Infinitesimally Yours

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

by Joseph Warren Dauben
Princeton University Press, 559 pp., $21.95 (paper)

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…

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