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The Jeweler's Velvet
Two ideas lie gleaming on the jeweler's velvet. The first is the calculus, the second, the algorithm. The calculus and the rich body of mathematical analysis to which it gave rise made modern science possible; but it has been the algorithm that has made possible the modern world.
They are utterly different, these ideas. The calculus serves the imperial vision of mathematical physics. It is a vision in which the real elements of the world are revealed to be its elementary constituents: particles, forces, fields, or even a strange fused combination of space and time. Written in the language of mathematics, a single set of fearfully compressed laws describes their secret nature. The universe that emerges from this description is alien, indifferent to human desires.
The great era of mathematical physics is now over. The three-hundred-year effort to represent the material world in mathematical terms has exhausted itself. The understanding that it was to provide is infinitely closer than it was when Isaac Newton wrote in the late seventeenth century, but it is still infinitely far away.
One man ages as another is born, and if time drives one idea from the field, it does so by welcoming another. The algorithm has come to occupy a central place in our imagination. It is the second great scientific idea of the West. There is no third.
An algorithm is an effective procedure, a way of getting something done in a finite number of discrete steps. Classical mathematics is, in part, the study of certain algorithms. In elementary algebra, for example, numbers are replaced by letters to achieve a certain degree of generality. The symbols are manipulated by means of firm, no-nonsense rules. The product of (a + b) and (a + b) is derived first by multiplying a by itself; second, by multiplying a by b twice; and third, by multiplying b by itself. The results are then added. The product is a2 + 2ab + b2 and that is the end of it. A machine could execute the appropriate steps. A machine can execute the appropriate steps. No art is involved. And none is needed.
In the wider world from which mathematics arises and to which the mathematician must like the rest of us return, an algorithm, speaking loosely, is a set of rules, a recipe, a prescription for action, a guide, a linked and controlled injunction, an adjuration, a code, an effort made to throw a complex verbal shawl over life's chattering chaos.
My dear boy, Lord Chesterfield begins, addressing his morganatic son, and there follows an extraordinary series of remarkably detailed letters, wise, witty, and occasionally tender, the homilies and exhortations given in English, French, Latin, and Greek. Dear boy is reminded to wash properly his teeth, to clean his linen, to manage his finances, and to discipline his temper; he needs to cultivate the social arts and to acquire the art of conversation and the elements of dance; he must, above all, learn to please. The graceful letters go on and on, the tone regretful if only because Lord Chesterfield must have known that he was volleying advice into an empty chamber, his son a dull, pimpled, rather loutish young man whose wish that his elegant father would for the love of God just stop talking throbs with dull persistence throughout his own obdurate silence.
The world the algorithm makes possible is retrograde in its nature to the world of mathematical physics. Its fundamental theoretical objects are symbols, and not muons, gluons, quarks, or space and time fused into a pliant knot. Algorithms are human artifacts. They belong to the world of memory and meaning, desire and design. The idea of an algorithm is as old as the dry humped hills, but it is also cunning, disguising itself in a thousand protean forms. With his commanding intelligence, the seventeenth-century philosopher and mathematician Gottfried Leibniz penetrated far into the future, seeing universal calculating machines and strange symbolic languages written in a universal script; but Leibniz was time's slave as well as her servant, unable to sharpen his most profound views, which like cities seen in dreams, rise up, hold their shape for a moment, and then vanish irretrievably.
Only in this century has the concept of an algorithm been coaxed completely into consciousness. The work was undertaken more than sixty years ago by a quartet of brilliant mathematical logicians: the subtle and enigmatic Kurt Godel; Alonzo Church, stout as a cathedral and as imposing; Emil Post, entombed, like Morris Raphael Cohen, in New York's City College; and, of course, the odd and utterly original A. M. Turing, whose lost eyes seem to roam anxiously over the second half of the twentieth century.
Mathematicians have loved mathematics because, like the graces of which Sappho wrote, the subject has wrists like wild roses. If it is beauty that governs the mathematicians' souls, it is truth and certainty that remind them of their duty. At the end of the nineteenth century, mathematicians anxious about the foundations of their subject asked themselves why mathematics was true and whether it was certain and to their alarm discovered that they could not say and did not know. Working mathematicians continued to work at mathematics, of course, but they worked at what they did with the sense that some sinister figure was creeping up the staircase of events. A number of redemptive schemes were introduced. Some mathematicians such as Gottlob Frege and Bertrand Russell argued that mathematics was a form of logic and heir thus to its presumptive certainty; following David Hilbert, others argued that mathematics was a formal game played with formal symbols. Every scheme seemed to embody some portion of the truth, but no scheme embodied it all. Caught between the crisis and its various correctives, logicians were forced to organize a new world to rival the abstract, cunning, and continuous world of the physical sciences, their work transforming the familiar and intuitive but hopelessly unclear concept of an algorithm into one both formal and precise.
Their story is rich in the unexpected. Unlike Andrew Wiles, who spent years searching for a proof of Fermat's last theorem, the logicians did not set out to find the concept that they found. They were simply sensitive enough to see what they spotted. But what they spotted was not entirely what they sought. In the end, the agenda to which they committed themselves was not met. At the beginning of the new millennium, we still do not know why mathematics is true and whether it is certain. But we know what we do not know in an immeasurably richer way than we did. And learning this has been a remarkable achievement--among the greatest and least-known of the modern era.
