God Plays Dice

Is massively collaborative mathematics possible?, asks Tim Gowers.

I cashed in 5.793 kilograms of assorted coins today at the bank. That's 115ドル.19, for a density of 19ドル.88 per kilogram; I know the weight because the bank's coin counting machine gave a receipt that had the numbers of each type of coin I received. I then immediately proceeded to the bookstore and purchased the The Princeton Companion to Mathematics (That was 106ドル with sales tax; perhaps I should have bought it from Amazon.com, where it's cheaper. Oh well, it's too late now. But at ten cents a page it's still a good deal.) I told myself I wouldn't buy it, but I'm doing some private tutoring on the side, and so there's a bit of extra money floating around, and I couldn't help myself...

I want to compliment the design of it; the side of the book which faces out on the shelf has "The Princeton Companion to" in small letters and then "Mathematics" in large letters, perhaps giving the impression that the book contains all of mathematics. This is not true, but from the portions of it I've seen online and the part I've read so far, it seems to admirably solve the optimization problem of squishing down mathematics to an object that only weighs five pounds. That's less than the coins I hauled to the bank to get the cash I paid for it! For links to various reviews, see this entry from Tim Gowers' blog. Gowers is the editor, and also wrote substantial parts of the book, although it has many other contributors; you can play the party game "how many of these names do I recognize?" I was interested to see that I recognize many more than I would have when I started grad school.

Yes, I'm actually trying to read it cover-to-cover. Like many mathematicians, there are a bunch of things that people seem to assume I'm familiar with, but I only heard about very briefly in some course in my first year of grad school in which I was holding on by the skin of my teeth. Indeed, this is one of the uses that's recommended in the introduction! I will resist the urge to turn this blog into a Companion to the Princeton Companion to Mathematics.

Tim Gowers on the forthcoming launch of the Tricki, essentially a wiki which will consist of mathematical problem-solving techniques or "tricks". I'm massively oversimplifying; read his post.

Here is an MP3 interview with Tim Gowers about The Princeton Companion to Mathematics, of which he is one of the editors.

It's 100ドル. Who wants to buy it for me?

From Doron Zeilberger's chapter on "Enumerative and Algebraic Combinatorics, to be included in the currently-in-preparation Princeton Companion to Mathematics

"The fundamental theorem of enumeration, independently discovered by several anonymous cave dwellers, states that
|A| = Σ_a∈A 1.
In words: the number of elements of A is the sum over all elements of A of the constant function 1."

Sounds kind of silly, but it's true. The whole chapter is a nice fourteen-page answer to "what is enumerative combinatorics?", mentioning most of the classic problems and most common methods of solution, which appears to be its raison d'être; I know most of this stuff but I can imagine how useful similar blurbs on subjects I'm not so familiar with would be, and indeed most of the book is intended to be at about the first-year undergraduate level; that's low enough that I should be able to read it without stopping for breath. (The guidelines for contributors say that the articles about various subjects should be something like the beginning of a very good colloquium talk, the sort where you really get the feeling that you know something about how some other area of mathematics works.) The PCM has a semi-official blog, which is Tim Gowers' blog. Several dozen of the component articles are available online, on a password-protected site; the password is in the linked-to post by Gowers. I suspect I'll have more to say about the PCM in the future.