Everything and More: A Compact History of Infinity.

I knew David Foster Wallace was brilliant when I read Infinite Jest, a wildly imaginative, sprawling novel that showcased DFW’s prodigious vocabulary as well as his deep knowledge of a variety of seemingly unrelated subjects. Even with that background, I was flabbergasted by Everything and More: A Compact History of Infinity, in which DFW delves into abstract set theory and other similarly abstruse topics from the history of math, explaining much of it lucidly and with humor until he gets too close to the finish to avoid relying on the reader to understand more of set theory than most readers will.

The book is less an explanation of the number infinity – which isn’t a single number, at least not in the sense that 1 or 5 or π or √2 – than the history of mathematicians’ attempts to deal with it. DFW starts with the Greeks, where most math stories begin anyway, even though the Greeks didn’t like or accept infinity or zero or the irrationals. (Zero came from Indian mathematicians, and reached Europe by way of Arab mathematicians quite a bit later.) The Greeks encountered questions around infinity, particularly in the famous paradoxes of Zeno, who liked to play semantic games around what we now refer to as convergent series – a sum of a series of terms that never ends but that approaches a specific limit as the number of terms grows. (In a related note, DFW fails to answer the question of how Zeno never got punched in the face for coming up with these paradoxes.) This discomfort with infinity continued through the writings of Aristotle and the Catholic Church’s influence over all manner of academic research, which included the idea that infinity was the sole province of God rather than of man, meaning we never got anywhere with infinity until the end of the Dark Ages and the separation of mathematics and religion during the Renaissance.

The pace of the narrative picks up at that point thanks to the explosion of advances in math and related areas of science. The empirical foundation that limited mathematical explorations until the 1600s is tossed aside in favor of more abstract thinking, with appearances by Kepler, Newton, and my homeboy Galileo, as trigonometry and eventually calculus displace geometry as the central philosophy guiding mathematical thinking and what we now think of as number theory. DFW presents an extraordinarily clear explanation of calculus, especially the infitesimals that underpin differentiation and integration and, as the name implies, connect it to the main topic of the book. The goal here is to get to Georg Cantor, the brilliant and mentally ill mathematician whose work remains the foundation of modern set theory and who was the first to recognize that there are different degrees of infinity (ℵ₀ and ℵ₁, at the least) but died unable to prove that those two infinities had no other infinities between them.

DFW’s writing is clear and witty thoughout the book, with many examples drawn from a former professor of his that help elucidate many of the more recondite concepts around infinity. His explanations of one-to-one mapping and Cantor’s diagonalization method of proving that real numbers are nondenumerable are outstanding, especially the latter, which I knew was true but still wanted to disbelieve because it just sounds impossible. Unfortunately, in the last 40-50 pages of the book, DFW gets so far down the set theory rabbit-hole that I found it increasingly hard to follow, such as discussions of ordinality versus cardinality and power sets of power sets. I got off the math train in college after multivariate calculus with vectors, in part because continuing meant pushing into more abstract areas – linear algebra was the next course, which starts the shift from empirical math to abstract – but that left me a little lost as Everything and More slid into Cantor’s work on the various infinities and work on numerability of sets.

Cantor’s transfinite numbers are the real goal of the narrative here, rather than what I would call the lay opinion of ∞ (what Cantor referred to as “absolute infinity”). A transfinite number is infinite in that it is greater than all of the finite numbers, but has some properties in common with the finites. If you’re familiar with the ℵ₀ I mentioned above – the first transfinite cardinal number, corresponding to the number of members (cardinality) of the set of natural numbers (non-negative integers). Cantor’s continuum hypothesis, which appeared first on the famous list of unsolved math problems David Hilbert presented in 1900, posited that there was no set with cardinality (number of members) between the natural numbers and the real numbers (the cardinality of which Cantor designated as ℵ₁). The hypothesis itself may be unprovable, at least within the confines of Zermelo-Fraenkel set theory … which DFW mentions but doesn’t explain, concluding instead with the explanation that later work by Kurt Gödel (the incompleteness guy) and Paul Cohen (who proved that the hypothesis and the ZFC’s axiom of choice were independent) set the question aside without really solving it. At least, I think that’s what he said, because I was just barely treading water by the final page. Which also made me wonder if all of these reviewers quoted as giving the book raves actually finished and understood the whole thing; I imagine the number of people who have sufficient math background to follow DFW down to the bitter end is pretty small.

Apropos of nothing else, the biggest laugh I got from the book was when DFW referred to a mathematician as a world-class pleonast, which is the pot writing a three-page letter to the editor about the mote in the kettle’s eye.

Next up: Ned Beauman’s 2012 novel The Teleportation Accident, recommended by a fellow bibliophile I met in New York in August.