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(No Other Thing Is More Enswathed in the Unknown)
The perfect symmetry of the whole apparatus—the wire in the middle, the two telephones at the ends of the wire, and the two gossips at the ends of the telephones—may be very fascinating to a mere mathematician.
—James Clerk Maxwell (1878)♦
A CURIOUS CHILD IN A COUNTRY TOWN in the 1920s might naturally form an interest in the sending of messages along wires, as Claude Shannon did in Gaylord, Michigan. He saw wires every day, fencing the pastures—double strands of steel, twisted and barbed, stretched from post to post. He scrounged what parts he could and jerry-rigged his own barbed-wire telegraph, tapping messages to another boy a half mile away. He used the code devised by Samuel F. B. Morse. That suited him. He liked the very idea of codes—not just secret codes, but codes in the more general sense, words or symbols standing in for other words or symbols. He was an inventive and playful spirit. The child stayed with the man. All his life, he played games and invented games. He was a gadgeteer. The grown-up Shannon juggled and devised theories about juggling. When researchers at the Massachusetts Institute of Technology or Bell Laboratories had to leap aside to let a unicycle pass, that was Shannon. He had more than his share of playfulness, and as a child he had a large portion of loneliness, too, which along with his tinkerer’s ingenuity helped motivate his barbed-wire telegraph.
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Gaylord amounted to little more than a few streets and stores interrupting the broad northern farmland of the Michigan peninsula.♦ Here and onward across the plains and prairie to the Rocky Mountains barbed wire had spread like a vine, begetting industrial fortunes though it was not a particularly glamorous technology amid the excitement of what was already called the Age of Electricity. Beginning in 1874, when an Illinois farmer received U. S. Patent No. 157,124 for “a new and valuable Improvement in Wire-Fences,” battles for ownership raged, ultimately reaching the Supreme Court, while the wire defined territory and closed the open range. At the peak, American farmers, ranchers, and railroads laid more than a million miles a year. Taken collectively the nation’s fence wire formed no web or network, just a broken lattice. Its purpose had been to separate, not to connect. For electricity it made a poor conductor even in dry weather. But wire was wire, and Claude Shannon was not the first to see this wide-ranging lattice as a potential communications grid. Thousands of farmers in remote places had the same idea. Unwilling to wait for the telephone companies to venture out from the cities, rural folk formed barbed-wire telephone cooperatives. They replaced metal staples with insulated fasteners. They attached dry batteries and speaking tubes and added spare wire to bridge the gaps. In the summer of 1895 The New York Times reported: “There can be no doubt that many rough-and-ready utilizations of the telephone are now being made. For instance, a number of South Dakota farmers have helped themselves to a telephone system covering eight miles of wire by supplying themselves with transmitters and making connections with the barb wire which constitutes the fence in that part of the country.” The reporter observed: “The idea is gaining ground that the day of cheap telephones for the million is at hand. Whether this impression is soundly based is an open question.”♦ Clearly people wanted the connections. Cattlemen who despised fences for making parcels of the free range now hooked up their speaking tubes to hear market quotations, weather reports, or just, crackling along the wires, the attenuated simulacrum of the human voice, a thrill in itself.
Three great waves of electrical communication crested in sequence: telegraphy, telephony, and radio. People began to feel that it was natural to possess machines dedicated to the sending and receiving of messages. These devices changed the topology—ripped the social fabric and reconnected it, added gateways and junctions where there had only been blank distance. Already at the turn of the twentieth century there was worry about unanticipated effects on social behavior. The superintendent of the line in Wisconsin fretted about young men and women “constantly sparking over the wire” between Eau Claire and Chippewa Falls. “This free use of the line for flirtation purposes has grown to an alarming extent,” he wrote, “and if it is to go on somebody must pay for it.” The Bell companies tried to discourage frivolous telephony, particularly by women and servants. A freer spirit prevailed at the farmer cooperatives, which avoided paying the telephone companies well into the 1920s. The Montana East Line Telephone Association—eight members—sent “up to the minute” news reports around its network, because the men also owned a radio.♦ Children wanted to play this game, too.
Claude Elwood Shannon, born in 1916, was given the full name of his father, a self-made businessman—furniture, undertaking, and real estate—and probate judge, already well into middle age. Claude’s grandfather, a farmer, had invented a machine for washing clothes: a waterproof tub, a wooden arm, and a plunger. Claude’s mother, Mabel Catherine Wolf, daughter of German immigrants, worked as a language teacher and sometime principal of the high school. His older sister, Catherine Wolf Shannon (the parents doled out names parsimoniously), studied mathematics and regularly entertained Claude with puzzles. They lived on Center Street a few blocks north of Main Street. The town of Gaylord boasted barely three thousand souls, but this was enough to support a band with Teutonic uniforms and shiny instruments, and in grade school Claude played an E-flat alto horn broader than his chest. He had Erector Sets and books. He made model planes and earned money delivering telegrams for the local Western Union office. He solved cryptograms. Left on his own, he read and reread books; a story he loved was Edgar Allan Poe’s “The Gold-Bug,” set on a remote southern island, featuring a peculiar William Legrand, a man with an “excitable brain” and “unusual powers of mind” but “subject to perverse moods of alternate enthusiasm and melancholy”♦—in other words, a version of his creator. Such ingenious protagonists were required by the times and duly conjured by Poe and other prescient writers, like Arthur Conan Doyle and H. G. Wells. The hero of “The Gold-Bug” finds buried treasure by deciphering a cryptograph written on parchment. Poe spells out the string of numerals and symbols (“rudely traced, in a red tint, between the death’s-head and the goat”)—53‡‡†305) )6* ;4826)4‡.)4‡) ;806* ;48†8¶60) )85;1‡( ;:‡*8†83(88) 5*‡ ;46(;88*96*?;8) *‡(;485) ;5*†2:*‡(;4956*2(5*–4) 8§8* ;4069285) ;)6†8)4‡‡;1 (‡9;48081 ;8:8‡1 ;48†85;4)485†528806*81 (‡9:48;(88;4 (‡?34;48)4‡;161;:188; ‡?;—and walks the reader through every twist of its construction and deconstruction. “Circumstances, and a certain bias of mind, have led me to take interest in such riddles,”♦ his dark hero proclaims, thrilling a reader who might have the same bias of mind. The solution leads to the gold, but no one cares about the gold, really. The thrill is in the code: mystery and transmutation.
Claude finished Gaylord High School in three years instead of four and went on in 1932 to the University of Michigan, where he studied electrical engineering and mathematics. Just before graduating, in 1936, he saw a postcard on a bulletin board advertising a graduate-student job at the Massachusetts Institute of Technology. Vannevar Bush, then the dean of engineering, was looking for a research assistant to run a new machine with a peculiar name: the Differential Analyzer. This was a 100-ton iron platform of rotating shafts and gears. In the newspapers it was being called a “mechanical brain” or “thinking machine”; a typical headline declared:
“Thinking Machine” Does Higher Mathematics;
Solves Equations That Take Humans Months♦
Charles Babbage’s Difference Engine and Analytical Engine loomed as ancestral ghosts, but despite the echoes of nomenclature and the similarity of purpose, the Differential Analyzer owed virtually nothing to Babbage. Bush had barely heard of him. Bush, like Babbage, hated the numbing, wasteful labor of mere calculation. “A mathematician is not a man who can readily manipulate figures; often he cannot,” Bush wrote. “He is primarily an individual who is skilled in the use of symbolic logic on a high plane, and especially he is a man of intuitive judgment.”♦
MIT in the years after World War I was one of the nation’s three focal points for the burgeoning practical science of electrical engineering, along with the Bell Telephone Laboratories and General Electric. It was also a place with a voracious need for the solving of equations—especially differential equations, and particularly differential equations of the second order. Differential equations express rates of change, as in ballistic projectiles and oscillating electric currents. Second-order differential equations concern rates of change in rates of change: from position to velocity to acceleration. They are hard to solve analytically, and they pop up everywhere. Bush designed his machine to handle this entire class of problems and thus the whole range of physical systems that generated them. Like Babbage’s machines, it was essentially mechanical, though it used electric motors to drive the weighty apparatus and, as it evolved, more and more electromechanical switches to control it.
Unlike Babbage’s machine, it did not manipulate numbers. It worked on quantities—generating curves, as Bush liked to say, to represent the future of a dynamical system. We would say now that it was analog rather than digital. Its wheels and disks were arranged to produce a physical analog of the differential equations. In a way it was a monstrous descendant of the planimeter, a little measuring contraption that translated the integration of curves into the motion of a wheel. Professors and students came to the Differential Analyzer as supplicants, and when it could solve their equations with 2 percent accuracy, the operator, Claude Shannon, was happy. In any case he was utterly captivated by this “computer,” and not just by the grinding, rasping, room-filling analog part, but by the nearly silent (save for the occasional click and tap) electrical controls.♦
THE DIFFERENTIAL ANALYZER OF VANNEVAR BUSH AT MIT (Illustration credit 6.1)
These were of two kinds: ordinary switches and the special switches called relays—the telegraph’s progeny. The relay was an electrical switch controlled by electricity (a looping idea). For the telegraph, the point was to reach across long distances by making a chain. For Shannon, the point was not distance but control. A hundred relays, intricately interconnected, switching on and off in particular sequence, coordinated the Differential Analyzer. The best experts on complex relay circuits were telephone engineers; relays controlled the routing of calls through telephone exchanges, as well as machinery on factory assembly lines. Relay circuitry was designed for each particular case. No one had thought to study the idea systematically, but Shannon was looking for a topic for his master’s thesis, and he saw a possibility. In his last year of college he had taken a course in symbolic logic, and, when he tried to make an orderly list of the possible arrangements of switching circuits, he had a sudden feeling of déjà vu. In a deeply abstract way, these problems lined up. The peculiar artificial notation of symbolic logic, Boole’s “algebra,” could be used to describe circuits.
This was an odd connection to make. The worlds of electricity and logic seemed incongruous. Yet, as Shannon realized, what a relay passes onward from one circuit to the next is not really electricity but rather a fact: the fact of whether the circuit is open or closed. If a circuit is open, then a relay may cause the next circuit to open. But the reverse arrangement is also possible, the negative arrangement: when a circuit is open, a relay may cause the next circuit to close. It was clumsy to describe the possibilities with words; simpler to reduce them to symbols, and natural, for a mathematician, to manipulate the symbols in equations. (Charles Babbage had taken steps down the same path with his mechanical notation, though Shannon knew nothing of this.)
“A calculus is developed for manipulating these equations by simple mathematical processes”—with this clarion call, Shannon began his thesis in 1937. So far the equations just represented combinations of circuits. Then, “the calculus is shown to be exactly analogous to the calculus of propositions used in the symbolic study of logic.” Like Boole, Shannon showed that he needed only two numbers for his equations: zero and one. Zero represented a closed circuit; one represented an open circuit. On or off. Yes or no. True or false. Shannon pursued the consequences. He began with simple cases: two-switch circuits, in series or in parallel. Circuits in series, he noted, corresponded to the logical connective and; whereas circuits in parallel had the effect of or. An operation of logic that could be matched electrically was negation, converting a value into its opposite. As in logic, he saw that circuitry could make “if … then” choices. Before he was done, he had analyzed “star” and “mesh” networks of increasing complexity, by setting down postulates and theorems to handle systems of simultaneous equations. He followed this tower of abstraction with practical examples—inventions, on paper, some practical and some just quirky. He diagrammed the design of an electric combination lock, to be made from five push-button switches. He laid out a circuit that would “automatically add two numbers, using only relays and switches”;♦ for convenience, he suggested arithmetic using base two. “It is possible to perform complex mathematical operations by means of relay circuits,” he wrote. “In fact, any operation that can be completely described in a finite number of steps using the words if, or, and, etc. can be done automatically with relays.” As a topic for a student in electrical engineering this was unheard of: a typical thesis concerned refinements to electric motors or transmission lines. There was no practical call for a machine that could solve puzzles of logic, but it pointed to the future. Logic circuits. Binary arithmetic. Here in a master’s thesis by a research assistant was the essence of the computer revolution yet to come.
Shannon spent a summer working at the Bell Telephone Laboratories in New York City and then, at Vannevar Bush’s suggestion, switched from electrical engineering to mathematics at MIT. Bush also suggested that he look into the possibility of applying an algebra of symbols—his “queer algebra”♦—to the nascent science of genetics, whose basic elements, genes and chromosomes, were just dimly understood. So Shannon began work on an ambitious doctoral dissertation to be called “An Algebra for Theoretical Genetics.”♦ Genes, as he noted, were a theoretical construct. They were thought to be carried in the rodlike bodies known as chromosomes, which could be seen under a microscope, but no one knew exactly how genes were structured or even if they were real. “Still,” as Shannon noted, “it is possible for our purposes to act as though they were.… We shall speak therefore as though the genes actually exist and as though our simple representation of hereditary phenomena were really true, since so far as we are concerned, this might just as well be so.” He devised an arrangement of letters and numbers to represent “genetic formulas” for an individual; for example, two chromosome pairs and four gene positions could be represented thus:
A1B2C3D5 E4F1G6H1
A3B1C4D3 E4F2G6H2
Then, the processes of genetic combination and cross-breeding could be predicted by a calculus of additions and multiplications. It was a sort of road map, far abstracted from the messy biological reality. He explained: “To non-mathematicians we point out that it is a commonplace of modern algebra for symbols to represent concepts other than numbers.” The result was complex, original, and quite detached from anything people in the field were doing.♦♦ He never bothered to publish it.
Meanwhile, late in the winter of 1939, he wrote Bush a long letter about an idea closer to his heart:
Off and on I have been working on an analysis of some of the fundamental properties of general systems for the transmission of intellegence, including telephony, radio, television, telegraphy, etc. Practically all systems of communication may be thrown into the following general form:♦
T and R were a transmitter and a receiver. They mediated three “functions of time,” f(t): the “intelligence to be transmitted,” the signal, and the final output, which, of course, was meant to be as nearly identical to the input as possible. (“In an ideal system it would be an exact replica.”) The problem, as Shannon saw it, was that real systems always suffer distortion—a term for which he proposed to give a rigorous definition in mathematical form. There was also noise (“e.g., static”). Shannon told Bush he was trying to prove some theorems. Also, and not incidentally, he was working on a machine for performing symbolic mathematical operations, to do the work of the Differential Analyzer and more, entirely by means of electric circuits. He had far to go. “Although I have made some progress in various outskirts of the problem I am still pretty much in the woods, as far as actual results are concerned,” he said.
I have a set of circuits drawn up which actually will perform symbolic differentiation and integration on most functions, but the method is not quite general or natural enough to be perfectly satisfactory. Some of the general philosophy underlying the machine seems to evade me completely.
He was painfully thin, almost gaunt. His ears stuck out a little from his close-trimmed wavy hair. In the fall of 1939, at a party in the Garden Street apartment he shared with two roommates, he was standing shyly in his own doorway, a jazz record playing on the phonograph, when a young woman started throwing popcorn at him. She was Norma Levor, an adventurous nineteen-year-old Radcliffe student from New York. She had left school to live in Paris that summer but returned when Nazi Germany invaded Poland; even at home, the looming war had begun to unsettle people’s lives. Claude struck her as dark in temperament and sparkling in intellect. They began to see each other every day; he wrote sonnets for her, uncapitalized in the style of E. E. Cummings. She loved the way he loved words, the way he said Boooooooolean algebra. By January they were married (Boston judge, no ceremony), and she followed him to Princeton, where he had received a postdoctoral fellowship.
The invention of writing had catalyzed logic, by making it possible to reason about reasoning—to hold a train of thought up before the eyes for examination—and now, all these centuries later, logic was reanimated with the invention of machinery that could work upon symbols. In logic and mathematics, the highest forms of reasoning, everything seemed to be coming together.
By melding logic and mathematics in a system of axioms, signs, formulas, and proofs, philosophers seemed within reach of a kind of perfection—a rigorous, formal certainty. This was the goal of Bertrand Russell and Alfred North Whitehead, the giants of English rationalism, who published their great work in three volumes from 1910 to 1913. Their title, Principia Mathematica, grandly echoed Isaac Newton; their ambition was nothing less than the perfection of all mathematics. This was finally possible, they claimed, through the instrument of symbolic logic, with its obsidian signs and implacable rules. Their mission was to prove every mathematical fact. The process of proof, when carried out properly, should be mechanical. In contrast to words, symbolism (they declared) enables “perfectly precise expression.” This elusive quarry had been pursued by Boole, and before him, Babbage, and long before either of them, Leibniz, all believing that the perfection of reasoning could come with the perfect encoding of thought. Leibniz could only imagine it: “a certain script of language,” he wrote in 1678, “that perfectly represents the relationships between our thoughts.”♦ With such encoding, logical falsehoods would be instantly exposed.
The characters would be quite different from what has been imagined up to now.… The characters of this script should serve invention and judgment as in algebra and arithmetic.… It will be impossible to write, using these characters, chimerical notions [chimères].
Russell and Whitehead explained that symbolism suits the “highly abstract processes and ideas”♦ used in logic, with its trains of reasoning. Ordinary language works better for the muck and mire of the ordinary world. A statement like a whale is big uses simple words to express “a complicated fact,” they observed, whereas one is a number “leads, in language, to an intolerable prolixity.” Understanding whales, and bigness, requires knowledge and experience of real things, but to manage 1, and number, and all their associated arithmetical operations, when properly expressed in desiccated symbols, should be automatic.
They had noticed some bumps along the way, though—some of the chimères that should have been impossible. “A very large part of the labour,” they said in their preface, “has been expended on the contradictions and paradoxes which have infected logic.” “Infected” was a strong word but barely adequate to express the agony of the paradoxes. They were a cancer.
Some had been known since ancient times:
Epimenides the Cretan said that all Cretans were liars, and all other statements made by Cretans were certainly lies. Was this a lie?♦
A cleaner formulation of Epimenides’ paradox—cleaner because one need not worry about Cretans and their attributes—is the liar’s paradox: This statement is false. The statement cannot be true, because then it is false. It cannot be false, because then it becomes true. It is neither true nor false, or it is both at once. But the discovery of this twisting, backfiring, mind-bending circularity does not bring life or language crashing to a halt—one grasps the idea and moves on—because life and language lack the perfection, the absolutes, that give them force. In real life, all Cretans cannot be liars. Even liars often tell the truth. The pain begins only with the attempt to build an airtight vessel. Russell and Whitehead aimed for perfection—for proof—otherwise the enterprise had little point. The more rigorously they built, the more paradoxes they found. “It was in the air,” Douglas Hofstadter has written, “that truly peculiar things could happen when modern cousins of various ancient paradoxes cropped up inside the rigorously logical world of numbers,… a pristine paradise in which no one had dreamt paradox might arise.”♦
One was Berry’s paradox, first suggested to Russell by G. G. Berry, a librarian at the Bodleian. It has to do with counting the syllables needed to specify each integer. Generally, of course, the larger the number the more syllables are required. In English, the smallest integer requiring two syllables is seven. The smallest requiring three syllables is eleven. The number 121 seems to require six syllables (“one hundred twenty-one”), but actually four will do the job, with some cleverness: “eleven squared.” Still, even with cleverness, there are only a finite number of possible syllables and therefore a finite number of names, and, as Russell put it, “Hence the names of some integers must consist of at least nineteen syllables, and among these there must be a least. Hence the least integer not nameable in fewer than nineteen syllables must denote a definite integer.”♦♦ Now comes the paradox. This phrase, the least integer not nameable in fewer than nineteen syllables, contains only eighteen syllables. So the least integer not nameable in fewer than nineteen syllables has just been named in fewer than nineteen syllables.
Another paradox of Russell’s is the Barber paradox. The barber is the man (let us say) who shaves all the men, and only those, who do not shave themselves. Does the barber shave himself? If he does he does not, and if he does not he does. Few people are troubled by such puzzles, because in real life the barber does as he likes and the world goes on. We tend to feel, as Russell put it, that “the whole form of words is just a noise without meaning.”♦ But the paradox cannot be dismissed so easily when a mathematician examines the subject known as set theory, or the theory of classes. Sets are groups of things—for example, integers. The set 0, 2, 4 has integers as its members. A set can also be a member of other sets. For example, the set 0, 2, 4 belongs to the set of sets of integers and the set of sets with three members but not the set of sets of prime numbers. So Russell defined a certain set this way:
S is the set of all sets that are not members of themselves.
This version is known as Russell’s paradox. It cannot be dismissed as noise.
To eliminate Russell’s paradox Russell took drastic measures. The enabling factor seemed to be the peculiar recursion within the offending statement: the idea of sets belonging to sets. Recursion was the oxygen feeding the flame. In the same way, the liar paradox relies on statements about statements. “This statement is false” is meta-language: language about language. Russell’s paradoxical set relies on a meta-set: a set of sets. So the problem was a crossing of levels, or, as Russell termed it, a mixing of types. His solution: declare it illegal, taboo, out of bounds. No mixing different levels of abstraction. No self-reference; no self-containment. The rules of symbolism in Principia Mathematica would not allow the reaching-back-around, snake-eating-its-tail feedback loop that seemed to turn on the possibility of self-contradiction. This was his firewall.
Enter Kurt Gödel.
He was born in 1906 in Brno, at the center of the Czech province of Moravia. He studied physics at the University of Vienna, seventy-five miles south, and as a twenty-year-old became part of the Vienna Circle, a group of philosophers and mathematicians who met regularly in smoky coffeehouses like the Café Josephinum and the Café Reichsrat to propound logic and realism as a bulwark against metaphysics—by which they meant spiritualism, phenomenology, irrationality. Gödel talked to them about the New Logic (this term was in the air) and before long about metamathematics—der Metamathematik. Metamathematics was not to mathematics what metaphysics was to physics. It was mathematics once removed—mathematics about mathematics—a formal system “looked at from the outside” (“äußerlich betrachtet”).♦ He was about to make the most important statement, prove the most important theorem about knowledge in the twentieth century. He was going to kill Russell’s dream of a perfect logical system. He was going to show that the paradoxes were not excrescences; they were fundamental.
Gödel praised the Russell and Whitehead project before he buried it: mathematical logic was, he wrote, “a science prior to all others, which contains the ideas and principles underlying all sciences.”♦ Principia Mathematica, the great opus, embodied a formal system that had become, in its brief lifetime, so comprehensive and so dominant that Gödel referred to it in shorthand: PM. By PM he meant the system, as opposed to the book. In PM, mathematics had been contained—a ship in a bottle, no longer buffeted and turned by the vast unruly seas. By 1930, when mathematicians proved something, they did it according to PM. In PM, as Gödel said, “one can prove any theorem using nothing but a few mechanical rules.”♦
Any theorem: for the system was, or claimed to be, complete. Mechanical rules: for the logic operated inexorably, with no room for varying human interpretation. Its symbols were drained of meaning. Anyone could verify a proof step by step, by following the rules, without understanding it. Calling this quality mechanical invoked the dreams of Charles Babbage and Ada Lovelace, machines grinding through numbers, and numbers standing for anything at all.
Amid the doomed culture of 1930 Vienna, listening to his new friends debate the New Logic, his manner reticent, his eyes magnified by black-framed round spectacles, the twenty-four-year-old Gödel believed in the perfection of the bottle that was PM but doubted whether mathematics could truly be contained. This slight young man turned his doubt into a great and horrifying discovery. He found that lurking within PM—and within any consistent system of logic—there must be monsters of a kind hitherto unconceived: statements that can never be proved, and yet can never be disproved. There must be truths, that is, that cannot be proved—and Gödel could prove it.
He accomplished this with iron rigor disguised as sleight of hand. He employed the formal rules of PM and, as he employed them, also approached them metamathematically—viewed them, that is, from the outside. As he explained, all the symbols of PM—numbers, operations of arithmetic, logical connectors, and punctuation—constituted a limited alphabet. Every statement or formula of PM was written in this alphabet. Likewise every proof comprised a finite sequence of formulas—just a longer passage written in the same alphabet. This is where metamathematics came in. Metamathematically, Gödel pointed out, one sign is as good as another; the choice of a particular alphabet is arbitrary. One could use the traditional assortment of numerals and glyphs (from arithmetic: +, −, =, ×; from logic: ¬, ∨, ⊃, ∃), or one could use letters, or one could use dots and dashes. It was a matter of encoding, slipping from one symbol set to another.
Gödel proposed to use numbers for all his signs. Numbers were his alphabet. And because numbers can be combined using arithmetic, any sequence of numbers amounts to one (possibly very large) number. So every statement, every formula of PM can be expressed as a single number, and so can every proof. Gödel outlined a rigorous scheme for doing the encoding—an algorithm, mechanical, just rules to follow, no intelligence necessary. It works forward and backward: given any formula, following the rules generates one number, and given any number, following the rules produces the corresponding formula.
Not every number translates into a correct formula, however. Some numbers decode back into gibberish, or formulas that are false within the rules of the system. The string of symbols “0 0 0 = = =” does not make a formula at all, though it translates to some number. The statement “0 = 1” is a recognizable formula, but it is false. The formula “0 + x = x + 0” is true, and it is provable.
This last quality—the property of being provable according to PM—was not meant to be expressible in the language of PM. It seems to be a statement from outside the system, a metamathematical statement. But Gödel’s encoding reeled it in. In the framework he constructed, the natural numbers led a double life, as numbers and also as statements. A statement could assert that a given number is even, or prime, or a perfect square, and a statement could also assert that a given number is a provable formula. Given the number 1,044,045,317,700, for example, one could make various statements and test their truth or falsity: this number is even, it is not a prime, it is not a perfect square, it is greater than 5, it is divisible by 121, and (when decoded according to the official rules) it is a provable formula.
Gödel laid all this out in a little paper in 1931. Making his proof watertight required complex logic, but the basic argument was simple and elegant. Gödel showed how to construct a formula that said A certain number, x, is not provable. That was easy: there were infinitely many such formulas. He then demonstrated that, in at least some cases, the number x would happen to represent that very formula. This was just the looping self-reference that Russell had tried to forbid in the rules of PM—
This statement is not provable
—and now Gödel showed that such statements must exist anyway. The Liar returned, and it could not be locked out by changing the rules. As Gödel explained (in one of history’s most pregnant footnotes),
Contrary to appearances, such a proposition involves no faulty circularity, for it only asserts that a certain well-defined formula … is unprovable. Only subsequently (and so to speak by chance) does it turn out that this formula is precisely the one by which the proposition itself was expressed.♦
Within PM, and within any consistent logical system capable of elementary arithmetic, there must always be such accursed statements, true but unprovable. Thus Gödel showed that a consistent formal system must be incomplete; no complete and consistent system can exist.
The paradoxes were back, nor were they mere quirks. Now they struck at the core of the enterprise. It was, as Gödel said afterward, an “amazing fact”—“that our logical intuitions (i.e., intuitions concerning such notions as: truth, concept, being, class, etc.) are self-contradictory.”♦ It was, as Douglas Hofstadter says, “a sudden thunderbolt from the bluest of skies,”♦ its power arising not from the edifice it struck down but the lesson it contained about numbers, about symbolism, about encoding:
Gödel’s conclusion sprang not from a weakness in PM but from a strength. That strength is the fact that numbers are so flexible or “chameleonic” that their patterns can mimic patterns of reasoning.… PM’s expressive power is what gives rise to its incompleteness.
The long-sought universal language, the characteristica universalis Leibniz had pretended to invent, had been there all along, in the numbers. Numbers could encode all of reasoning. They could represent any form of knowledge.
Gödel’s first public mention of his discovery, on the third and last day of a philosophical conference in Königsberg in 1930, drew no response; only one person seems to have heard him at all, a Hungarian named Neumann János. This young mathematician was in the process of moving to the United States, where he would soon and for the rest of his life be called John von Neumann. He understood Gödel’s import at once; it stunned him, but he studied it and was persuaded. No sooner did Gödel’s paper appear than von Neumann was presenting it to the mathematics colloquium at Princeton. Incompleteness was real. It meant that mathematics could never be proved free of self-contradiction. And “the important point,” von Neumann said, “is that this is not a philosophical principle or a plausible intellectual attitude, but the result of a rigorous mathematical proof of an extremely sophisticated kind.”♦ Either you believed in mathematics or you did not.
Bertrand Russell (who, of course, did) had moved on to more gentle sorts of philosophy. Much later, as an old man, he admitted that Gödel had troubled him: “It made me glad that I was no longer working at mathematical logic. If a given set of axioms leads to a contradiction, it is clear that at least one of the axioms must be false.”♦ On the other hand, Vienna’s most famous philosopher, Ludwig Wittgenstein (who, fundamentally, did not), dismissed the incompleteness theorem as trickery (“Kunststücken”) and boasted that rather than try to refute it, he would simply pass it by:
Mathematics cannot be incomplete; any more than a sense can be incomplete. Whatever I can understand, I must completely understand.♦
Gödel’s retort took care of them both. “Russell evidently misinterprets my result; however, he does so in a very interesting manner,” he wrote. “In contradistinction Wittgenstein … advances a completely trivial and uninteresting misinterpretation.”♦
In 1933 the newly formed Institute for Advanced Study, with John von Neumann and Albert Einstein among its first faculty members, invited Gödel to Princeton for the year. He crossed the Atlantic several more times that decade, as fascism rose and the brief glory of Vienna began to fade. Gödel, ignorant of politics and naïve about history, suffered depressive breakdowns and bouts of hypochondria that forced him into sanatoria. Princeton beckoned but Gödel vacillated. He stayed in Vienna in 1938, through the Anschluss, as the Vienna Circle ceased to be, its members murdered or exiled, and even in 1939, when Hitler’s army occupied his native Czechoslovakia. He was not a Jew, but mathematics was verjudet enough. He finally managed to leave in January 1940 by way of the Trans-Siberian Railway, Japan, and a ship to San Francisco. His name was recoded by the telephone company as “K. Goedel” when he arrived in Princeton, this time to stay.♦
Claude Shannon had also arrived at the Institute for Advanced Study, to spend a postdoctoral year. He found it a lonely place, occupying a new red-brick building with clocktower and cupola framed by elms on a former farm a mile from Princeton University. The first of its fifteen or so professors was Einstein, whose office was at the back of the first floor; Shannon seldom laid eyes on him. Gödel, who had arrived in March, hardly spoke to anyone but Einstein. Shannon’s nominal supervisor was Hermann Weyl, another German exile, the most formidable mathematical theorist of the new quantum mechanics. Weyl was only mildly interested in Shannon’s thesis on genetics—“your bio-mathematical problems”♦—but thought Shannon might find common ground with the institute’s other great young mathematician, von Neumann. Mostly Shannon stayed moodily in his room in Palmer Square. His twenty-year-old wife, having left Radcliffe to be with him, found it increasingly grim, staying home while Claude played clarinet accompaniment to his Bix Beiderbecke record on the phonograph. Norma thought he was depressed and wanted him to see a psychiatrist. Meeting Einstein was nice, but the thrill wore off. Their marriage was over; she was gone by the end of the year.
Nor could Shannon stay in Princeton. He wanted to pursue the transmission of intelligence, a notion poorly defined and yet more pragmatic than the heady theoretical physics that dominated the institute’s agenda. Furthermore, war approached. Research agendas were changing everywhere. Vannevar Bush was now heading the National Defense Research Committee, which assigned Shannon “Project 7”:♦ the mathematics of fire-control mechanisms for antiaircraft guns—“the job,” as the NDRC reported dryly, “of applying corrections to the gun control so that the shell and the target will arrive at the same position at the same time.”♦ Airplanes had suddenly rendered obsolete almost all the mathematics used in ballistics: for the first time, the targets were moving at speeds not much less than the missiles themselves. The problem was complex and critical, on ships and on land. London was organizing batteries of heavy guns firing 3.7-inch shells. Aiming projectiles at fast-moving aircraft needed either intuition and luck or a vast amount of implicit computation by gears and linkages and servos. Shannon analyzed physical problems as well as computational problems: the machinery had to track rapid paths in three dimensions, with shafts and gears controlled by rate finders and integrators. An antiaircraft gun in itself behaved as a dynamical system, subject to “backlash” and oscillations that might or might not be predictable. (Where the differential equations were nonlinear, Shannon made little headway and knew it.)
He had spent two of his summers working for Bell Telephone Laboratories in New York; its mathematics department was also taking on the fire-control project and asked Shannon to join. This was work for which the Differential Analyzer had prepared him well. An automated antiaircraft gun was already an analog computer: it had to convert what were, in effect, second-order differential equations into mechanical motion; it had to accept input from rangefinder sightings or new, experimental radar; and it had to smooth and filter this data, to compensate for errors.
At Bell Labs, the last part of this problem looked familiar. It resembled an issue that plagued communication by telephone. The noisy data looked like static on the line. “There is an obvious analogy,” Shannon and his colleagues reported, “between the problem of smoothing the data to eliminate or reduce the effect of tracking errors and the problem of separating a signal from interfering noise in communications systems.”♦ The data constituted a signal; the whole problem was “a special case of the transmission, manipulation, and utilization of intelligence.” Their specialty, at Bell Labs.
Transformative as the telegraph had been, miraculous as the wireless radio now seemed, electrical communication now meant the telephone. The “electrical speaking telephone” first appeared in the United States with the establishment of a few experimental circuits in the 1870s. By the turn of the century, the telephone industry surpassed the telegraph by every measure—number of messages, miles of wire, capital invested—and telephone usage was doubling every few years. There was no mystery about why: anyone could use a telephone. The only skills required were talking and listening: no writing, no codes, no keypads. Everyone responded to the sound of the human voice; it conveyed not just words but feeling.
The advantages were obvious—but not to everyone. Elisha Gray, a telegraph man who came close to trumping Alexander Graham Bell as inventor of the telephone, told his own patent lawyer in 1875 that the work was hardly worthwhile: “Bell seems to be spending all his energies in [the] talking telegraph. While this is very interesting scientifically it has no commercial value at present, for they can do much more business over a line by methods already in use.”♦ Three years later, when Theodore N. Vail quit the Post Office Department to become the first general manager (and only salaried officer) of the new Bell Telephone Company, the assistant postmaster general wrote angrily, “I can scarce believe that a man of your sound judgment … should throw it up for a d——d old Yankee notion (a piece of wire with two Texan steer horns attached to the ends, with an arrangement to make the concern blate like a calf) called a telephone!”♦ The next year, in England, the chief engineer of the General Post Office, William Preece, reported to Parliament: “I fancy the descriptions we get of its use in America are a little exaggerated, though there are conditions in America which necessitate the use of such instruments more than here. Here we have a superabundance of messengers, errand boys and things of that kind.… I have one in my office, but more for show. If I want to send a message—I use a sounder or employ a boy to take it.”♦
One reason for these misguesses was just the usual failure of imagination in the face of a radically new technology. The telegraph lay in plain view, but its lessons did not extrapolate well to this new device. The telegraph demanded literacy; the telephone embraced orality. A message sent by telegraph had first to be written, encoded, and tapped out by a trained intermediary. To employ the telephone, one just talked. A child could use it. For that very reason it seemed like a toy. In fact, it seemed like a familiar toy, made from tin cylinders and string. The telephone left no permanent record. The Telephone had no future as a newspaper name. Business people thought it unserious. Where the telegraph dealt in facts and numbers, the telephone appealed to emotions.
The new Bell company had little trouble turning this into a selling point. Its promoters liked to quote Pliny, “The living voice is that which sways the soul,” and Thomas Middleton, “How sweetly sounds the voice of a good woman.” On the other hand, there was anxiety about the notion of capturing and reifying voices—the phonograph, too, had just arrived. As one commentator said, “No matter to what extent a man may close his doors and windows, and hermetically seal his key-holes and furnace-registers with towels and blankets, whatever he may say, either to himself or a companion, will be overheard.”♦ Voices, hitherto, had remained mostly private.
The new contraption had to be explained, and generally this began by comparison to telegraphy. There were a transmitter and receiver, and wires connected them, and something was carried along the wire in the form of electricity. In the case of the telephone, that thing was sound, simply converted from waves of pressure in the air to waves of electric current. One advantage was apparent: the telephone would surely be useful to musicians. Bell himself, traveling around the country as impresario for the new technology, encouraged this way of thinking, giving demonstrations in concert halls, where full orchestras and choruses played “America” and “Auld Lang Syne” into his gadgetry. He encouraged people to think of the telephone as a broadcasting device, to send music and sermons across long distances, bringing the concert hall and the church into the living room. Newspapers and commentators mostly went along. That is what comes of analyzing a technology in the abstract. As soon as people laid their hands on telephones, they worked out what to do. They talked.
In a lecture at Cambridge, the physicist James Clerk Maxwell offered a scientific description of the telephone conversation: “The speaker talks to the transmitter at one end of the line, and at the other end of the line the listener puts his ear to the receiver, and hears what the speaker said. The process in its two extreme states is so exactly similar to the old-fashioned method of speaking and hearing that no preparatory practice is required on the part of either operator.”♦ He, too, had noticed its ease of use.
So by 1880, four years after Bell conveyed the words “Mr. Watson, come here, I want to see you,” and three years after the first pair of telephones rented for twenty dollars, more than sixty thousand telephones were in use in the United States. The first customers bought pairs of telephones for communication point to point: between a factory and its business office, for example. Queen Victoria installed one at Windsor Castle and one at Buckingham Palace (fabricated in ivory; a gift from the savvy Bell). The topology changed when the number of sets reachable by other sets passed a critical threshold, and that happened surprisingly soon. Then community networks arose, their multiple connections managed through a new apparatus called a switch-board.
The initial phase of ignorance and skepticism passed in an eyeblink. The second phase of amusement and entertainment did not last much longer. Businesses quickly forgot their qualms about the device’s seriousness. Anyone could be a telephone prophet now—some of the same predictions had already been heard in regard to the telegraph—but the most prescient comments came from those who focused on the exponential power of interconnection. Scientific American assessed “The Future of the Telephone” as early as 1880 and emphasized the forming of “little clusters of telephonic communicants.” The larger the network and the more diverse its interests, the greater its potential would be.
What the telegraph accomplished in years the telephone has done in months. One year it was a scientific toy, with infinite possibilities of practical use; the next it was the basis of a system of communication the most rapidly expanding, intricate, and convenient that the world has known.… Soon it will be the rule and not the exception for business houses, indeed for the dwellings of well-to-do people as well, to be interlocked by means of telephone exchange, not merely in our cities, but in all outlying regions. The result can be nothing less than a new organization of society—a state of things in which every individual, however secluded, will have at call every other individual in the community, to the saving of no end of social and business complications, of needless goings to and fro, of disappointments, delays, and a countless host of those great and little evils and annoyances.
The time is close at hand when the scattered members of civilized communities will be as closely united, so far as instant telephonic communication is concerned, as the various members of the body now are by the nervous system.♦
The scattered members using telephones numbered half a million by 1890; by 1914, 10 million. The telephone was already thought, correctly, to be responsible for rapid industrial progress. The case could hardly be overstated. The areas depending on “instantaneous communication across space”♦ were listed by the United States Commerce Department in 1907: “agriculture, mining, commerce, manufacturing, transportation, and, in fact, all the various branches of production and distribution of natural and artificial resources.” Not to mention “cobblers, cleaners of clothing, and even laundresses.” In other words, every cog in the engine of the economy. “Existence of telephone traffic is essentially an indication that time is being saved,” the department commented. It observed changes in the structure of life and society that would still seem new a century later: “The last few years have seen such an extension of telephone lines through the various summer-resort districts of the country that it has become practicable for business men to leave their offices for several days at a time, and yet keep in close touch with their offices.” In 1908 John J. Carty, who became the first head of the Bell Laboratories, offered an information-based analysis to show how the telephone had shaped the New York skyline—arguing that the telephone, as much as the elevator, had made skyscrapers possible.
It may sound ridiculous to say that Bell and his successors were the fathers of modern commercial architecture—of the skyscraper. But wait a minute. Take the Singer Building, the Flatiron Building, the Broad Exchange, the Trinity, or any of the giant office buildings. How many messages do you suppose go in and out of those buildings every day? Suppose there was no telephone and every message had to be carried by a personal messenger? How much room do you think the necessary elevators would leave for offices? Such structures would be an economic impossibility.♦
To enable the fast expansion of this extraordinary network, the telephone demanded new technologies and new science. They were broadly of two kinds. One had to do with electricity itself: measuring electrical quantities; controlling the electromagnetic wave, as it was now understood—its modulation in amplitude and in frequency. Maxwell had established in the 1860s that electrical pulses and magnetism and light itself were all manifestations of a single force: “affectations of the same substance,” light being one more case of “an electromagnetic disturbance propagated through the field according to electromagnetic laws.”♦ These were the laws that electrical engineers now had to apply, unifying telephone and radio among other technologies. Even the telegraph employed a simple kind of amplitude modulation, in which only two values mattered, a maximum for “on” and a minimum for “off.” To convey sound required far stronger current, far more delicately controlled. The engineers had to understand feedback: a coupling of the output of a power amplifier, such as a telephone mouthpiece, with its input. They had to design vacuum-tube repeaters to carry the electric current over long distance, making possible the first transcontinental line in 1914, between New York and San Francisco, 3,400 miles of wire suspended from 130,000 poles. The engineers also discovered how to modulate individual currents so as to combine them in a single channel—multiplexing—without losing their identity. By 1918 they could get four conversations into a single pair of wires. But it was not currents that preserved identity. Before the engineers quite realized it, they were thinking in terms of the transmission of a signal, an abstract entity, quite distinct from the electrical waves in which it was embodied.
A second, less well defined sort of science concerned the organizing of connections—switching, numbering, and logic. This branch descended from Bell’s original realization, dating from 1877, that telephones need not be sold in pairs; that each individual telephone could be connected to many other telephones, not by direct wires but through a central “exchange.” George W. Coy, a telegraph man in New Haven, Connecticut, built the first “switch-board” there, complete with “switch-pins” and “switch-plugs” made from carriage bolts and wire from discarded bustles. He patented it and served as the world’s first telephone “operator.” With all the making and breaking of connections, switch-pins wore out quickly. An early improvement was a hinged two-inch plate resembling a jackknife: the “jack-knife switch,” or as it was soon called, the “jack.” In January 1878, Coy’s switchboard could manage two simultaneous conversations between any of the exchange’s twenty-one customers. In February, Coy published a list of subscribers: himself and some friends; several physicians and dentists; the post office, police station, and mercantile club; and some meat and fish markets. This has been called the world’s first telephone directory, but it was hardly that: one page, not alphabetized, and no numbers associated with the names. The telephone number had yet to be invented.
That innovation came the next year in Lowell, Massachusetts, where by the end of 1879 four operators managed the connections among two hundred subscribers by shouting to one another across the switchboard room. An epidemic of measles broke out, and Dr. Moses Greeley Parker worried that if the operators succumbed, they would be hard to replace. He suggested identifying each telephone by number. He also suggested listing the numbers in an alphabetical directory of subscribers. These ideas could not be patented and arose again in telephone exchanges across the country, where the burgeoning networks were creating clusters of data in need of organization. Telephone books soon represented the most comprehensive listings of, and directories to, human populations ever attempted. (They became the thickest and densest of the world’s books—four volumes for London; a 2,600-page tome for Chicago—and seemed a permanent, indispensable part of the world’s information ecology until, suddenly, they were not. They went obsolete, effectively, at the turn of the twenty-first century. American telephone companies were officially phasing them out by 2010; in New York, the end of automatic delivery of telephone directories was estimated to save 5,000 tons of paper.)
At first, customers resented the impersonality of telephone numbers, and engineers doubted whether people could remember a number of more than four or five digits. The Bell Company finally had to insist. The first telephone operators were teenage boys, cheaply hired from the ranks of telegraph messengers, but exchanges everywhere discovered that boys were wild, given to clowning and practical jokes, and more likely to be found wrestling on the floor than sitting on stools to perform the exacting, repetitive work of a switchboard operator.♦ A new source of cheap labor was available, and by 1881 virtually every telephone operator was a woman. In Cincinnati, for example, W. H. Eckert reported hiring sixty-six “young ladies” who were “very much superior” to boys: “They are steadier, do not drink beer, and are always on hand.”♦ He hardly needed to add that the company could pay a woman as little as or less than a teenage boy. It was challenging work that soon required training. Operators had to be quick in distinguishing many different voices and accents, had to maintain a polite equilibrium in the face of impatience and rudeness, as they engaged in long hours of athletic upper-body exercise, wearing headsets like harnesses. Some men thought this was good for them. “The action of stretching her arms up above her head, and to the right and left of her, develops her chest and arms,” said Every Woman’s Encyclopedia, “and turns thin and weedy girls into strong ones. There are no anaemic, unhealthy looking girls in the operating rooms.”♦ Along with another new technology, the typewriter, the telephone switchboard catalyzed the introduction of women into the white-collar workforce, but battalions of human operators could not sustain a network on the scale now arising. Switching would have to be performed automatically.
This meant a mechanical linkage to take from callers not just the sound of their voice but also a number—identifying a person, or at least another telephone. The challenge of converting a number into electrical form still required ingenuity: first push buttons were tried, then an awkward-seeming rotary dial, with ten finger positions for the decimal digits, sending pulses down the line. Then the coded pulses served as an agent of control at the central exchange, where another mechanism selected from an array of circuits and set up a connection. Altogether this made for an unprecedented degree of complexity in the translations between human and machine, number and circuitry. The point was not lost on the company, which liked to promote its automatic switches as “electrical brains.” Having borrowed from telegraphy the electromechanical relay—using one circuit to control another—the telephone companies had reduced it in size and weight to less than four ounces and now manufactured several million each year.
“The telephone remains the acme of electrical marvels,” wrote a historian in 1910—a historian of the telephone, already. “No other thing does so much with so little energy. No other thing is more enswathed in the unknown.”♦ New York City had several hundred thousand listed telephone customers, and Scribner’s Magazine highlighted this astounding fact: “Any two of that large number can, in five seconds, be placed in communication with each other, so well has engineering science kept pace with public needs.”♦ To make the connections, the switchboard had grown to a monster of 2 million soldered parts, 4,000 miles of wire, and 15,000 signal lamps.♦ By 1925, when an assortment of telephone research groups were formally organized into the Bell Telephone Laboratories, a mechanical “line finder” with a capacity of 400 lines was replacing 22-point electromechanical rotary switches. The American Telephone & Telegraph Company was consolidating its monopoly. Engineers struggled to minimize the hunt time. At first, long-distance calling required reaching a second, “toll” operator and waiting for a call back; soon the interconnection of local exchanges would have to allow for automatic dialing. The complexities multiplied. Bell Labs needed mathematicians.
What began as the Mathematics Consulting Department grew into a center of practical mathematics like none other. It was not like the prestigious citadels, Harvard and Princeton. To the academic world it was barely visible. Its first director, Thornton C. Fry, enjoyed the tension between theory and practice—the clashing cultures. “For the mathematician, an argument is either perfect in every detail or else it is wrong,” he wrote in 1941. “He calls this ‘rigorous thinking.’ The typical engineer calls it ‘hair-splitting.’ ”♦
The mathematician also tends to idealize any situation with which he is confronted. His gases are “ideal,” his conductors “perfect,” his surfaces “smooth.” He calls this “getting down to essentials.” The engineer is likely to dub it “ignoring the facts.”
In other words, the mathematicians and engineers could not do without each other. Every electrical engineer could now handle the basic analysis of waves treated as sinusoidal signals. But new difficulties arose in understanding the action of networks; network theorems were devised to handle these mathematically. Mathematicians applied queuing theory to usage conflicts; developed graphs and trees to manage issues of intercity trunks and lines; and used combinatorial analysis to break down telephone probability problems.
Then there was noise. This did not at first (to Alexander Graham Bell, for example) seem like a problem for theorists. It was just there, always crowding the line—pops, hisses, crackles interfering with, or degrading, the voice that had entered the mouthpiece. It plagued radio, too. At best it stayed in the background and people hardly noticed; at worst the weedy profusion spurred the customers’ imaginations:
There was sputtering and bubbling, jerking and rasping, whistling and screaming. There was the rustling of leaves, the croaking of frogs, the hissing of steam, and the flapping of birds’ wings. There were clicks from telegraph wires, scraps of talk from other telephones, curious little squeals that were unlike any known sound.… The night was noisier than the day, and at the ghostly hour of midnight, for what strange reasons no one knows, the babel was at its height.♦
But engineers could now see the noise on their oscilloscopes, interfering with and degrading their clean waveforms, and naturally they wanted to measure it, even if there was something quixotic about measuring a nuisance so random and ghostly. There was a way, in fact, and Albert Einstein had shown what it was.
In 1905, his finest year, Einstein published a paper on Brownian motion, the random, jittery motion of tiny particles suspended in a fluid. Antony van Leeuwenhoek had discovered it with his early microscope, and the phenomenon was named after Robert Brown, the Scottish botanist who studied it carefully in 1827: first pollen in water, then soot and powdered rock. Brown convinced himself that these particles were not alive—they were not animalcules—yet they would not sit still. In a mathematical tour de force, Einstein explained this as a consequence of the heat energy of molecules, whose existence he thereby proved. Microscopically visible particles, like pollen, are bombarded by molecular collisions and are light enough to be jolted randomly this way and that. The fluctuations of the particles, individually unpredictable, collectively express the laws of statistical mechanics. Although the fluid may be at rest and the system in thermodynamic equilibrium, the irregular motion perseveres, as long as the temperature is above absolute zero. By the same token, he showed that random thermal agitation would also affect free electrons in any electrical conductor—making noise.
Physicists paid little attention to the electrical aspects of Einstein’s work, and it was not until 1927 that thermal noise in circuits was put on a rigorous mathematical footing, by two Swedes working at Bell Labs. John B. Johnson was the first to measure what he realized was noise intrinsic to the circuit, as opposed to evidence of flawed design. Then Harry Nyquist explained it, deriving formulas for the fluctuations in current and in voltage in an idealized network. Nyquist was the son of a farmer and shoemaker who was originally called Lars Jonsson but had to find a new name because his mail was getting mixed up with another Lars Jonsson’s. The Nyquists immigrated to the United States when Harry was a teenager; he made his way from North Dakota to Bell Labs by way of Yale, where he got a doctorate in physics. He always seemed to have his eye on the big picture—which did not mean telephony per se. As early as 1918, he began working on a method for transmitting pictures by wire: “telephotography.” His idea was to mount a photograph on a spinning drum, scan it, and generate currents proportional to the lightness or darkness of the image. By 1924 the company had a working prototype that could send a five-by-seven-inch picture in seven minutes. But Nyquist meanwhile was looking backward, too, and that same year, at an electrical engineers’ convention in Philadelphia, gave a talk with the modest title “Certain Factors Affecting Telegraph Speed.”
It had been known since the dawn of telegraphy that the fundamental units of messaging were discrete: dots and dashes. It became equally obvious in the telephone era that, on the contrary, useful information was continuous: sounds and colors, shading into one another, blending seamlessly along a spectrum of frequencies. So which was it? Physicists like Nyquist were dealing with electric currents as waveforms, even when they were conveying discrete telegraph signals. Nowadays most of the current in a telegraph line was being wasted. In Nyquist’s way of thinking, if those continuous signals could represent anything as complex as voices, then the simple stuff of telegraphy was just a special case. Specifically, it was a special case of amplitude modulation, in which the only interesting amplitudes were on and off. By treating the telegraph signals as pulses in the shape of waveforms, engineers could speed their transmission and could combine them in a single circuit—could combine them, too, with voice channels. Nyquist wanted to know how much—how much telegraph data, how fast. To answer that question he found an ingenious approach to converting continuous waves into data that was discrete, or “digital.” Nyquist’s method was to sample the waves at intervals, in effect converting them into countable pieces.
A circuit carried waves of many different frequencies: a “band” of waves, engineers would say. The range of frequencies—the width of that band, or “band width”—served as a measure of the capacity of the circuit. A telephone line could handle frequencies from about 400 to 3,400 hertz, or waves per second, for a bandwidth of 3,000 hertz. (That would cover most of the sound from an orchestra, but the high notes of the piccolo would be cut off.) Nyquist wanted to put this as generally as he could. He calculated a formula for the “speed of transmission of intelligence.”♦ To transmit intelligence at a certain speed, he showed, a channel needs a certain, measurable bandwidth. If the bandwidth was too small, it would be necessary to slow down the transmission. (But with time and ingenuity, it was realized later, even complex messages could be sent across a channel of very small bandwidth: a drum, for example, beaten by hand, sounding notes of only two pitches.)
Nyquist’s colleague Ralph Hartley, who had begun his career as an expert on radio receivers, extended these results in a presentation in the summer of 1927, at an international congress on the shore of Lake Como, Italy. Hartley used a different word, “information.” It was a good occasion for grand ideas. Scientists had gathered from around the world for the centennial of Alessandro Volta’s death. Niels Bohr spoke on the new quantum theory and introduced for the first time his concept of complementarity. Hartley offered his listeners both a fundamental theorem and a new set of definitions.
The theorem was an extension of Nyquist’s formula, and it could be expressed in words: the most information that can be transmitted in any given time is proportional to the available frequency range (he did not yet use the term bandwidth). Hartley was bringing into the open a set of ideas and assumptions that were becoming part of the unconscious culture of electrical engineering, and the culture of Bell Labs especially. First was the idea of information itself. He needed to pin a butterfly to the board. “As commonly used,” he said, “information is a very elastic term.”♦ It is the stuff of communication—which, in turn, can be direct speech, writing, or anything else. Communication takes place by means of symbols—Hartley cited for example “words” and “dots and dashes.” The symbols, by common agreement, convey “meaning.” So far, this was one slippery concept after another. If the goal was to “eliminate the psychological factors involved” and to establish a measure “in terms of purely physical quantities,” Hartley needed something definite and countable. He began by counting symbols—never mind what they meant. Any transmission contained a countable number of symbols. Each symbol represented a choice; each was selected from a certain set of possible symbols—an alphabet, for example—and the number of possibilities, too, was countable. The number of possible words is not so easy to count, but even in ordinary language, each word represents a selection from a set of possibilities:
For example, in the sentence, “Apples are red,” the first word eliminated other kinds of fruit and all other objects in general. The second directs attention to some property or condition of apples, and the third eliminates other possible colors.…
The number of symbols available at any one selection obviously varies widely with the type of symbols used, with the particular communicators and with the degree of previous understanding existing between them.♦
Hartley had to admit that some symbols might convey more information, as the word was commonly understood, than others. “For example, the single word ‘yes’ or ‘no,’ when coming at the end of a protracted discussion, may have an extraordinarily great significance.” His listeners could think of their own examples. But the point was to subtract human knowledge from the equation. Telegraphs and telephones are, after all, stupid.
It seemed intuitively clear that the amount of information should be proportional to the number of symbols: twice as many symbols, twice as much information. But a dot or dash—a symbol in a set with just two members—carries less information than a letter of the alphabet and much less information than a word chosen from a thousand-word dictionary. The more possible symbols, the more information each selection carries. How much more? The equation, as Hartley wrote it, was this:
H = n log s
where H is the amount of information, n is the number of symbols transmitted, and s is the size of the alphabet. In a dot-dash system, s is just 2. A single Chinese character carries so much more weight than a Morse dot or dash; it is so much more valuable. In a system with a symbol for every word in a thousand-word dictionary, s would be 1,000.
The amount of information is not proportional to the alphabet size, however. That relationship is logarithmic: to double the amount of information, it is necessary to quadruple the alphabet size. Hartley illustrated this in terms of a printing telegraph—one of the hodgepodge of devices, from obsolete to newfangled, being hooked up to electrical circuits. Such telegraphs used keypads arranged according to a system devised in France by Émile Baudot. The human operators used keypads, that is—the device translated these key presses, as usual, into the opening and closing of telegraph contacts. The Baudot code used five units to transmit each character, so the number of possible characters was 25 or 32. In terms of information content, each such character was five times as valuable—not thirty-two times—as its basic binary units.
Telephones, meanwhile, were sending their human voices across the network in happy, curvaceous analog waves. Where were the symbols in those? How could they be counted?
Hartley followed Nyquist in arguing that the continuous curve should be thought of as the limit approached by a succession of discrete steps, and that the steps could be recovered, in effect, by sampling the waveform at intervals. That way telephony could be made subject to the same mathematical treatment as telegraphy. By a crude but convincing analysis, he showed that in both cases the total amount of information would depend on two factors: the time available for transmission and the bandwidth of the channel. Phonograph records and motion pictures could be analyzed the same way.
These odd papers by Nyquist and Hartley attracted little immediate attention. They were hardly suitable for any prestigious journal of mathematics or physics, but Bell Labs had its own, The Bell System Technical Journal, and Claude Shannon read them there. He absorbed the mathematical insights, sketchy though they were—first awkward steps toward a shadowy goal. He noted also the difficulties both men had in defining their terms. “By the speed of transmission of intelligence is meant the number of characters, representing different letters, figures, etc., which can be transmitted in a given length of time.”♦ Characters, letters, figures: hard to count. There were concepts, too, for which terms had yet to be invented: “the capacity of a system to transmit a particular sequence of symbols …”♦
THE BAUDOT CODE
Shannon felt the promise of unification. The communications engineers were talking not just about wires but also the air, the “ether,” and even punched tape. They were contemplating not just words but also sounds and images. They were representing the whole world as symbols, in electricity.
♦ In an evaluation forty years later the geneticist James F. Crow wrote: “It seems to have been written in complete isolation from the population genetics community.…[Shannon] discovered principles that were rediscovered later.… My regret is that [it] did not become widely known in 1940. It would have changed the history of the subject substantially, I think.”
♦ In standard English, as Russell noted, it is one hundred and eleven thousand seven hundred and seventy-seven.
