## Less Than Zero

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*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.