Less than zero.(Infinite Ascent: A Short History of Mathematics)(Book Review)

David Berlinski Infinite Ascent: A Short History of Mathematics. Modern Library, 224 pages, $21.95

The relevant library shelves in the Courant Institute of Mathematical Sciences hold no fewer than eighteen different general histories of mathematics in English. The author setting out to write yet another such book must therefore have an angle (so to speak), some original approach to the topic. What is David Berlinski's angle? "Short" doesn't cut it: Dirk Struik cornered that market fifty years ago with his Concise History of Mathematics. A philosophical point of view like Morris Kline's adds interest, but Berlinski seems not to be in thrall to any strong philosophical conviction. So what's his angle?

He has two. The first is structural. While the narrative of Infinite Ascent proceeds more or less chronologically, each period is represented by just one major topic. For the Greeks we have "Proof," for the later seventeenth century "The Calculus," for the later nineteenth "Sets," and so on. There are nine of these topics, each with its own chapter. A final chapter deals discursively with recent developments. This approach has a lot to be said for it up to about 1800, since no historical period prior to that date could bring forth more than one great mathematical advance. It is not well suited to the abundance of the nineteenth century, though, and--as the author implicitly acknowledges--breaks down completely for the twentieth.

Berlinski's second angle is stylistic. I had never read any of his books before picking up Infinite Ascent, but I had heard about his odd style of writing and was curious to encounter it. My curiosity was quickly satisfied. The oddities of Berlinski's prose are not of the interesting kind. I should like to say that they brought to mind Dr. Johnson's censure of the metaphysical poets--"heterogenous ideas yoked by violence together" etc., etc.--but Berlinski does not belong in the company of poets, metaphysical or otherwise. His conceits are not imaginative, only whimsical; he is straining at effects he cannot attain; and in straining, he all too often stumbles over simple points of fact or grammar. This is not style, it is poshlust.

Some samples: "Like two immense polar bears, they [Newton and Leibniz] remain for ever frozen on the tundra of time." The whole point of polar bears is that they do not freeze--not even on solid ice, let alone on tundra. "The cathedral of math has increased in size but not in its inner nature." Of what does a cathedral's inner nature consist, and how would it increase? "The theory of complex numbers and their functions has broken men's hearts." Has it? Whose? "What can be said about mathematical objects is more interesting than the objects themselves." Say what?

Berlinski has, in fact, a tin ear for the English language. On encountering a clanger like "immured in his own immature fury," one's normal reaction would be: "There but for the grace of God ..." Having just read a hundred pages of Berlinski, though, it is hard not to suspect that our author believes he has brought off a fine alliteration. And then, what are "vein-mined hands," and how does Berlinski know that Pythagoras had such hands? I am aware that galaxies collide and pass through each other, but are there really instances of them merging? Does the author know the difference between a kiwi (bird) and a kiwi fruit (fruit)? I can certainly believe that Cantor may have laid linoleum, plans, or down the law, but could he really have "laid low"?

Berlinski's mathematical expositions, when they can be glimpsed through the Creative Writing vapors, are actually not bad. Even here, though, there are some vexations. After being told, with appropriate italics, that: "It is the integers and the operation of addition that taken together comprise a group," just two pages later we read that: "the even integers are ... a group in their own right." I don't know what Berlinski means by "the diameter of a triangle." And what is this about "a very well-known contemporary text, Counter-Examples in Analysis" comprising "a series of misleading proofs supporting theorems that ...