INFINITE ASCENT

A Short History of Mathematics

By David Berlinski

Modern Library. 197 pp. $21.95

David Berlinski's new book calls itself "a short history of mathematics," but it's really more of a greatest-hits collection. Instead of trying to sum up several millennia of intellectual history in just under 200 pages, Berlinski wisely chooses to concentrate on 10 key developments that have shaped our species's mathematical progress so far.

Mathematics is an omnivorous discipline: The story, again and again, is that some subject of interest starts out beyond the reach of mathematics, then is studied by mathematical means and at last is gobbled up and made part of mathematics. Berlinski's chosen topics illustrate this point nicely: Euclid's geometry incorporated our intuitions about space and distance into a rigorous mathematical framework, as did calculus for the physics of continuous motion, as did Evariste Galois's theory of groups for the notion of symmetry, as did Georg Cantor's set theory for the infinite, and as did the work of David Hilbert, Bertrand Russell and Kurt Godel for the notion of mathematical proof itself. (One notable absence from Berlinski's list is the theory of probability, which made the study of randomness and chance into respectable mathematical pursuits and laid the groundwork for the modern theory of statistics; this story is beautifully told in Ian Hacking's The Emergence of Probability.)

Berlinski's expository style is unorthodox. His treatments of mathematical ideas and their histories come replete with imagined dips into the minds of historical protagonists, direct addresses to the reader, narrative interludes and the occasional broad ethnic gag. Unorthodox doesn't have to mean unserious, of course; in the hands of a capable writer, this sort of playful business can be just the thing to convince a math-leery reader to persevere.

Unfortunately, Berlinski is not a capable writer. He is prone to similes, sometimes three or four to a page, that are so unilluminating as to seem chosen at random. In a typical example, he describes a metric tensor (a key ingredient in the mathematical description of curved spaces) as "a general prescription, one that, like a high-school guidance counselor, indicates what kind of relationships these points may enjoy and under what circumstances," a comparison that is simultaneously unrevealing about tensors and inconsistent with my knowledge of guidance counselors.

Berlinski's prose jerks disorientingly between incoherent pomp ("But neither my own verbal gestures, nor any table of examples, succeed in providing what is really needed, and that is a moment of complete intellectual clarity, circumstances that may be read backward into the late seventeenth century, as Gottfried Leibniz used one concept that he could not precisely define to explore other concepts that he could not precisely see") and drab filler of the my-term-paper-is-due-in-three-hours type ("The largest mathematical personality of his time, Euler was born in 1707 and died in 1783").

All these infelicities would be secondary if Berlinski offered clear and accessible accounts of the great ideas themselves. But he is only a little kinder to mathematics than he is to the English language. Most of what he says about mathematics is close to correct, but his treatment is typically so vague that even those already familiar with the material will have to work to make out what he's trying to say. Sometimes symbols and notations are used before being defined; sometimes they're never defined.

Yes, it's hard to explain mathematics, but it's not impossible: Classics like Douglas R. Hofstadter's Godel, Escher, Bach and more recent books like John Derbyshire's Prime Obsession demonstrate that a conscientious author can tell clear stories about deep and difficult mathematics, with room for some literary flourishes along the way.

One can't help feeling that Berlinski hardly tried. Here is his version of the famous anecdote about the great German mathematician Carl Friedrich Gauss as a young student, using a clever trick to compute the sum of the numbers from 1 to 100: "Gauss was able to turn down his tablet at once, the correct answer inscribed on slate, even as the dutiful donkeys in the room, chubby farm children of no intellectual distinction, scratched away industriously." If Berlinski's book succeeds at anything, it is in presenting a picture of mathematics as an impenetrable activity carried out by geniuses like Gauss, Leibniz and Bernhard Riemann. The proper role of the "dutiful donkeys" in this picture is to nod mute agreement to Berlinski's paeans and superlatives -- certainly not to think carefully about mathematics, let alone try doing it themselves. Learning that lesson about mathematics is worse than learning nothing at all. *

Jordan Ellenberg is an assistant professor of mathematics at the University of Wisconsin, Madison. He is the author of a novel, "The Grasshopper King," and writes the "Do the Math" column in Slate.