This summer, with no job to speak of, I'm finding myself with a lot of time on my hands. This is not something I'm used to, but I'm somehow managing. I've been doing some light reading, going through books at a good clip. At this point, I've read most of Cory Doctorow's novels (still have his short stories to go), and I'll probably write another post about those once I make it through them all. Today, though, I'm writing about the book Everything and More: A Compact History of Infinity, by David Foster Wallace.
A couple of things to start off: First, David Foster Wallace is best known (to me, at least) as the author of Infinite Jest, a massive novel about addiction, consumption, family, tennis, and a few other things besides. It plays hard with narrative form, toying with chronology and containing about 100 pages of endnotes which are crucial to the story and must be read. I really enjoyed it, and it even got me watching tennis (although I mostly just watch the majors). Second, the book is about the mathematical development of infinity, from the Greeks to Georg Cantor, the mathematician who discovered/created most of the important proofs related to infinity that allow mathematicians to work with infinite sets in meaningful ways. So, a book about math by an author I dig? Of course I picked it up. These two things (i.e. the book's author and subject) likely will not motivate you as strongly as they did me, so I am attempting to add one more reason to that list with a positive review.
The book's content is strong, with a good overview of the development of infinity and how various problems of interest over the years spurred mathematicians to deal with (or in many cases, ignore) infinity and related concepts1, and how prevailing attitudes towards the infinite affected these developments2. So it's not just math; you get a good helping of history and philosophy as well. In this way it is similar to the book Zero: The Biography of a Dangerous Idea. I didn't like that book very much, though; it read too much like a series of magazine articles that sought to hype their subject material unrealistically. I can only stomach so many statements about the terrible power of the Void, i.e. zero. Anyway, this book is better, and less worried about the importance of its central concept (even though it is important).
And the math isn't too bad. The book was written with a reader in mind who had done well in high school math, but didn't necessarily do math in college. I would say that if you have taken a first-semester calculus course, you definitely have enough math background to read this book. Even if you haven't, as long as your eyes don't completely glaze over when somebody mentions math, you can read this book with a little effort. I mean, Zeno's Paradox is the problem given the most time in this book, and I'm pretty sure I saw that explained in a Meg Ryan movie once. Not that I watch Meg Ryan movies. Damn.
Of course, this very reduction in complexity to make the book more palatable to the general reader can be somewhat frustrating to someone who is familiar with the math being discussed. It can seem that important details are being glossed over; still, one gets the impressions that such decisions were made carefully after much tweaking to see which details could be pared down. Wallace acknowledges this glossing in at least one example, so he's not unaware. Overall, the math comes off well (most importantly, Cantor's diagonalization proof, central to his results on the relative size of certain infinities, and related proofs are presented well), but I did find the discussion of Cantor's work on the Uniqueness Theorem for Trig Series3 to be murky. (To be fair, part of this is undoubtedly my tendency to confuse sequences and series; there is a difference.)
But what makes the book great is Wallace's style. It's conversational, humorous, and knowledgeable. It reads like a series of entertaining lectures by a cool professor given to small asides and allusions to interesting facts that he just doesn't have time to cover. Wallace spins off footnotes like nobody's business4, and this is often where he sneaks the funniest remarks. He's not shy with parenthetical remarks, either. I would often count 3 right parentheses at the end of a sentence, meaning he had doubly nested parenthetical statements. It was the first time I could think of prose that reminded me of LISP in any way. In short, it's written in the sort of focused but cross-referenced (so to speak) style I like.
So, if you're curious about infinity, and want to read a book that traces it from the Greeks and Zeno to roughly the present day, I recommend this one. If someone ever told you to read the book Zero: A Biography..., read this one instead. And hey, I'll probably have another post up in a week or two about some fiction, if that's more your thing.
1: Good example here: the creation of calculus to deal with a wide variety of physical problems. Calculus requires one to use infinitely small quantities, but there was about a century of hand-waving about these before mathematicians figured out exactly why they could do math using such quantities.
2: Another good example: The Greeks as a culture didn't have a concept of infinity (or zero), and this was part of they reason they didn't manage to create calculus about 2 millenia before it ultimately came about. At least one Greek mathematician, Eudoxus, had the right idea, just no way to even acknowledge the infinities involved.
3: Don't worry if that sounds crazy, it's the hardest math in the entire book. It's not typical.
4: A device I'm rather fond of, clearly.
A couple of things to start off: First, David Foster Wallace is best known (to me, at least) as the author of Infinite Jest, a massive novel about addiction, consumption, family, tennis, and a few other things besides. It plays hard with narrative form, toying with chronology and containing about 100 pages of endnotes which are crucial to the story and must be read. I really enjoyed it, and it even got me watching tennis (although I mostly just watch the majors). Second, the book is about the mathematical development of infinity, from the Greeks to Georg Cantor, the mathematician who discovered/created most of the important proofs related to infinity that allow mathematicians to work with infinite sets in meaningful ways. So, a book about math by an author I dig? Of course I picked it up. These two things (i.e. the book's author and subject) likely will not motivate you as strongly as they did me, so I am attempting to add one more reason to that list with a positive review.
The book's content is strong, with a good overview of the development of infinity and how various problems of interest over the years spurred mathematicians to deal with (or in many cases, ignore) infinity and related concepts1, and how prevailing attitudes towards the infinite affected these developments2. So it's not just math; you get a good helping of history and philosophy as well. In this way it is similar to the book Zero: The Biography of a Dangerous Idea. I didn't like that book very much, though; it read too much like a series of magazine articles that sought to hype their subject material unrealistically. I can only stomach so many statements about the terrible power of the Void, i.e. zero. Anyway, this book is better, and less worried about the importance of its central concept (even though it is important).
And the math isn't too bad. The book was written with a reader in mind who had done well in high school math, but didn't necessarily do math in college. I would say that if you have taken a first-semester calculus course, you definitely have enough math background to read this book. Even if you haven't, as long as your eyes don't completely glaze over when somebody mentions math, you can read this book with a little effort. I mean, Zeno's Paradox is the problem given the most time in this book, and I'm pretty sure I saw that explained in a Meg Ryan movie once. Not that I watch Meg Ryan movies. Damn.
Of course, this very reduction in complexity to make the book more palatable to the general reader can be somewhat frustrating to someone who is familiar with the math being discussed. It can seem that important details are being glossed over; still, one gets the impressions that such decisions were made carefully after much tweaking to see which details could be pared down. Wallace acknowledges this glossing in at least one example, so he's not unaware. Overall, the math comes off well (most importantly, Cantor's diagonalization proof, central to his results on the relative size of certain infinities, and related proofs are presented well), but I did find the discussion of Cantor's work on the Uniqueness Theorem for Trig Series3 to be murky. (To be fair, part of this is undoubtedly my tendency to confuse sequences and series; there is a difference.)
But what makes the book great is Wallace's style. It's conversational, humorous, and knowledgeable. It reads like a series of entertaining lectures by a cool professor given to small asides and allusions to interesting facts that he just doesn't have time to cover. Wallace spins off footnotes like nobody's business4, and this is often where he sneaks the funniest remarks. He's not shy with parenthetical remarks, either. I would often count 3 right parentheses at the end of a sentence, meaning he had doubly nested parenthetical statements. It was the first time I could think of prose that reminded me of LISP in any way. In short, it's written in the sort of focused but cross-referenced (so to speak) style I like.
So, if you're curious about infinity, and want to read a book that traces it from the Greeks and Zeno to roughly the present day, I recommend this one. If someone ever told you to read the book Zero: A Biography..., read this one instead. And hey, I'll probably have another post up in a week or two about some fiction, if that's more your thing.
1: Good example here: the creation of calculus to deal with a wide variety of physical problems. Calculus requires one to use infinitely small quantities, but there was about a century of hand-waving about these before mathematicians figured out exactly why they could do math using such quantities.
2: Another good example: The Greeks as a culture didn't have a concept of infinity (or zero), and this was part of they reason they didn't manage to create calculus about 2 millenia before it ultimately came about. At least one Greek mathematician, Eudoxus, had the right idea, just no way to even acknowledge the infinities involved.
3: Don't worry if that sounds crazy, it's the hardest math in the entire book. It's not typical.
4: A device I'm rather fond of, clearly.
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