Less Than Zero
Infinite Ascent: A Short History of Mathematics
by David Berlinski
Modern Library; 224 pp. $21.95
The relevant library shelves in the Courant Institute of Mathematical Sciences hold no less 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 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-ruined 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" being comprised of "a series of misleading proofs supporting theorems that are not theorems"? The only book known to me, or to the Internet, under anything like that title is Gelbaum and Olmsted's Counterexamples [sic] in Analysis, of which a description more wrong-headed than Berlinski's could hardly be imagined. Berlinski's history is shaky, too. He describes Omar Khayyam as "a Persian among Arabs"; actually, he was a Persian among Persians, under Turkish rule. The German empire was not contracting in 1916; it was on the brink of a tremendous (though admittedly short-lived) eastward expansion.
Chapter Six, titled "Groups," is representative of the book's faults and occasional virtues. Its account of the life and death of Évariste Galois — who was killed in a duel at age 20, after getting mixed up in revolutionary politics — owes less to modern scholarship than to the romantic inventions in E.T. Bell's 1937 classic Men of Mathematics. Bell is great fun to read, but deeply unreliable on points of fact. The night before Galois died, writes Berlinski, retailing Bell:
[H]e sat at his desk and proposed to commit to posterity the teeming and obsessive mathematical ideas that he had until then kept locked within his skull.
Had he, though? Galois's most important ideas had in fact been submitted in a paper to the French Academy three years earlier. Cauchy, the greatest French mathematician of the age, had reviewed them, and thought highly of them. Nine months later Galois had submitted his work for the Academy's Grand Prix, very likely on Cauchy's encouragement. "Locked within his skull," eh?
As with Galois's work, so with the man. Berlinski's statement that "At the age of twenty, Galois lost his virginity along with his heart" is a rehash of Bell's "Some worthless girl initiated him." Bell's only source for that was a conjecture by Paul Dupuy, based on an allusion by François Raspail — seven years after the event! — to a mumbled remark Galois made while in a drunken stupor. Not even Tom Petsinis, operating with a novelist's license (The French Mathematician, 1997) thinks that Galois lost his virginity to Stéphanie Dumotel, and from what we know of the principals, and of manners in that place and time, it seems very improbable. Nor are there any grounds for Bell's characterization of Stéphanie as "worthless," or for Berlinski's traducing her as being of "uncertain reputation." I know of no evidence that Stéphanie was other than perfectly respectable. She eventually married a university professor. (Well…) It was not her fault that the naïve and introverted young Galois fell in love with her. She may indeed have teased him, as in Petsinis's story. Women, even respectable ones, do that.
I cannot be quite so hard on Berlinski's attempt to explain Galois's most important concept, that of a normal subgroup. I don't think he has pulled it off, but then, neither has anyone else, and Berlinski's is a nice try. This is the pons asinorum for readers of pop-math expositions. Galois Theory is very beautiful, but hard. All the more reason to honor the genius of the unhappy and (for my money) unattractive young Galois by trying to get the facts about him as correct as they can be gotten, on the fragmentary evidence we have. Berlinski's opinion that Galois was "as interesting as the young Byron" is, to be blunt about it, preposterous. "The young Shelley" might be closer to the mark, given Galois's social failure and political inclinations; but really, aside from his math, Galois was not interesting at all.
In his final chapter, titled "The Present," the author redeems himself to a degree. The portentous piffle is at a minimum here, and Berlinski makes a couple of good points — about the faddishness of later-20th-century math, for example. I actually felt the stirring of a mild urge to hear more of what he thinks about the current state of affairs. All in all, though, I believe that the nonmathematical reader seeking enlightenment on this topic will be better off with the leisurely Carl Boyer, or with the philosophical Kline, or even with the scenery-chewing E.T. Bell — or, if brevity is the key criterion, with Struik — than with David Berlinski.