Review: Everything and More

Everything and More: A Compact History of Infinity
by David Foster Wallace
Weidenfeld & Nicolson, 2003, 336pp

I bought this book in a second-hand bookshop a few weeks ago. It looked intriguing enough for me to pick up and flip through the pages, probably because of the mental connection to Infinite Jest one naturally makes. And also because I’ve always found Wallace’s non-fiction more interesting than his fiction.

First, on the intended audience. That’s people who

1. care enough about the history of infinity or philosophy of mathematics,
2. and have enough exposure to math to be able to follow simple proofs but ideally not enough to have rigorously dealt with infinite sets.

I’m in that target audience, though I’m not sure how many others like me are out there. If the cultural impact of Gödel, Escher, Bach (which I feel is aimed at the same type of reader) is any indication, probably quite a few.

Next, the writing style. I see how page-long footnotes can seem gimmicky¹, but I’ve never really minded them. No concept can be naturally explained and no story can be naturally told in a fully linear fashion; footnotes are one of the few tools that writers have to present text nonlinearly. Of course, Everything and More also has abbreviation tables, emergency glossaries, optional and required interpolations, all of which can seem somewhat silly but, in my view, are quite useful—especially in a book like this.

Now, the book subject itself: infinity—or, more accurately, its history. The book is laid out loosely chronologically with occasional tangents (=footnotes/interpolations) to explain math concepts in more depth. The destination is clear from the start though—Wallace opens by crowning Georg Cantor “the most important mathematician of the nineteenth century”, and everything that follows builds toward Cantor’s theories of sets and infinities.

The general history—Babylonians, Greeks, Aquinas, Newton/Leibniz, Weierstrass—is interesting enough, but it’s the last quarter of the book where he dives into the development of set theory (and its problems) that is the most exciting. Wallace has the ability to reveal just enough information to make math feel like a good mystery novel. When he explains the “size” of infinite sets, he shows that the set of real numbers is larger than the set of integers. A curious reader may be tempted to ask about other sets smaller than the reals, but Wallace holds that back until much later, when you learn it was exactly the sort of question Cantor obsessed over, and that it resisted solution for decades. [THE REST OF THE PARAGRAPH MAY BE CONSIDERED A MATH SPOILER] In fact, mathematicians eventually figure out that this statement can’t be proved or disproved from the standard axioms of set theory. I was aware of Gödel’s incompleteness theorems before, but this was the first time I encountered it not just theoretically, but through a problem that I understood.

Everything and More occupies a strange place. It’s not quite popular math, but it isn’t a textbook either. Though you can (and I’d say, should) treat it as a textbook—making notes and proving statements—if you want to truly learn something, not just in a pop-mathy/sciency way. And if you’re enthusiastic enough about the subject, you’ll find yourself caught up in this math mystery novel, trying to anticipate the next move.