more useful.

This is the sort of thing that I was talking about, to some extent, in this post about "companions to books". I gave there as an example of a companion to a textbook Bergman's A Companion To Lang's Algebra, wich were written by a professor to supplement the textbook for the course he was teaching; this is different in that it's written by the author.

From what I can gather, Enderton's book is one of the standard textbooks for an introductory logic course; presumably the many people who have taught courses based on this book have had similar sets of thoughts (although perhaps less extensive, as teaching a course takes less time than writing a book) and it would be nice to see all of those in some central place.

I said in my previous post that with this sort of companion I did not mean something like Wikipedia, because the original text would be inviolable. But in one sense I do mean something like Wikipedia -- no particular contributor's contribution would be all that valuable but they'd add up to something. The question, though, is who would use that something. The student wouldn't read it, because students are by nature lazy. Would professors teaching the courses care enough to read such a work? And more importantly, would they contribute to it?

I'd also like to point out something that cwitty mentioned in a comment here, namely the apparently dead project called "Fermat's Last Margin". Shae Erisson wrote in 2004:

The idea behind Fermat's Last Margin is...
I write a lot margin notes in books that I own, and research papers that I print out. To me, a research paper is an unfinished discussion. I like to argue, exclaim, deride, doodle, and generally get completely comfortable with academic publications that I read.
There is a downside. Andrew Bromage once made some questioning comments about a Comonads paper on the #haskell irc channel. Six months later, someone else made the same questioning comments. I realized that we can't share our margin notes! Who knows what brilliant ideas we've missed because we had one half, and someone else had the other half? Fermat's Last Margin is my answer.
The names comes from the story of Fermat's Last Theorem. The story is that Fermat wrote in the margin of a book that he had thought of a novel proof for this theorem, but the margin was too small, the proof would not fit. If Fermat had this margin I am making, it would be the last margin he would ever need.

Unfortunately that doesn't seem to have quite gotten off the ground. The point, though, is that there don't seem to be channels for communicating results on a level smaller than the research paper or the conference presentation, which causes unnecessary duplication of effort. Imagine if you couldn't share a body of mathematical ideas until you had enough of them to fill up a semester-long course or a book; chances are math never would have gotten off the ground if that were the case. Sure, there are ways for people to share ideas that aren't enough for a paper or a talk (for example, just talking to each other), but I don't think more of them would hurt, and I think the mathematical community ought to explore how to foster those sorts of interaction using the Web 2.0 paradigm. (I was trying to avoid using the words "Web 2.0" but they're a convenient shorthand for what I want to say.)