A Tour of the Calculus
by David Berlinski
Pantheon. 331 pp. $27.50

Calculus Is a remarkable intellectual achievement—and a deeply consequential one, for it has made modern science, and especially modern physics, possible. Its essentials have been refined, over three centuries, into a structure of elegant economy. David Berlinski’s Tour walks the general reader through this modern edifice, displaying the logical frame that supports it and the grandeur of the completed whole.

The excursion begins with the “fantastic” idea that “the world may be understood in terms of real numbers”; Berlinski uses “fantastic” precisely here, to mean a product of visionary imagination. The real numbers are those that measure continuous quantities such as time and space; they include not only the counting numbers and their ratios (fractions) but the so-called irrational numbers like pi. And the “terms” are those laid down by Galilean science.

The modern empirical mode of inquiry established by Galileo asked not why but how. It sought not the reason a body falls, but rules relating the distance and the duration of its descent. This project triumphed spectacularly with Newton’s Principia (1687). Newton is Berlinski’s presiding deity, for to construct his System of the World, Newton had first to devise a proper mathematical basis for Galilean science: namely, the model of motion-and-change we call calculus.

This new world of mathematics, discovered independently by Leibniz (1646-1714), took two centuries of wanderings to map; the heart of Berlinski’s project is a rationalized account of the cartographic journeys, visiting landmarks found in any modern text—such as analytic geometry and real numbers, functions, limits, continuity—and impelled forward by fundamental questions: can we describe the world’s scenery in quantitative terms at all? And how, in those terms, can we understand its processes of change?

The first of these questions leads to analytic geometry and to deep problems concerning the notion of number itself. Analytic geometry marries geometry to algebra. It permits us to represent space, or anything we can picture geometrically, within a powerful engine of calculation. But the real numbers with which algebra calculates become a problem as soon as we stop taking them for granted. We can measure any whole or fractional number of inches by counting. Astonishingly, however, such measurements do not exhaust the whole continuum of lengths. They do not, for example, allow us to measure the diagonal of a one-inch square. We can postulate a number, the square root of 2, as the length, but what does that mean and what is its justification? For Berlinski, who is fundamentally concerned with the nature of continuity, this question about the continuum of real numbers functions like the first statement of a musical theme.

The second question—how are we to understand change?—introduces the most difficult technical ideas in the book, function and limit. (Both originated early in the 19th century with Cauchy, “the first modern mathematician.”) A function is a new kind of mathematical entity corresponding to Galileo’s notion of how—it is a pure correlation of time and appearance, agnostic about cause. After 1 second, a ball has fallen 16 feet; after 1.5 seconds, 36 feet; after 2 seconds, 64. But how can a mere heap of separate correlations capture what is a seamless process? The connection is made by the subtle notion of limit, which allows Berlinski to develop his discussion of continuity, enlarging it to include the interplay between the discrete (individual correlations) and the continuous (process).

The story culminates in calculus’s central result, “the Fundamental Theorem,” a finding as deserving of its definite article and honorific capitalization as was Aristotle of his medieval title “the Philosopher.” Very roughly, the Fundamental Theorem formulates a precise, and pregnant, equivalence between a function describing a process’s rate of change (such as the speed of a falling object) and one describing its accumulated change (the distance the object travels). It makes a pleasing capstone to Berlinski’s discussion, not only because of its intrinsic importance but because it puts to work what has gone before: the dialogue between algebra and geometry and the taming of continuity, area, and speed by the notion of limit.

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A Tour of the Calculus offers much to admire. It is a popular but honest explanation of beautiful and deep mathematics, refusing to patronize or demoralize readers with winks to reassure them that “getting it” does not matter. Berlinski clearly savors the comedy of teaching, and—just as in his essays in COMMENTARY¹—is consistently full of high spirits, whether he is opinionizing, salting his text with literary jokes, or otherwise cutting up. Indeed, throughout, the expedition is conducted as serious play, with Berlinski displaying a swagger plainly modeled on Vladimir Nabokov’s Lectures on Literature, down to the same fondness for alliteration (“moist mystery,” “poignant particularity”) and cinematic tricks (“the bright blue water of the pool . . . rushing up to meet that tender tummy”).

A Tour of the Calculus thus aspires to, and often attains, artfulness. It is also, however, marred by a tendency too often in evidence in popular scientific writing—a tendency to confuse mystery with mystification, intelligent wonder with Oh, wow! So, for example, difficulties about the real number continuum are introduced thus:

The square root of 2 . . . is not there, it cannot be found, it is not a part of the furniture of this or any other world.

And the rigorous theory of real numbers concludes with this cadenza:

Where before there was nothing more than an emptiness answering to the square root of 2, a new number now appears, a Dark Prince, an object utterly unlike any rational number, one flushed from the shadows and full of brooding mystery.

Berlinski knows when he is horsing around, but for an innocent reader his book offers too many occasions of sin.

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A coda to A Tour of the Calculus speculates that the story it tells is indeed over, that the reign of physics and its accompanying mathematical disciplines, among which calculus is preeminent, is coming to a close. Biology, Berlinski suggests, will take center stage, less because of the practical and political consequences of biotechnology than because the terms of physics have proved inadequate to human experience. Where physics is reductive, theoretical, and abstract, biology embraces variety, appearance, and arbitrariness. We are ready, Berlinski believes, to trade analytical depth for descriptive adequacy, for a science requiring none of the “mathematical analysis” in which calculus plays a central role.

Perhaps. But the story so far, a world-transforming adventure, is enduringly important. Years ago, when a friend asked what need she had of calculus, I blustered, why do you need King Lear? A Tour of the Calculus, despite its idiosyncrasies, is a spirited and welcome attempt at a more adequate reply.

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¹ “The Soul of Man Under Physics” (January 1996) and “The Deniable Darwin” (June 1996).

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