The Digital Bureaucrat
More than sixty years ago, mathematical logicians, by defining precisely the concept of an algorithm, gave content to the ancient human idea of an effective calculation. Their definitions led to the creation of the digital computer, an interesting example of thought bending matter to its ends.
The first computer appeared in the 1940s and has like certain insects undergone pupation, materializing first as an oddity, and then in the 1950s and 1960s as a specter. In a famous cartoon from the New Yorker, a computer, when asked whether or not there was a God, responded that there was one now. Some sense remains that, like the sorcerer's apprentice, we have appropriated a device we do not understand and cannot control, but curiously enough, as the digital computer has become more powerful, it has also become less intimidating. After shedding several of its earlier incarnations, the computer has acquired the role it was destined all along to assume. It is essentially an enabling device, one serving to amplify the low babble of human needs, indispensable without being invaluable.
The digital computer is a machine and, like every material object, a captive in the end of various bleak laws of thermodynamics. Having run out of time, it runs out of steam. As does the computer programmer stabbing at the computer keyboard with the tips of two tense fingers. As do we all. An algorithm is otherwise. Occupying the space between the pin-prick of desire and the resulting bubble of satisfaction, it is an abstract instrument of coordination, supplying procedural means to various ends. Contrived signs and symbols, algorithms, like thoughts, reside in a world beyond time.
Every computer divides itself into its hardware and its software, the machine host to its algorithm, the human being to his mind. It is hardly surprising that men and women have done what computers now do long before computers could do anything at all. The dissociation between mind and matter in men and machines is very striking; it suggests that almost any stable and reliable organization of material objects can execute an algorithm and so come to command some form of intelligence.
In this regard, the apparatus of the modern digital computer is convenient but hardly necessary. To see this point is almost at once to see it confirmed. After all, what is a bureaucracy but a social organization that has since at least the time of the ancient Chinese patiently undertaken the execution of complicated algorithms?
If a bureaucracy resembles a computer at the level of social organization, the living cell, if it resembles anything in our experience, resembles a computer at the level of molecular organization. The metaphor is irresistible and few biologists have resisted it. And for good reason. No other metaphor conveys the intricacies of cellular replication, transcription, and translation; and, for all that we can tell, nothing besides an algorithm can handle the administration of the biological molecules.
These reflections might indicate that the digital computer represents less of a bright, bursting novelty in human experience than is generally imagined. Although true, this is, of course, a conclusion too reassuring to be completely true. There is a considerable difference between the execution of an algorithm by a social bureaucracy or even a bacterial cell and the execution of an algorithm by a digital computer. Having coaxed the concept of an algorithm into self-consciousness, the logicians have made possible the creation of algorithms of matchless power, elegance, concision, and reliability. A digital computer may well do what a bureaucracy has done, but it does it with astonishing speed, the digital computer possessing an altogether remarkable ability to compress the otherwise sluggish stream of time. This has made all the difference in the world.
Five hundred years have passed since Magellan sailed around the world, and the sun still comes up like thunder out of China in the east; yet the old heavy grunting physical sense of a world that must be physically circumnavigated in order for a human exchange to be accomplished - this has quite vanished. Dawn kisses the continents one after the other, and as it does, a series of coded communications hustles itself along the surface of the earth, relayed form point to point by fiber-optic cables, or bouncing in a triangle from the earth to synchronous satellites, serene in the cloudless sky. There is good news in Lisbon and bad news in Seoul, or the reverse; mountaineers reaching the summit of K2 send messages to their fearful spouses and then slip into sleep, laptops beeping until their batteries (and their owners) go dead; there is data everywhere, and information on every conceivable topic: the way in which raisins are made in the Sudan, the history of the late Sung dynasty, telephone numbers of dominatrices in Los Angeles, and pictures, too. A man may be whipped, scourged, and scoured without ever leaving cyberspace; he may satisfy his curiosity or his appetites, read widely in French literature, decline verbs in Sanskrit or scan an interlinear translation of the Iliad, discovering the Greek for "greave" or "grieve"; he may scout the waters off Cap Ferrat - somewhat gray with pollution as I recall - or see the spot where treasure lies buried in the wine-dark sea off the coast of Crete. He may arrange for his own cremation on the Internet or search out remedies for obscure diseases; he may make contact with covens in South Carolina, or exchange messages with chat groups who believe that Princess Diana was murdered on instructions tendered by the house of Winsor, the dark, demented old Queen herself sending the orders that sealed her fate. For all the great dreams profitlessly invested in the digital computer, it is nonetheless true that not since the framers of the American Constitution took seriously the idea that all men are created equal has an idea so transformed the material condition of life, the expectations of the race.
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 a + 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 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 Gödel; 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 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.