Something like an oil painting of what looks like a monk and a scowling dude in a suit, walking together through a pastoral scene.
There’s great promise in this book, which it fails to live up to. It offers too little mathematics to appeal to even recreational mathematicians, and more mathematics than non-mathematically-inclined readers would be able to stomach. At the same time, it offers basically no interesting theology. It does, on the other hand, spend loads of time name-dropping Russian and French mathematicians for no good reason, and tells us all about persecution of academics during the Soviet era. It ought to be called “Russian Mathematicians: Some Things That Happened To Them For 75 Years Or So.”

The book’s fundamental assertion is that religious mysticism in Russia helped fuel the Russians’ great mathematical discoveries, and drove Russia ahead of France by the early part of the 20th century. That’s fine as far as it goes, but it obviously needs more than just some correlation to prove it. All Graham and Kantor have to offer is correlation: this one episode of mysticism happened, and this other episode of Russian mathematical flowering happened. We’re meant to conclude that the one caused the other.

It’s seemed to me, on the other hand, that at least since the great Kolmogorov, Russians have been particularly fond of abstraction for its own sake. This has allowed them to go off in mathematical directions of which the French daren’t dream. Did the Russian talent for abstraction drive their mathematical accomplishments as much as mysticism did? It seems to me that there could be many causes for the Russian mathematical flowering; Graham and Kantor focus on just one.

The reason they focus on just the one is that they’re interested in a particular Russian sect of Christianity, persecuted before and during the Soviet era, called the Name Worshippers. The Name Worshippers apparently believed that one could reach a trance-like state and come closer to god by repeating a particular prayer endlessly. The mathematics of infinity, pioneered by George Cantor, seemed to them a natural fit, for reasons that Graham and Kantor never make clear.

Cantor is most famous for his demonstration that certain infinities are larger than others — an idea that is very hard for many people to stomach, but which is rather easy to demonstrate and utterly commonplace in mathematics today. It was Bertrand Russell, I believe, who used a very straightforward example to demonstrate this. Suppose, said Russell, that you have a village in which a) every man has exactly one wife, b) every wife has exactly one husband, and c) no one marries outside the town. Then it doesn’t matter whether there are finitely many or infinitely many people in the town — you can be certain that there are just as many women as men. Mathematicians say that there is a “bijection” between the set of men in the town and the set of women in the town. If, on the other hand, there were an unmarried woman in the town and no unmarried men, then you could conclude that there were more women than men.

That’s where Cantor starts, in a discussion of very general objects called “sets.” A set is a collection of things bearing some properties in common — say, the set of all people alive and standing within the legal boundaries of Cambridge, Massachusetts at 11:28:00 a.m. on August 12, 2010, along with their couches. Two sets have the same “cardinality” if there exists a bijection between them. Cantor goes on to establish that can exist no bijection between the set of integers and the set of real numbers — that is, that the real numbers (rational numbers plus irrational numbers) is larger than the set of integers. He uses a very clever trick called the diagonal argument in which he first supposes that there *does* exist a bijection between the reals and the integers, then shows that such a bijection always leaves out some real numbers. It is unavoidably true that there are more reals than integers, which is to say that the reals are larger. Which is a special case, finally, of the assertion that some infinities are larger than others.

A lot of mathematicians at the time had trouble with this argument; abstract objects like sets seemed so far removed from the world of human experience that there were bound to be weird paradoxes and strange infinities. Better to stick with objects that were indubitably real, like the integers, and leave the abstract madness out of mathematics. Graham and Kantor blame French and German unwillingness to embrace abstraction for their falling behind the Russians. (The German mathematician Leopold Kronecker is famous for saying that God created the integers and that all else is the work of man.) It was Germans, Russians, and Hungarians, however — the German Gauss, the Russian Lobachevsky, and the Hungarian Bolyai — who had discovered non-Euclidean geometries 40 or 50 years before. Doesn’t this show two things? First, Germans and Hungarians were capable of letting their imaginations roam free, by casting off the Euclidean restraints that humanity had held sacred for 2000 years; they didn’t need Russian mysticism to make it happen. Second, did the Russian Lobachevsky need mysticism to achieve his breakthroughs? If not, then much of Graham and Kantor’s book reduces to “Some Russians needed mysticism. Others did not.” It becomes a documentary about an interesting coincidence, rather than something with any causal importance. Graham and Kantor clearly believe (see the final chapter in particular) that the Name-Worshipping episode is important to the development of mathematics, whereas to me it looks like a coincidence.

Returning to our story, though: the Russian Cantor’s set theory revolutionized all of mathematics. You can’t do serious mathematics now without encountering axioms posed in terms of sets. Look at the definition of a “topology”, for instance. Or what defines a “measure” (a mathematical generalization that covers ideas like “distance” and “weight” and “volume” and many others). You can compute the probability that a billion coin tosses in a row will all come up heads using only finite mathematics; but if you want to answer complicated questions like “what is the probability that a Brownian-motion path is continuous?” [1], you need infinities.

Graham and Kantor name-drop mathematicians and mathematical concepts, but rarely try to explain them; they gesture generally toward “functions of Baire class n” on a few occasions, without ever making it clear what these things are or why we should care. One gathers that a Baire class has something to do with discontinuous functions (the most pathological example of which is f(x) = 1 when x is rational, f(x) = 0 when x is irrational). And on a few occasions, Graham and Kantor suggest that the Russians found discontinuous functions freeing — that continuous functions somehow confine human will. Forgive me if I see absolutely no connection between the two. Graham and Kantor don’t help clarify what the connection might be.

So we get mathematical terms without mathematical understanding, and mysticism that’s mostly mystification. I could do without this book.

__P.S.__: I see that my friend Cosma Shalizi has also reviewed [book: Naming Infinity], but I’ve not read his review yet.

[1] — Brownian motion shows up all over the place. It’s named after Robert Brown, who was studying the motion of pollen particles in water. Einstein used it to help estimate Avogadro’s number. Being physical objects, it’s reasonable to argue that pollen particles must move continuously (i.e., if they go from point A to point B, they must move through every point between A and B — they can’t just jump from one to the other). So then, if Brownian motion is an accurate model for the motion of pollen particles, you’d want to show that the probability of a Brownian-motion path’s being continuous is 1 — that is, that if you picked a Brownian-motion path “at random,” that you’d never pick a path wherein the pollen particle takes discontinuous jumps.

There are obviously infinitely many paths that a particle of pollen could follow; in fact, there are *uncountably* many paths. One is then forced, when judging models with the appropriate degree of skepticism, to hit up against Cantor’s infinities.