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Sparring with the Great Geometer

THE KING OF INFINITE SPACE: Euclid and His Elements. David Berlinski. xiv + 172 pp. Basic Books, 2013. $24.

Euclid is universally acclaimed great,” writes David Berlinski in the preface to The King of Infinite Space. He adds that Euclid’s Elements “is not simply a great book in mathematics: it is a great book.”

“Universally acclaimed” is a phrase that just begs for contradiction. Among all the generations of geometry students, is there not one dissenter? What about Martin Gardner, who wrote (in the October 1981 issue of Scientific American): “Euclid’s Elements is dull, long-winded and does not make explicit the fact that two circles can intersect, that a circle has an outside and an inside, that triangles can be turned over and other assumptions essential to his system”?

As it turns out, Berlinski himself is not an uncritical Euclid fanboy. He discusses in depth some of the same deficiencies that Gardner mentioned, and quite a few others as well. He pokes at crumbling stones in the foundations of Euclid’s geometric edifice, in which 467 theorems stand upon 5 “common notions,” 23 definitions and 5 axioms. In the end he also acknowledges that classical Euclidean geometry is “an exhausted discipline,” emphatically not a growth industry. All the same, it’s abundantly clear that Berlinski has come to praise Euclid, not to bury him.

The King of Infinite Space is not a crib for the lazy student who can’t be bothered to read all 13 books of the Elements. Neither is it a line-by-line exegesis for the diligent student who wants help with specific propositions in Euclid. Instead Berlinski offers a meditative monologue on Euclid’s place in the history of mathematics and the history of ideas. Berlinski speaks to you one-on-one, taking you into his confidence, never preachy or teachy.

Often in these pages Berlinski spars with Euclid. He points out that some of the common notions—five ideas that everyone is supposed to accept without question—are anything but obvious. “The whole is greater than the part,” says one of those notions. But when we compare the set of all natural numbers (1, 2, 3, 4, . . .) with the set of even numbers (2, 4, 6, 8, . . .), the whole is no larger than the part; the elements of the two sets can be put in one-to-one correspondence.

The axioms are also suspect. The first of them says that a straight line can be drawn between any two points. Berlinski responds:

It is surely false that any two points can be joined by a straight line, for unless one thinks of a point as the shrunken head of a straight line, no straight line can join a point to itself. Should one say instead that any two distinct points may be joined by a straight line? What makes points distinct? It can be nothing about their internal properties. They have none. To say that two points are distinct only if they are separated in space is to invite the question what separates them? If the answer is a straight line, nothing has been gained.

Debating with himself as well as with Euclid, Berlinski takes these puzzles very seriously. He does not pretend to resolve them, but in the end he comes down on the side of pragmatic mathematics rather than doctrinaire philosophy. Let us get on with the game, he decides, rather than argue forever about the rules.

Berlinski studied both philosophy and mathematics, but some years ago he left academic life and now calls himself “a writer, thinker, and raconteur.” One of his early books was A Tour of the Calculus, a kind of anti-textbook, a rear-guard attempt to transform a dreaded course into something beautiful and even fun. This new book is in the same genre. Berlinski is again telling us to eat our broccoli not because it’s good for us but because it’s good.

Much of Berlinski’s recent work is overtly polemical, weighing in on conflicts between religion and the sciences, and criticizing the “scientific overconfidence” of Darwinism. The King of Infinite Space is not one of these diatribes meant to provoke controversy or overturn an orthodoxy; and yet even here Berlinski finds occasion for playing the curmudgeon. A longstanding tension in mathematics—going back at least as far as Euclid—sets geometry against arithmetic, measuring against counting, the continuous against the discrete. Euclid covers both, but, as Berlinski says, “it is to geometry that his heart owes its allegiance.” Berlinski’s heart is in the same place. He laments the triumph of the discrete in many aspects of modern life, such as the division of time into fixed units: “The result has been a culture that in comparison to the ancient world is numerically sophisticated but visually disgusting. We count, they saw.”

My only serious misgivings about Berlinski’s commentary concern history rather than mathematics. The Elements was composed roughly 2,300 years ago and probably includes material that is older still. It is a document contemporaneous with some of the books of the Hebrew Bible, and it has comparable uncertainties of provenance. Not one scrap of the original text has survived; we have only translations of translations. Nothing whatever is known of the author. Berlinski is well aware of these facts; he mentions them explicitly. Nevertheless, he offers fanciful descriptions of Euclid “at the height of his powers—alert, vibrant and commanding.” He speculates that Euclid “may well have attended the academy that Plato founded, mingling with the philosophers and inserting himself gregariously in their gathering gossip.” “The man himself remains invisible,” Berlinski writes, but three lines later he offers us a scene from a costume drama:

He must have been a man of heavy architecture, and at some point in his concourse with those endlessly gabbling philosophers, he gathered up his robes and, with a dawning sense of his powers, determined that he had something to offer that they had not seen and could not express.

The fabricated cinematic detail is distracting and silly, but Euclid did have something to offer. Here is how Berlinski sums up the legacy:

The Euclidean style endures. It is vital, an ideal, a moral advantage, a corrective to whatever is spongy, soft, indistinct, slovenly, half-hidden, half-formed, half-baked, or only half-right, the mind in full possession of its powers, straight as an arrow, hard as a stone, uncompromising as a bank.

Brian Hayes is senior writer for American Scientist. He is the author of Group Theory in the Bedroom, and Other Mathematical Diversions (Hill and Wang, 2008) and Infrastructure: A Field Guide to the Industrial Landscape (W. W. Norton, 2005).

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