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. Ironically, it was to save the natural world from unreality that the infinitesimal was invoked in the first place. Only one question about it remains open: Is it real? It now stands as the epitome of a philosophical conundrum fully resolved. 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. 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. Bertrand Russell scouted it as “unnecessary, erroneous, and self-contradictory.” David Hume declared it to be more shocking to common sense than any priestly dogma. 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. 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.”įrom 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. Nor, one might add, has the poetical imagination been much help. 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, they have all stumbled.” The infinitely great lies without, at the circumference of all things the infinitesimal lies within, at the center of all things. Pascal, in the seventy-second of his Pensées, pictured nature’s “double infinity” as a pair of abysses between which finite man is poised. Properly speaking, as all the books under review agree, the infinitesimal is every bit as remote from us as the infinitely great is. “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.”) ![]() ![]() 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. ![]() (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. ![]() 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. In everyday parlance, “infinitesimal” is loosely used to refer to things that are extremely tiny by human standards, too small to be worth measuring. That is the infinitely small, or the infinitesimal. There is, however, another kind of infinity that is quite different from these, though just as marvelous in its own way. When people talk about “the infinite,” they usually mean the infinitely great: inconceivable vastness, world without end, boundless power, the Absolute.
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