The MagazineHere's Looking at EuclidWhy geometry matters in the life of the mind.Feb 18, 2013, Vol. 18, No. 22
• By DAVID GUASPARI
Many ancient societies knew important mathematical facts, but only one discovered mathematics—which is not a collection of accurate rules of thumb, but a body of knowledge organized deductively, by the radical notion of proof. And Euclid is its prophet. 'Portrait of Fra Luca Pacioli with a student' by Jacapo de' Barbari (1495) The Elements, composed in about 300 b.c., is a landmark of human thought. It has exerted a profound influence on science and philosophy by serving as both a basic geometry textbook (widely used for more than 2,000 years) and an archetype of rigorous knowledge. Accordingly, the ultimate subject of The King of Infinite Space is the life of the mind. Written with David Berlinski’s characteristic mix of hothouse prose and standup comedy, it is aimed at the Intelligent General Reader—though I often found myself wondering what that admirable creature might make of it. Berlinski begins with Euclid’s foundations—definitions, axioms, and common notions (general logical principles)— and devotes much attention to their shortcomings. For example, Euclid appeals to the notion of figures “coinciding,” meaning that one figure can be placed on another so that their sides match up exactly. Are we now in a bind? If the figures in question are physical objects, we understand what it means to move one and place it on another; but physical objects will never match exactly. If, on the other hand, they are nonphysical, perceived only by the mind, then the notions of moving and placing are metaphorical. So what, exactly, do they mean? Euclid shows, Berlinski says, “a willingness to repose his confidence in things he could neither explain nor justify.” Is the author condescending to his subject? Is he suggesting that an interest in Euclid is some embarrassing affectation, like an enthusiasm for Renaissance fairs (excuse me, faires)? In fact, the point of this discussion is to show how bold Euclid’s project was and how remarkable was its success. Without benefit of certain refined logical tools and distinctions (that would be unavailable for another 2,200 years), Euclid erected an enduring architectural marvel: an axiomatic system, the first, a rich body of results supported on a tidy footing of “evident” principles by timbers of the strongest possible kind, proofs that compel belief. We are next led through a sample of these proofs, taken from the first of Euclid’s 12 books—including the proof of its most famous result, the Pythagorean Theorem. To understand the life of a certain kind of mind, we must see that mind in action. Berlinski adds an ahistorical gloss claiming that Euclid didn’t really “get” the Pythagorean Theorem, which ought to be understood as an algebraic statement about the numerical lengths of the sides of a right triangle: a2+b2=c2. Euclid’s formulation, about the areas of squares drawn on the sides of the triangle, is a clumsy second best. Berlinski does acknowledge that a notion of number applicable not only to the counting numbers (such as 2 and 12), but also to lengths and areas and volumes as well, wasn’t developed until nearly two millennia after Euclid, and that a few additional centuries were then required to make rigorous sense of the maneuver. But that concession feels grudging. The remarkable fact, which deserves celebration, is that Euclid’s account of the quantitative aspects of geometry was rigorous from the get-go. For Euclid, multitudes (collections of things that can be numbered) are distinct from magnitudes (continuously variable entities, such as lines). And magnitudes themselves are of different kinds. We can add a number to a number but not to amagnitude. We can add like magnitudes (line to line, or area to area) but not unlike magnitudes. We can “multiply” a line by a number—by adding together two or three or four copies of it—but we cannot multiply a length by a length. Multiplicity is reserved for multitudes. This makes perfect sense. The need for these careful distinctions was reaffirmed by the surprising and unwelcome discovery that the side and the diagonal of a square have no “common measure”: There exists no unit of length of which both are exact multiples. (An algebraist would say that the square root of two is irrational.)This caused metaphysical problems for the Pythagoreans, who seem to have believed that numbers were in some sense the ultimate constituents of the world. It also presented a serious obstacle to the creation of a rigorous account of geometry. To explain the concept of similarity—of having the same shape, if not the same size—we need an account of ratios and proportions. This is easily provided for multitudes—or for magnitudes with a common measure, so that proportions among them can be understood as proportions among multitudes of that measure. The solution to this very difficult problem, presented in Book V, is the most brilliant technical achievement in the Elements. And it is attributed not to Euclid but to Eudoxus. A bit more than half of The King of Infinite Space is spent chewing slowly on Book I. The rest ranges beyond Euclid proper, the pace accelerating through discussions of geometry and algebra, non‑Euclidean and Riemannian geometries, Gauss’s intrinsic curvature, Hilbert’s reformulation of Euclidean geometry, groups and fields (ordered Archimedean fields, to be precise), and whips past, at the speed of light, Felix Klein’s “Erlangen Program.” These are deep subjects. I fear that readers will find much of the account gnomic, partly because so much is compressed into so small a space, and partly because of the author’s strenuous manufacture of lapidary phrases. Many succeed: It is apt to call mathematics a form of “mountain‑climbing pastoral,” a quest for worlds of ideal beauty accessible only by grueling and dangerous journeys. Some seem merely to have had tailfins slapped on, as when railroad tracks in a perspective drawing are said to “converge toward a distant, soundlessly spinning point.” Would any insight be lost if that point were “angrily motionless” or “mysteriously aloof”— or left unadorned? And some want to conjure drama where none exists: Noting that a proposition proven in Book I is equivalent to one of Euclid’s axioms, Berlinski says that, in proving it, “Euclid had demonstrated what he had already assumed.” From that point of view, every proposition Euclid proves is merely a restatement of something already assumed, since it follows from an axiom. Non-Euclidean geometries occupy much of the book’s latter half, and their discovery is a fascinating episode of intellectual history. Euclid’s parallel postulate amounts to saying that, given a line and a point, exactly one line can be drawn parallel to that line through the point. (Euclid formulates it differently.) To succeeding generations, this seemed both “evidently” true and more complicated than Euclid’s other axioms. And centuries of futile effort were spent trying to prove it from the others. In the 19th century, Janos Bolyai and Nikolai Lobachevsky independently developed consistent and rich theories of “geometry” in which the parallel postulate is false. They made themselves at home in a new mental world—one that was, from some points of view, literally unthinkable. Euclid’s geometry had been the paradigm of certain knowledge. How could denying one of its basic principles lead to anything other than incoherence? What exactly was geometry true of ? The ensuing story satisfies Berlinski’s taste for drama. He quotes a letter to Bolyai from his father, who had himself spent many years trying to prove the parallel postulate and was alarmed by his son’s growing interest in the problem:
It has been a long time since any professional mathematician worked in classical Euclidean geometry, searching for theorems that Euclid overlooked. So why study it? Once a theorem is proven, why should anyone pore over the proof and learn how to prove it all over again? According to a famous story, Euclid believed the work to be its own reward, dismissing a request from the pharaoh for a less arduous way to learn mathematics, telling him that “there is no Royal Road to geometry.” What Euclid offers, Berlinski says, is “a method of proof and so a way of life.” Could it become obsolete? Berlinski cites a major achievement of modern mathematics, officially called the “classification of finite simple groups.” It is a consequence of results published over decades by more than a hundred mathematicians and totaling more than 10,000 densely packed journal pages. It seems hard to believe that those pages are entirely free from error. (Errors have been found, but have turned out to be correctible.) The authors are retiring and dying off. A generation from now, will there be anyone who understands the proof? Is this the wave of the future? No matter: “The Euclidean academy . . . confers a form of immortality on its academicians. It is the immortality in having participated in one of the arts of civilization.” David Guaspari is a writer in Ithaca, N.Y. |