296pp
10/18/07 - 11/11/07
When I was an undergrad, I read a lot more science than I do now: popular science writing, science biographies, and philosophy of science. This, in hindsight, is one of the few ways I'll acknowledge that going to an engineering school actually molded me. Not that my own studies had anything to do with any real engineering, science or math, but my friends mostly came from the computer science and physics schools. I don't know that I was aware of it at the time, but the people I surrounded myself with shared a peculiar set of reading habits. Very few people read literature--probably a good many would have told you that fiction doesn't matter--and everyone else seemed to pick their books from this limited pool of science books. James Gleick's Chaos was ubiquitous (I still remember the student bookstore's huge yellow piles of his FSTR, when that book came out), and so many people were reading the 25th anniversary edition of Gödel, Escher, Bach, that I'd assumed it had been assigned for some freshman lit class. And so for a couple of years there I let my peers' reading habits rub off on me. I suppose that if I'd been at a liberal arts college, I'd've been reading Dante, or whatever one reads to look deep and impress girls (which might be a nutshell summation of my understanding of a liberal arts education). Obviously, my current reading preferences have turned toward literature, and postmodern fiction in particular, but I probably have more of a taste for science writing than I would've had I not gone to Georgia Tech.
My choice of this particular book was prompted by two things: I'm currently in the middle of an audiobook version of American Prometheus: The Triumph and Tragedy of J. Robert Oppenheimer (incidentally I don't blog about audiobooks. Nor comic books or law books), and I came across this interview with R. Goldstein.
Despite having gotten about a third to halfway through the aforementioned GEB--the usual length for that book--I could not have told you a damned thing about the Incompleteness Theorems of Kurt Gödel, aside from how to pronounce his name. Maybe. Having read this book, I'm in a somewhat better position. Six months from now I will have forgotten entirely.
To be sure, this is 99% my own fault. This kind of explicatory writing about abstruse subjects cannot be read casually, or in short bursts. Often I would pick up the book, and spend ten or fifteen minutes refreshing myself as to, say, the technical meaning of "consistency" or Euclid's fifth postulate. Had I really dedicated myself to seriously reading Goldstein's explanation of incompleteness, I would not have so often lost her train of thought.
Having spoken in defense of Goldstein's writing, I can now proceed to offer criticism, hoping that Steven Pinker will not come to my apartment to beat me up.
Right in the heart of the densest, most difficult part of the book, when Goldstein's explaining the First Incompleteness Theorem, the reader--which reader obviously came to the table hoping to wrap his head around this central piece of 20th Cent mathematics--is asked to take a pretty huge leap of faith. After performing the admirable task of making Gödel numbering clear, Goldstein tells us that Pr(x), the property of "provability", is "a formally expressibly arithmetical property, albeit one that is extremely complicated, not anything that we can explicitly give here." Really? Not even, like, a little bit? I get that this is way too much math for my little head, I really do, but a sentence like that, stuck in this kind of book, bears a whiff of prestidigitation. Goldstein is in the process of blowing my mind by telling me (I think), that so abstract a property as the provability of a proposition, given the arbitrary system of Gödel numbers, can be expressed as a function, when she then tells me I'll just have to trust her (& 20th Cent mathematics) on that one. Which, I mean, I do. But I'd really appreciate it if as great and lucid writer as Goldstein at least made an attempt at my halfway understanding that particular point.
There's a lot more to this book than just Goldstein's accessible explanation of that remote area of math, even though that's the part (pp 150-188) that readers might be most interested in. It starts with a good bit of intellectual history, describing the Wiener Kreis to set the stage for a discussion of the philosophical implications of the Incompleteness Theorems. There's a really charming part at the end, where Goldstein describes her time as a grad student at Princeton, fascinated by the luminary in her own backyard. She then briefly and respectfully moves on to Gödel's paranoid demise; unfortunately the man was so withdrawn that I doubt anyone will ever be able to credibly write at length about the no-doubt fascinating world of his inner delusions.
This book comes to us from the "Great Discoveries" imprint, a hit-or-miss series of "Great Thinkers Thinking about Great Thinkers" books. I'm a big fan of DFW's Everything & More from this line, and Incompleteness could almost be read as the sequel to that book, or the next book in modern mathematics syllabus.
Strangely, between this book and American Prometheus I am accidentally getting a history of the Institute for Advanced Studies.
