## Infinitesimally Yours

#### Reasoning with the Infinite: From the Closed World to the Mathematical Universe

#### The First Moderns: Profiles in the Origins of Twentieth-Century Thought

#### Abraham Robinson: The Creation of Nonstandard Analysis, a Personal and Mathematical Odyssey

#### Non-standard Analysis

When people talk about “the infinite,” they usually mean the infinitely great: inconceivable vastness, world without end, boundless power, the Absolute. There is, however, another kind of infinity that is quite different from these, though just as marvelous in its own way. That is the infinitely small, or the *infinitesimal*.

In everyday parlance, “infinitesimal” is loosely used to refer to things that are extremely tiny by human standards, too small to be worth measuring. It tends to be a term of contempt. In his biography of Frederick the Great, Carlyle tells us that when Leibniz offered to explain the infinitely small to Queen Sophia Charlotte of Prussia, she replied that on that subject she needed no instruction: the behavior of her courtiers made her all too familiar with it. (About the only nonpejorative use of “infinitesimal” I have come across occurs in Truman Capote’s unfinished novel *Answered Prayers*, when the narrator is talking about the exquisite vegetables served at the tables of the really rich: “The greenest petits pois, infinitesimal carrots…” Then there are the abundant malapropisms. Some years back, *The New Yorker* reprinted a bit from an interview with a Hollywood starlet in which she was describing how she took advantage of filming delays on the set to balance her checkbook, catch up on her mail, and so forth. “If you really organize your time,” she observed, “it’s almost infinitesimal what you can accomplish.” To which *The New Yorker* ruefully added: “We know.”)

Properly speaking, as all the books under review agree, the infinitesimal is every bit as remote from us as the infinitely great is. Pascal, in the seventy-second of his *Pensées*, pictured nature’s “double infinity” as a pair of abysses between which finite man is poised. The infinitely great lies without, at the circumference of all things; the infinitesimal lies within, at the center of all things. These two extremes “touch and join by going in opposite directions, and they meet in God and God alone.” The infinitely small is even more difficult for us to comprehend than the infinitely great, Pascal observed: “Philosophers have much oftener claimed to have reached it, [but] they have all stumbled.”

Nor, one might add, has the poetical imagination been much help. There have been many attempts in literature to envisage the infinitely great: Father Arnall’s sermon on eternity in *A* *Portrait of the Artist as a Young Man*, Borges’s infinite “Library of Babel.” For the infinitesimal, though, there is only vague talk from Blake about an infinity you can hold “in the palm of your hand,” or, perhaps more helpful, these lines from Swift: “So, naturalists observe, a flea/Hath small fleas on him prey;/ And these have smaller fleas to bite ‘em,/And so proceed *ad infinitum*.”

From the time it was conceived, the idea of the infinitely small has been regarded with deep misgiving, even more so than that of the infinitely great. How can something be smaller than any given finite thing and not be simply nothing at all? Aristotle tried to ban the notion of the infinitesimal on the grounds that it was an absurdity. David Hume declared it to be more shocking to common sense than any priestly dogma. Bertrand Russell scouted it as “unnecessary, erroneous, and self-contradictory.”

Yet for all the bashing it has endured, the infinitesimal has proved itself to be the most powerful device ever deployed in the discovery of physical truth, the key to the scientific revolution that ushered in the Enlightenment. And, in one of the more bizarre twists in the history of ideas, the infinitesimal—after being stuffed into the oubliette seemingly for good at the end of the nineteenth century—was decisively rehabilitated in the 1960s. It now stands as the epitome of a philosophical conundrum fully resolved. Only one question about it remains open: Is it real?

Ironically, it was to save the natural world from unreality that the infinitesimal was invoked in the first place. The idea seems to have appeared in Greek thought sometime in the fifth century BCE, surfacing in the great metaphysical debate over the nature of being. On one side of this debate stood the monists—Parmenides and his followers—who argued that being was indivisible and that all change was illusion. On the other stood the pluralists—including Democritus and his fellow Atomists, as well as the Pythagoreans—who upheld the genuineness of change, which they understood as a rearrangement of the parts of reality.

But when you start parsing reality, breaking up the One into the Many, where do you stop? Democritus held that matter could be analyzed into tiny units—“atoms”—that, though finite in size, could not be further cut up. But space, the theater of change, was another question. There seemed to be no reason why the process of dividing it up into smaller and smaller bits could not be carried on forever. Therefore its ultimate parts must be smaller than any finite size.

This conclusion got the pluralists into a terrible bind, thanks to Parmenides’ cleverest disciple, Zeno of Elea. Irritated (according to Plato) by those who ridiculed his master, Zeno composed no fewer than forty dialectical proofs of the oneness and changelessness of reality. The most famous of these are his four paradoxes of motion, two of which—the “dichotomy” and “Achilles and the Tortoise”—attack the infinite divisibility of space. Take the dichotomy paradox. In order to complete any journey, you must first travel half the distance. But before you can do that, you must travel a quarter of the distance, and before that an eighth, and so on. In other words, you must complete an infinite number of subjourneys in reverse order. So you can never get started.

A story has it that when Zeno told this paradox to Diogenes the Cynic, Diogenes “refuted” it by getting up and walking away. But Zeno’s paradoxes are far from trivial. Bertrand Russell called them “immeasurably subtle and profound,” and even today there is doubt among philosophers whether they have been completely resolved. Aristotle dismissed them as fallacies, but he was unable to disprove them; instead he tried to block their conclusions by denying that there could be any actual infinity in nature. You could divide up space as finely as you pleased, Aristotle said, but you could never reduce it to an infinite number of parts.

Aristotle’s abhorrence of the actual infinite came to pervade Greek thought, and a century later Euclid’s *Elements* barred infinitesimal reasoning from geometry. This was disastrous for Greek science. The idea of the infinitely small had offered to bridge the conceptual gap between number and form, between the static and the dynamic. Consider the problem of finding the area of a circle. It is a straightforward matter to determine the area of a figure bounded by straight lines, such as a square or triangle. But how do you proceed when the boundary of the figure is curvilinear, as with a circle? The clever thing to do is to pretend the circle is a polygon made up of infinitely many straight line segments, each of infinitesimal length. It was by approaching the problem in this way that Archimedes, late in the third century BCE, was able to establish the modern formula for circular area involving π. Owing to Euclid’s strictures, however, Archimedes had to disavow his use of the infinite. He was forced to frame his demonstration as a *reductio ad absurdum*—a double reductio, no less—in which the circle was approximated by finite polygons with greater and greater numbers of sides. This cumbersome form of argument became known as the method of exhaustion, because it involved “exhausting” the area of a curved figure by fitting it with a finer and finer mesh of straight-edged figures.

For static geometry, the method of exhaustion worked well enough as an alternative to the forbidden infinitesimal. But it proved sterile in dealing with problems of dynamics, in which both space and time must be sliced to infinity. An object falling to earth, for example, is being continuously accelerated by the force of gravity. It has no fixed velocity for any finite interval of time, even one as brief as a thousandth of a second; every “instant” its speed is changing. Aristotle denied the meaningfulness of instantaneous speed, and Euclidean axiomatics could get no purchase on it. Only full-blooded infinitesimal reasoning could make sense of continuously accelerated motion. Yet that was just the sort of reasoning the Greeks fought shy of, because of the *horror infiniti* that was Zeno’s legacy. Thus was Greek science debarred from attacking phenomena of matter in motion mathematically. Under Aristotle’s influence, physics became a qualitative pursuit, and the Pythagorean goal of understanding the world by number was abandoned. The Greeks may have amassed much particular knowledge of nature, but their love of rigor held them back from discovering a single scientific law.

Though ostracized by Aristotle and Euclid, the infinitesimal did not entirely disappear from Western thought. Thanks to the enduring influence of Plato—who, unlike Aristotle, did not limit existence to what is found in the world of the senses—the infinitesimal continued to have a murky career as the object of transcendental speculation. Neo-Platonists like Plotinus and early Christian theologians like Saint Augustine restored the infinite to respectability by identifying it with God. Medieval philosophers spent even more time engaged in disputation over the infinitely small than over the infinitely great.

With the revival of Platonism during the Renaissance, the infinitesimal began to creep back into mathematics, albeit in a somewhat mystical way. For Johann Kepler, the infinitely small existed as a divinely given “bridge of continuity” between the curved and the straight. Untroubled by logical niceties—“Nature teaches geometry by instinct alone, even without ratiocination,” he wrote—Kepler employed infinitesimals in 1612 to calculate the ideal proportions for, of all things, a wine cask. And his calculation was correct.

Kepler’s friendliness toward the infinitesimal was shared by Galileo and Fermat. All three were edging away from the barren structure of Euclidean geometry toward a fertile, if freewheeling and unrigorous, science of motion, one that represented bodies as moving through infinitely divisible space and time. But there was a certain theological nettle to be grasped by these natural philosophers, as Michel Blay observes in the introduction to his *Reasoning with the Infinite*: “How could one conceive of a real infinite, present in the world, when it was exactly the conception of the infinite that was supposed to be reserved to the Creator of the world—when speaking the name of the infinite was reserved to God alone?” It was Blaise Pascal who was most galvanized by this question. None of his contemporaries embraced the idea of the infinite more passionately than did Pascal. And no one has ever written with more conviction of the awe that the infinite vastness and minuteness of nature can evoke. Nature proposes the two infinities to us as mysteries “not to understand, but to admire,” Pascal wrote—and to use in our reasoning, he might have added. For Pascal was also a mathematician, and he freely introduced infinitely small quantities into his calculations of the areas of curvilinear forms. His trick was to omit them as negligible once the desired finite answer was obtained. This offended the logical sensibilities of contemporaries like Descartes, but Pascal replied to criticism by saying, in essence, that what reason cannot grasp the heart makes clear.