Editor’s note: We are delighted to welcome Science After Babel, the latest book from mathematician and philosopher David Berlinski. This article is adapted from Chapter 7.
The calculus and the rich body of mathematical analysis to which it gave rise made modern science possible, but it was the algorithm that 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 17th 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 recipe, a computer program — a way of getting something done in a finite number of discrete steps. Classical mathematics contains algorithms for virtually every elementary operation. Over the course of centuries, the complex (and counterintuitive) operations of addition, multiplication, subtraction, and division have been subordinated to fixed routines. Arithmetic algorithms now exist in mechanical form; what was once an intellectual artifice has become an instrumental artifact.
Old as the Hills, and Cunning
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. It was only in this century that the concept of an algorithm was coaxed completely into consciousness. The work was undertaken more than sixty years ago by a quartet of brilliant mathematical logicians: Kurt Gödel, Alonzo Church, Emil Post, and A. M. Turing, whose lost eyes seem to roam anxiously over the second half of the 20th century.
If it is beauty that governs the mathematician’s soul, it is truth and certainty that remind him of his duty. At the end of the 19th 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. Caught between mathematical crises and their 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 the algorithm into one both formal and precise.
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. 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.
Serene in the Cloudless Sky
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 from point to point by fiber-optic cables, or bouncing in a triangle from the earth to synchronous satellites, serene in the cloudless sky, and back to earth again, the great global network of computers moving chunks of data at the speed of light: stock-market indices, currency prices, gold and silver futures, news of cotton crops, rumors of war, strange tales of sexual scandal, images of men in starched white shirts stabbing at keyboards with stubby fingers or looking upward at luminescent monitors, beads of perspiration on their tensed lips. E-mail flashes from server to server, the circle of affection or adultery closing in an electronic braid; there is good news in Lisbon and bad news in Saigon. There is data everywhere and information on every conceivable topic: the way 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 search out remedies for obscure diseases, make contact with covens in South Carolina, or exchange messages with people in chat groups who believe that Princess Diana was murdered on instructions tendered by the House of Windsor, the dark demented devious old Queen herself sending the order that sealed her fate.
All of this is very interesting and very new — indeed, interesting because new — but however much we may feel that our senses are brimming with the debris of data, the causal nexus that has made the modern world extends in a simple line from the idea of an algorithm, as logicians conceived it in the 1930s, directly to the ever-present always-moving now; and 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 conditions of life, the expectations of the race.
It is the algorithm that rules the world itself, insinuating itself into every device and every discussion or diagnosis, offering advice and making decisions, maintaining its presence in every transaction, carrying out dizzying computations, arming and then aiming cruise missiles, bringing the dinosaurs back to life on film, and, like blind Tiresias, foretelling the extinction of the universe either in a cosmic crunch or in one of those flaccid affairs in which after a long time things just peter out.
Our Fantastic and Artificial World
The algorithm has made the fantastic and artificial world that many of us now inhabit. It also seems to have made much of the natural world, at least that part of it that is alive. The fundamental act of biological creation, the most meaningful of moist mysteries among the great manifold of moist mysteries, is the construction of an organism from a single cell. Look at it backward so that things appear in reverse (I am giving you my own perspective): Viagra discarded, hair returned, skin tightened, that unfortunate marriage zipping backward, teeth uncapped, memories of a radiant young woman running through a field of lilacs, a bicycle with fat tires, skinned knees, Kool-Aid, and New Hampshire afternoons. But where memory fades in a glimpse of the noonday sun seen from a crib in winter, the biological drama only begins, for the rosy fat and cooing creature loitering at the beginning of the journey, whose existence I’m now inferring, the one improbably responding to kitchy kitchy coo, has come into the world as the result of a spectacular nine-month adventure, one beginning with a spot no larger than a pinhead and passing by means of repeated but controlled cellular divisions into an organism of rarified and intricately coordinated structures, these held together in systems, the systems in turn animated and controlled by a rich biochemical apparatus, the process of biological creation like no other seen anywhere in the universe, strange but disarmingly familiar, for when the details are stripped away, the revealed miracle seems cognate to miracles of a more familiar kind, as when something is read and understood.
Meaning in Molecules
Much of the schedule by which this spectacular nine-month construction is orchestrated lies resident in DNA — and “schedule” is the appropriate word, for while the outcome of the drama is a surprise, the offspring proving to resemble his maternal uncle and his great-aunt (red hair, prominent ears), the process itself proceeds inexorably from one state to the next, and processes of this sort, which are combinatorial (cells divide), finite (it comes to an end in the noble and lovely creature answering to my name), and discrete (cells are cells), would seem to be essentially algorithmic in nature, the algorithm now making and marking its advent within the very bowels of life itself.
DNA is a double helix — this everyone now knows, the image as familiar as Marilyn Monroe — two separate strands linked to one another by a succession of steps so that the molecule itself looks like an ordinary ladder seen under water, the strands themselves curved and waving. Information is stored on each strand by means of four bases — A, T, G, and C; these are by nature chemicals, but they function as symbols, the instruments by which a genetic message is conveyed.
A library is in place, one that stores information, and far away, where the organism itself carries on, one sees the purposes to which the information is put, an inaccessible algorithm ostensibly orchestrating the entire affair. Meaning is inscribed in molecules, and so there is something that reads and something that is read; but they are, those strings, richer by far than the richest of novels, for while Tolstoy’s Anna Karenina can only suggest the woman, her black hair swept into a chignon, the same message carrying the same meaning, when read by the right biochemical agencies, can bring the woman to vibrant and complaining life, reading now restored to its rightful place as a supreme act of creation.
The mechanism is simple, lucid, compelling, extraordinary. In transcription, the molecule faces outward to control the proteins. In replication, it is the internal structure of DNA that conveys secrets, not from one molecule to another but from the past into the future. At some point in the life of a cell, double-stranded DNA is cleaved, so that instead of a single ladder, two separate strands may be found waving gently, like seaweed, the bond between base pairs broken. As in the ancient stories in which human beings originally were hermaphroditic, each strand finds itself longingly incomplete, its bases unsatisfied because unbound. In time, bases attract chemical complements from the ambient broth in which they are floating, so that if a single strand of DNA contains first A and then C, chemical activity prompts a vagrant T to migrate to A, and ditto for G, which moves to C, so that ultimately the single strand acquires its full complementary base pairs. Where there was only one strand of DNA, there are now two. Naked but alive, the molecule carries on the work of humping and slithering its way into the future.
There, That Is What Intelligence Is
A general biological property, intelligence is exhibited in varying degrees by everything that lives, and it is intelligence that immerses living creatures in time, allowing the cat and the cockroach alike to peep into the future and remember the past. The lowly paramecium is intelligent, learning gradually to respond to electrical shocks, this quite without a brain let alone a nervous system. But like so many other psychological properties, intelligence remains elusive without an objective correlative, some public set of circumstances to which one can point with the intention of saying, There, that is what intelligence is or what intelligence is like.
The stony soil between mental and mathematical concepts is not usually thought efflorescent, but in the idea of an algorithm modern mathematics does offer an obliging witness to the very idea of intelligence. Like almost everything in mathematics, algorithms arise from an old wrinkled class of human artifacts, things so familiar in collective memory as to pass unnoticed. By now, the ideas elaborated by Gödel, Church, Turing, and Post have passed entirely into the body of mathematics, where themes and dreams and definitions are all immured, but the essential idea of an algorithm blazes forth from any digital computer, the unfolding of genius having passed inexorably from Gödel’s incompleteness theorem to Space Invaders VII rattling on an arcade Atari, a progression suggesting something both melancholy and exuberant about our culture.
The computer is a machine, and so belongs to the class of things in nature that do something; but the computer is also a device dividing itself into aspects, symbols set into software to the left, the hardware needed to read, store, and manipulate the software to the right. This division of labor is unique among man-made artifacts: it suggests the mind immersed within the brain, the soul within the body, the presence anywhere of spirit in matter. An algorithm is thus an ambidextrous artifact, residing at the heart of both artificial and human intelligence. Computer science and the computational theory of mind appeal to precisely the same garden of branching forks to explain what computers do or what men can do or what in the tide of time they have done.
A Combinatorial System
Molecular biology has revealed that whatever else it may be, a living creature is also a combinatorial system, its organization controlled by a strange, hidden, and obscure text, one written in a biochemical code. It is an algorithm that lies at the humming heart of life, ferrying information from one set of symbols (the nucleic acids) to another (the proteins).
The complexity of human artifacts, the things that human beings make, finds its explanation in human intelligence. The intelligence responsible for the construction of complex artifacts — watches, computers, military campaigns, federal budgets, this very essay — finds its explanation in biology. Yet however invigorating it is to see the algorithmic pattern appear and reappear, especially on the molecular biological level, it is important to remember, if only because it is so often forgotten, that in very large measure we have no idea how the pattern is amplified. Yet the explanation of complexity that biology affords is largely ceremonial. At the very heart of molecular biology, a great mystery is vividly in evidence, as those symbolic forms bring an organism into existence, control its morphology and development, and slip a copy of themselves into the future.
The transaction hides a process never seen among purely physical objects, one that is characteristic of the world where computers hum and human beings attend to one another. In that world intelligence is always relative to intelligence itself, systems of symbols gaining their point from having their point gained. This is not a paradox. It is simply the way things are. Two hundred years ago the French biologist Charles Bonnet asked for an account of the “mechanics which will preside over the formation of a brain, a heart, a lung, and so many other organs.” No account in terms of mechanics is yet available. Information passes from the genome to the organism. Something is given and something read; something ordered and something done. But just who is doing the reading and who is executing the orders, this remains unclear.