Pondering the Infinite

PONDERING THE INFINITE….I finished reading Everything and More today, David Foster Wallace’s first new book since Infinite Jest in 1996. It’s not a novel, though, it’s a survey of the mathematical problems that ultimately led to Georg Cantor’s development of modern set theory and transfinite analysis. Some miscellaneous comments:

• No index. That’s a real bummer for a work of nonfiction.

• Karl Weierstrass, a heavy-duty 19th century mathematician, did not allow his students to take notes in class.

• Odd factoid according to Wallace: “Almost all history’s great philosophers never married. Heidegger’s the only real exception. The great mathematicians are nuptially split about 50/50, still way below the civilian average. No cogent explanation on record; feel free to hypothesize.”

• I continue to not really get Cantor’s diagonal proof. It seems like (a) it could be applied just as well to the rationals as to the reals, which is obviously wrong, and (b) it doesn’t really work anyway since the diagonal number can be made arbitrarily close to one of the numbers already on the list ? although I admit that this second objection is a little slippery. I don’t know if my lack of understanding is due to relying on popular explications rather than the actual technical proof itself, or is simply because I’m missing something.

  I realize this paragraph is probably meaningless to nearly everyone reading it. Apologies.

  UPDATE: Ah, the power of Google! The answer to my first objection is here, and my second objection is probably just logical thumbsucking anyway. I guess that Cantor fellow was a pretty smart cookie.

• Wallace attended Urbana University Sr. High School in Urbana, and the text of the book makes it clear that his math classes there covered calculus all the way through partial differential equations plus discussion of set theory and transfinite arithmetic. Even granting that these were AP classes, holy cow. Do they really teach this kind of stuff in high school these days?

• Typically for Wallace, Everything and More is filled with footnotes. Unfortunately, due to a quirk of typography, they seemed to blend in with the text so much that I kept missing them. Then I had to go back and try to figure out where the footnote was before I turned the page. Annoying.

• Wallace insists in several places that you really don’t need any kind of college math background to understand his book. He’s lying. It’s actually surprisingly technical for a popular account, but that also means it’s symbol-laden enough to scare off anyone who finds the routine use of capital Greek letters offputting.

Transfinite numbers, of course, are merely the 19th century’s contribution to the menagerie of numbers that no one believed in at first but then eventually did, following in the rich tradition of irrationals, zero, negative numbers, imaginary numbers, and transcendentals. Eventually all of these things came to be widely accepted, putatively because they were finally put on a rigorous basis but really because the old guys died off and everyone else just clapped their hands and decided to believe. After all, it’s quite a coincidence that every type of number that’s actually useful has eventually found a supposedly rigorous treatment, isn’t it? One might be excused for thinking that usefulness itself is the only thing that anyone really cares about.

Hell, after a thousand years we still can’t divide by zero. How rigorous is that?