Everything and More: A Compact History of \(\infty\)

David Foster Wallace is probably best known for Infinite Jest, a sprawling, ambitious novel (which are the words that reviewers are supposed to use about books that break the 1000 page barrier, even when we haven't read them). He has also published several collections of short stories and a collection of essays, among other works. His latest book, Everything and More: A Compact History of ∞ is a work of nonfiction which is designed to introduce the reader to the philosophical and mathematical underpinnings of the infinite, ranging from Zeno and ancient Greece up through nineteenth century analysis and Cantorian set theory. It is the first book in WW Norton's new "Great Discoveries" series, in which non-science writers write about seminal works in science.

We should get this out of the way early on: David Foster Wallace's style of writing is not for everyone. While his fans, of which I consider myself one, like to use words such as "quirky" or "clever" to describe his writing, there are many others who would prefer to use words such as "annoying" or "pretentious". If you are the type of readers who finds mid-text interpolations, footnotes within footnotes, or prose such as "Let us assert, in a very calm and low-key way, that if you think you can detect a whiff of Aristotelianism ... you are not smelling things" annoying, then you may find this book to be more trouble than it is worth. Despite the fact that he continually refers to other parts of the book, there is no table of contents. There is no index. And the fact that the title of the book contains the lemniscate symbol rather than the word "infinity" is not an anomaly — the author consistently uses symbols and acronyms rather than spelling things out, in a way that makes it unclear whether he is doing it for abbreviation or just to be cute.

But enough about style. The substance of Everything and More covers a wide range of topics related both to things which are "infinitely large" — such as the number of integers — and things that are "infinitely small" — such as the infinitesimals needed for a rigorous theory of calculus. The book starts with a general discussion of abstraction in mathematics, the idea of proof, and the myth of the "mad genius" which Wallace thinks (and this reviewer agrees) that too many popular representations of mathematicians cling to because their authors are too lazy to see any other storylines. He also introduces the well-known paradox of Zeno as well as Galileo's paradox, which points out that two circles with the same center can have different circumferences even though their points can easily be put in bijection with one another.

§3 gives the reader a timeline of ways in which the infinite and infinitesimals appear in mathematics before the development of calculus. He then gives a mid-book "Emergency Glossary" which defines terms that any mathematician will be familiar with, such as Fourier Series and Integrals, but that the general reader will not be. §4 discusses the development of calculus (though Wallace makes a point of saying "we're not going to get too much into the Newton-v.-Liebniz thing") and §5 describes "the juggernaut-like momentum of the pre-Cantor mathematical context" which includes discussions of both the mathematics and the lives of Dedekind, Weierstrass, and Riemann. There is also some biographical information on Cantor as well as a discussion of epsilon-delta proofs that any of our calculus students would find illuminating if not completely rigorous. The final two chapters contain the mathematical meat of the book, including long discussions of Dedekind cuts, power sets, Cantor's diagonalization proof and much much more. The book alludes a bit to Gödel's Incompleteness Theorem, but does not go into much depth, possibly because Rebecca Goldstein will be writing a book on it later in the series.

The natural question one asks when one picks up a book such as this — as a reader and even moreso as a reviewer — is "What is the target audience?" We all write very differently if we are writing for specialists, for a general mathematical audience, or for a truly general audience. The answer to this question is rarely obvious, but figuring out the intended audience for the book under review was particularly perplexing to this reviewer. In interviews, Wallace has said that he views this book as accessible to the general public, doing for (parts of) mathematics what Stephen Pinker does for linguistics or what Dawkins and Gould do for biology. Furthermore, he claims that readers with essentially no mathematical background will be able to follow his book. I don't believe this claim. While I think he presents the ideas from calculus well (for the most part), it seems to me that it would be very difficult for a reader without at least two solid semesters of calculus — and ideally a bit more than that — to follow the exposition. However, I imagine very few readers of this review have not taught calculus, let alone taken it, so I will not belabor this point.

On the other end of the spectrum, Wallace certainly did not write this book with a target audience of mathematicians in mind. Most mathematicians will already know all of the mathematics that he covers in this book. Many mathematicians will also know the various historical and philosophical details that he covers, but many will not. More specifically, I did not. And it is these aspects of the book that I recommend. If one skims over large chunks of the mathematics you are left with an engaging brief history and philosophy of one aspect of mathematics which is far more literate and exciting than many of the books in this genre.

In the previous paragraph, I argued that Wallace has succeeded in writing a book of history that a mathematically sophisticated audience will appreciate. But he did not set out to write a history book and there is much mathematical content in Everything and More. For the most part he gets the mathematical content correct. There are a handful of places where he makes incorrect mathematical statements, or explains things in ways that do not seem the most natural to a mathematician. But most of the time he admits up front that he is oversimplifying ("Technically, it's a lot more complicated than that. But it's really not", he writes. See above comments about cleverness-v.-pretension). It is often said that in order to write expository mathematics one needs to lie, but should lie honestly. I think that this is exactly what Wallace does. (It should be pointed out that several reviewers, including Jim Holt writing in The New Yorker, do not agree with this assessment and find Wallace's lies and oversimplifications to be troublesome.) If one wanted to write a review picking apart the technical bits of the book, it would not be difficult to do so. But I think such a review would do Wallace and the reader an injustice.

As far as pop-math books go, this one is certainly not as deep or rigorous as Gödel, Escher, Bach, which may disappoint some readers but also makes it a book that is eminently more accessible. On the other hand, it is not as light and easy (and sloppy) as Amir Aczel's books. It falls somewhere in the middle of this range, and perhaps this is where the book's target audience can be found — in the middle ground between the dichotomy of professional mathematician and layperson. In particular, I think this book would work extremely well for those students who have just completed a calculus sequence before they start a Rudinesque "Introduction to Analysis" course. The kind of student that knows what an epsilon-delta proof looks like but doesn't really get it. Wallace is able to motivate the rigor of analysis and set theory both historically and philosophically while explaining just enough of the actual mathematics to be fair. And, hopefully, to whet their appetite for more. I am not sure that this target audience is large enough to make a book such as this one successful, but I hope it does, as I would welcome more books in this style on my bookshelf

Darren Glass is a VIGRE Assistant Professor of Mathematics at Columbia University. His research interests include number theory, algebraic geometry, and cryptography. He can be reached at glass@math.columbia.edu.