The Essential Message:
Claude Shannon and the Making of Information Theory
by
Erico Marui Guizzo
B.S., Electrical Engineering
University of Sao Paulo, Brazil, 1999
Submitted to the Program in Writing and Humanistic Studies
in Partial Fulfillment of the Requirements for the Degree of
Master of Sciencein Science Writing
at the
Massachusetts Institute of Technology
September 2003
MASSACHUSETTSINSTITUTE
OFTECHNOLOGY
JUN 2 7 2003
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© 2003 Erico Marui Guizzo. All rights reserved.
The author hereby grants to MITpermission to reproduce and to distribute publicly
paper and electronic copies of this thesis document in whole or in part.
Signatureof Author:.................. ...................... s{.. udies
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June 16, 2003
Certified and Accepted by:.......................................... ...............................
Robert Kanigel
Thesis Advisor
Professor of Science Writing
Director, Graduate Programin Science Writing
ARCHIVES
K"
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The Essential Message:
Claude Shannon and the Making of Information Theory
by
Erico Marui Guizzo
Submitted to the Program in Writing and Humanistic Studies
on June 16, 2003 in Partial Fulfillment of the Requirements
for the Degree of Master of Science in Science Writing
ABSTRACT
In 1948, Claude Shannon, a young engineer and mathematician working at the Bell
Telephone Laboratories, published "A Mathematical Theory of Communication," a
seminal paper that marked the birth of information theory. In that paper, Shannon defined
what the once fuzzy concept of "information" meant for communication engineers and
proposed a precise way to quantify it-in his theory, the fundamental unit of information is
the bit. He also showed how data could be "compressed" before transmission and how
virtuallyerror-free communication could be achieved. The concepts Shannon developed in
his paper are at the heart of today's digital information technology. CDs, DVDs, cell
phones, fax machines, modems, computer networks, hard drives, memory chips,
encryptionschemes, MP3music, optical communication, high-definition television-all
these things embody many of Shannon's ideas and others inspired by him.
But despite the importance of his work and its influence on everyday life, Claude Shannon
is still unknown to most people. Many papers, theses, books, and articles on information
theory have been published, but none have explored in detail and in accessible language
aimed at a general audience what the theory is about, how it changed the world of
communication, and-most importantly-what path led Shannon to his revolutionary
ideas. "The Essential Message" presents an account of the making of information theory
based on papers, letters, interviews with Shannon and his colleagues, and other sources. It
describes the context in which Shannon was immersed, the main ideas in his 1948 paperand
the reaction to it-and how his theory shaped the technologiesthat changed one of the
most fundamental activities in our lives: communication.
Thesis Supervisor: Robert Kanigel
Title: Professorof Science Writing
Director, Graduate Program in Science Writing
2
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Charles Dickens, David Copperfield*
* From a piece of paper with quotations kept by Shannon. Claude Elwood Shannon Papers,
Manuscript Division, Libraryof Congress, Washington, D.C.
TWO MEN TALK about the past. The conversation is animated. They drink beer.
It is an early evening in the summer of 1982. The two men sit in the living room
of a large stuccoed house overlooking a lake in the suburbs of Boston. Bob Price
is interested in things that happened more than thirty years ago. He wants to know
about the origins of information theory. And Claude Shannon is the right person
to ask: he invented the theory.' Shannon's information theory transformedone of
the most fundamental activities in our lives: communication. His stunning new
ideas made possible the information age that most of humanity lives in today.
"I'm looking at this 1945 cryptographyreport," Price says, "it's got the
words 'information theory' in it. It says you're next going to get around to write
up information theory. It sounds-"
"Oh, did it say that in there?" Shannon asks reluctantly.
"Yes, it sounds as though the cryptography gave you the mysterious link
that made the whole... Well, the fan diagrams, for instance, if it hadn't been for
cryptography would you havethose fan diagrams?"
"What fan diagrams?"
Price points to diagrams in a book that resemble paper fans: "These,
these."
"Oh, those are the fan diagrams," Shannon says laughing.
"Well, do you think that would have come out of cryptography? Or you
had that already without the cryptography?"
Shannon pauses for a second. And then, lowering his voice, he says:
"Well, I have no idea."
"OK, sure, well it's a long time ago."
"But not only that, Bob. You ask questions of where would things have
come from... These complex hypotheticalquestions."
On that evening in 1982, Price tried hard to get onto the mind of the gray
haired man sitting next to him. But the man resisted. Shannon wasn't rude or
stubborn. Quite the contrary, he was known for his sharp sense of humor and even
his self-deprecating nature. He never took himself too seriously. 2 How did he
6
come up with information theory? What about those fan diagrams? Well, it
seemed Shannon couldn't care less.
MEMORABLE BREAKTHROUGHS IN science often have an origin myth, usually
an epiphany that crystallizes in history that particular achievement destined to
change the world. Archimedes in the bathtub. Newton under the apple tree.
Einstein riding a beam of light.3 Like a picture, static in time and lirrited in space,
an origin myth doesn't tell the whole story--and the story it tells sometimes
weaves fact and fiction. But an origin myth helps to popularize scientific
achievements-and science heroes-that people otherwise wouldn't know about.
Information theory doesn't have an origin myth. And Shannon always
repeated his creation was not the result of a single moment of clarity at a bathtub
or under a tree. Didn't he have a "eureka moment"? "I would have," once he said
jokingly, "but I didn't know how to spell the word."4 Maybe that is why, despite
the importance of his work and its influence on everyday life, Claude Shannon is
still unknown to most people.5 Today we use digital cell phones and send
messages over the Internet but we know little about the ideas that contributed to
make these things possible. Many of these ideas are the result of Shannon's
theory, which sought the mathematical laws governing systems designed to
transmit and manipulate informatioin.
But if information theory doesn't have an origin myth, it has a very clear
beginning. The field was founded in 1948when Shannon published the paper
considered his masterwork, "A Mathematical Theory of Communication."6 The
fundamental problem of communication, he wrote in the second paragraph, is that
of reproducing at one point a message selected at another point. A message could
be a letter, a word, a number, speech, music, images, video-anything we want to
transmit to another place. To do that, we need a transmission system; we need to
send the message over a communication channel. But how fastcan we send these
7
messages? Can we transmit, say, a high-resolution picture over a telephone line?
How long that will take? Is there a best way to do it?
Before Shannon, engineers had no clear answers to these questions. At
that time, a wild zoo of technologies was in operation, each with a life of its
own--telephone, telegraph, radio, television, radar, and a number of other
systems developed during the war.7 Shannon came up with a unifying, general
theoryof communication. It didn't matter whether you transmitted signals using a
copper wire, an optical fiber, or a parabolic dish. It didn't matter if you were
transmitting text, voice, or images. Shannon envisioned communication in
abstract, mathematical terms; he defined what the once fuzzy concept of
"information" meant for communication engineers and proposed a precise way to
quantify it. According to him, the information content of any kind of message
could be measured in binary digits, or just bits-a name suggested by a colleague
at Bell Labs. Shannon took the bit as the fundamental unit in information theory.
It was the first time that the term appeared in print.8
In his paper, Shannon showed that every channel has a maximum rate for
transmitting electronic data reliably, which he called the channel capacity. Try to
send information at a rate greater than this threshold and you will always lose part
of your message. This ultimate limit, measured in bits per second, became an
essential benchmark for communication engineers. Before, they developed
systems without knowing the physical limitations. Nowthey werenot working in
the dark anymore; with the channel capacity they knew where they could go-and
where they couldn't.
But the paper contained still one more astounding revelation. Shannon
demonstrated, contrary to what was commonly believed, that engineers could beat
their worst enemy ever: transmission errors-or in their technicaljargon, "noise."
Noise is anything that disturbs communication. It can be an electric signal in a
telephone wire that causes crosstalk in an adjacent wire, a thunderstorm static that
perturbs TV signals distorting the image on the screen, or a failure in network
equipment that corrupts Internet data. At that time, the usual way to overcome
8
noise was to increasethe energy of the transmission signals or send the same
message repeatedly-much as when, in a crowded pub, you have to shout for a
beer several times. Shannon showed a better way to avoid errors without wasting
so much energy and time: coding.
Coding is at the heart of information theory. All communication processes
need some sort of coding. The telephone system transforms the spoken voice into
electrical signals. In Morse code, letters are transmitted with combinations of dots
and dashes. The DNA molecule specifies a protein's structure with four types of
genetic bases. Digital communication systems use bits to represent-or encodeinformation.
Each letter of the alphabet, for example, can be represented with a
group of bits, a sequence of zeroes and ones. You can assign any number of bits
to each letter and arrange the bits in any way you want. In other words, you can
create as many codes as desired. But is there a best code we should use? Shannon
showed that with specially designed codes engineers could do two things: first,
they could squish the messages-thus saving transmission time; also, they could
protect data from noise and achieve virtually error-free communication using the
whole capacity of achannel-perfect communication at full speed, something no
communication specialist had ever dreamed possible.
Measuring the information content of a message; channels with limited
capacity; transmitting information with bits; compressing data; error-free
communication. Shannon's concepts and results were not just surprising but
counterintuitive. They went against some of the most fundamental beliefs in the
communication field. Veteran engineers hardly welcomed--or believed-the
theory.9 Some thought the paper was confusing and badly written.'0 One
mathematician was totally skeptical." Even some of Shannon's colleagues didn't
immediately understand his ideas; they found them interesting but not very
useful.'2
But at the same time, others at Bell Labs and places such as the
Massachusetts Institute of Technology did recognize the significance of the work.
And they were stunned. "I can't think of anybody that could ever have guessed
9
that such a theory existed," says Robert Fano, an emeritus professor of computer
science at MIT and a pioneer in the field. "It's just an intellectual jump, it's very
profound." 3 Fano and others showed that Shannon's information theory was not
only correct-it was revolutionary.
And a revolution did come. Not in the 1940s. But today. CDs, DVDs, cell
phones, fax machines, modems, computer networks, hard drives, memory chips,
encryption schemes, MP3music, optical communication, high-definition
television. All these things embody many of Shannon's ideas and others inspired
by him. Every time we save a file in our hard drives, play a disc on a CD player,
send an email, or talk on our cell phones we are relying on the concepts of bits
and codes and channels originated in Shannon's 1948 paper.14
But information theory, as Shannon has always said, didn't come from
scratch in a single flash of inspiration. It was the result of several years of work in
different places and contexts. He circulated among some of the brightest scientific
minds of the twentieth century. As a student at the MIT, he worked under
Vannevar Bush, the man who directed U.S. science efforts during World War II.
Also at MIT, Shannon met and took a course with the brilliant-and eccentricmathematician
Norbert Wiener, the father of cybernetics. At the Institute for
Advanced Study in Princeton, Shannon worked under another great
mathematician, Herman Weyl, and met figures such as John von Neumann, Kurt
Godel, and Albert Einstein-Einstein once showed up for one of Shannon's
lectures, but was apparently looking for the tea room and left.'5 During the war, at
the Bell Telephone Laboratories, one of the top research organizations in the
world, Shannon worked with the godfathers of control engineering, Hendrik Bode
and Harry Nyquist. He also worked on cryptography and had lunch several times
with a leading figure in the field, the British mathematician Alan Turing.
What path led Shannon to information theory? This was the question that
Bob Price, a communication engineer from MIT, wanted to clarify. He left
Shannon's house on that evening in 1982 with only one certainty: the answer was
much more complex-and fascinating-than he expected.
10
* * *
"THE MOST MATHEMATICAL of the engineering sciences." That is how
Shannon once defined the study of communication systems, for it combined both
fields-mathematics and engineering-in a unique way.'6 Entering the University
of Michigan in 1932 he still wasn't sure which he liked the best.'7 As a boy, he
used to play with radios, remote-controlled models, and other electrical
equipment. He once set up a telegraph line to a friend's house half a mile away
using the barbed wires of a nearby pasture.1 8 But math and abstract problems such
as the cryptograms of Edgar Allan Poe's The Gold Bug also very much interested
him. So while an undergraduate at Michigan he took courses in both electrical
engineering and mathematics and graduated with two diplomas. 9 One day in
1936, wondering about a job, Shannon saw a notice for a position of research
assistant at the Massachusetts Institute of Technology. 20 A group headed by the
dean of engineering Vannevar Bush needed someone to operate an early type of
mechanical computer, amachine known as the differential analyzer. Thejob
seemed ideal for Shannon's skills and interests.2 ' He applied and was accepted.
Shannon's work consisted in setting up differential equations into the
machine for MIT's mathematicians, physicists,and visiting researchers. The
differential analyzerwas partly controlled by over a hundred relays,
electromechanical switches largely used in the telephone system to route calls.
Telephone engineers had to design and test intricate circuits containing thousands
of relays, a complicated and tedious task that used to overwhelm them. In 1937,
Shannon spent the summer working at Bell Labs, the research arm of the
American Telephone and Telegraph Company,or just AT&T,the largest
communication company in the country. Immersed in that environment, where the
relay played a crucial role, and drawing on his experience with the differential
analyzer at MIT, Shannon started thinking of a better way to study and design
relay circuits.
II
The solution became his master's thesis, in which he showed how the
algebra of logic invented by nineteenth century mathematician George Boole
could facilitate enormously the designof relay circuits.2 2 Shannon noticed that
this kind of circuit is formed by switches that could be either on or off, while in
Boolean algebra a complex statement-reasoning, Boole argued-was formed by
simpler statements that could be either true or false. For Shannon, the connection
was clear: on and off, true and false. The symbolic operations devised by Boole to
manipulate a set of true-and-false statements could be used by engineers to design
circuits with on-and-off relays-or any kind of switch, for that matter. To
represent closed and open switches, he chose two symbols: 0 and 1. Using
Boolean algebra, Shannon transformed the design, analysis, and test of
complicated relay circuits into an abstract, mathematical manipulation of zeroes
and ones-something that engineers could easily do with pencil and paper.
In 1938, Shannon submitted this work to a conference of the American
Institute of Electrical Engineers. His mentor, Vannevar Bush, soon received a
letter from the organizers: the paper had been accepted and very much impressed
the reviewers, one of which said: "To the best of myknowledge, this is the first
application of the methods of symbolic logic to so practical an engineering
problem. From the point of view of originality I rate the paper as outstanding."2 3
Indeed, the paper surprisedeven Shannon's closest colleagues. "We used to talk
about switching circuits, and I showed him some telephone diagrams I had," says
Amos Joel, a friend of Shannon at MIT. "But all of a sudden-I really don't know
how-he came with this whole idea of using Boolean algebra."24 The importance
of the work was immediately recognized, and Shannon was awarded the
prestigious Alfred Noble Prize (an engineering award, not Sweden's Nobel Prize).
His work, once called "one of the most important master's theses ever written,"25
laid the basis of digital circuit design, an essential tool for the microelectronic
industry-nineteenth-century logic made possible today's twenty-first century
information technology.
12
At MIT, besides the recognition from the faculty and colleagues,
Shannon's mathematical talent brought him also some unexpected situations. His
enrollment in a flight training program raised concerns in the professor in charge
of the course. The professor found Shannon "unusual" and went to talk to other
members of the faculty who knew the young student. "From these conversations,"
the professor wrote in a letter to MIT President Karl Compton, "I am convinced
that Shannon is not only unusual but is in fact a near-genious [sic] of most
unusual promise." The professor asked Compton if Shannon should be withdrawn
for any life risk however small wasn't justified in his case.2 6 "Somehow I doubt
the advisability of urging a young man to refrain from flying or arbitrarily to take
the opportunity awayfrom him, on the ground of his beingintellectually
superior," answered Compton. "I doubt whether it would be good for the
development of his own character and personality."27
Once he had finished his master's thesis, Shannon began to look for a
topic for a PhD dissertation. At about that time, Bush had been named president
of the Carnegie Institution, a private, nonprofit research organization. The
Institution had a department of genetics at Cold Spring Harbor in New York and
Bush suggested that Shannon spend a summer there. Perhaps he could do for
genetics what he had done for circuit switching. Shannon then worked on a PhD
dissertation on theoretical genetics, which he completed in less than a year. He
enjoyed being a "geneticist" for some time but didn't plan to stay in the field.28
Shannon was driven by an endless curiosity and his interests were very broad.
While studying genetics and learning how to fly, Shannon continued to work on
Boolean algebra and relay circuits; his main project was to build a calculating
machine to perform symbolic mathematical operations.29
But among all these things, Shannon still found time to work on a subject
that had always interested him: the problems of communication-"the most
mathematical of the engineering sciences," as he put it. "Off and on," he wrote in
a letter to Bush in February 1939, "I have been working on an analysis of some of
the fundamental properties of general systems for the transmission of intelligence,
13
including telephony, radio, television, telegraphy, etc." "Intelligence" was how
engineers called the variouselectrical signals that flowed in the communication
systems-a term they would soon replace for another: information. In his letter,
Shannon included a series of equations and concluded, "There are several other
theorems at the foundation of communication engineering whichhave not been
thoroughly investigated. " 30
The place Shannon found to explore these ideas further was the Institute
for Advanced Study at Princeton. Before graduating from MIT,he had applied for
a one-year fellowship there. In the spring of 1940, he received the good news: he
had been accepted.3
' Arriving at Institute later that year, Shannon went to his
advisor, the German mathematician Herman Weyl, and said he wanted to work on
problems related to the transmission of information.3 Weyl, a disciple of David
Hilbert-considered one of the greatest mathematicians of the twentieth
century-had left Germany with the Nazis' rise to power in 1933. Weyl showed
interest in Shannon's ideas, and soon they were discussing analogies between the
transmission of information and the Heisenberg uncertainty principle.33At
Princeton, at age 24, Shannon met several famed scientists. John von Neumann, in
particular, impressed him a lot-"the smartest person I've ever met," he would
say years later.3 Einstein was also there. In the morning, driving to the Institute,
Shannon used to see the German physicist walking in his bedroom sleepers;
Shannon then used to waive at him--and Einstein waved back. "He didn't know
really who I was," Shannon recalled later. "Probably he thought I was some kind
of weirdo."3
Shannon attended many seminars at Piinceton, not alwaysrelated to the
kind of thing that he was interested in. The mathematics department was oriented
to pure mathematics or physics, not to engineering problems like
communication. 6 Nevertheless, it was at Princeton, in 1940, that the first ideas of
information theory began to consolidate in Shannon's mind. He seemed ready to
work full time on the subject that interested him so much. Perhaps in a year or so
he would work out and publish some significant results. But then the war came.
14
* * *
THE BELLTELEPHONELaboratories at Murray Hill, in New Jersey, consist of a
dozen new and old interconnected five-story buildings spread over an area of
about a hundred acres. The redbrick style and the green areas make the place look
like a college campus. Long corridors full of large pale-painteddoors spread like
a labyrinth all over the buildings. The complex was built during World War II to
expand the research activities of the laboratories, then held almost entirely at its
headquarters at 463 West Street in Manhattan. If New York focused on
engineering, Murray Hill was created to focus on science37, a successful initiative
that would transform Bell Labsinto one of the best industrial laboratories in the
world. Follow ing the end of the war, the organization experienced a very intense
and lively period of activity, now seen as the golden years of the labs.
Shannon came to the Bell Labs by the summer of 1941. As he recalled
later, he didn't fancy the idea of going to war and thought he could contribute a
lot more working full time for the science and military joint effort. The war was
under way in Europe and the involvement of the U.S. was imminent. In fact, in
Junel940, President Franklin Roosevelt established the National Defense
Research Committee. 39 Directed by Vannevar Bush, the committee's objective
was to mobilize U.S. science to the war efforts, and one of the priorities was to
address what Bush called "the antiaircraft problem." Airplaneswere flying higher
and faster, and traditional gunnery wouldn't be able to shoot them down. More
agile and reliable gunfire control systems had to be developed. Bush was
convinced that this technology wasn't receiving the necessary attention.40
To address this problem he called Warren Weaver, a mathematician and
also a skillful administrator, who was working as a director for the Rockefeller
Foundation, a major source of funding for innovative science. Weaver was put in
charge of a NDRCdivision on gunfire control systems and quickly set dozens of
groups to work on the problem, including industrial laboratories. companies, and
15
universities. Bell Labs received a major contract and became one of the largest
groups. But why was a band of communication engineers suddenly working on
antiaircraft technology?
The group at Bell Labs had realized that the aim of a gun and a telephone
call had a lot in common. Not long before the war, engineers began to consider
text, speech, and images as a single entity-electrical signals-that could flow in
the telephone network and had to be transformed, switched, and amplified. Now
the same approach applied to fire control: the coordinates of an enemy airplane
had to be transformed into electrical signals that could flow inside the gunfire
control systems and had to be manipulated.4'
The Bell Labs teamwas working on a kind of electronic computer that
could track a plane, estimate its future position, and aim a gun, a fraction of a mile
ahead, so that the shell had time to get there-and shoot down the target. The
coordinates of the plane were supplied by optical equipment devices similar to
telescopes or by radar. The problem was that these devices were not perfectly
accurate. The coordinates they provided had errors and deviations-"noise." If
plotted in a graph, the coordinates wouldn't form a smooth line, but a wildly
zigzagging curve. If these coordinates were used to aim a gun, the shell would
hardly destroy the target.
The problem of predicting the future position of a plane with noisy
coordinates was solved by Norbert Wiener, at MIT. A child prodigy, Wiener
received his B.A. in mathematics from Tufts College in Boston in 1909 at age
15.42In the fall of the same year he entered Harvard as a graduate student and four
years later received his PhD degree with a dissertation on mathematical logic.
Wiener continued his studies inEurope, where he worked under Bertrand Russell,
G. H. Hardy, and David Hilbert. After World War I, he came back to the United
States andjoined the mathematics department at MIT.43 Wiener realized that the
prediction problem required a statistical treatment because the coordinates varied
in an unpredictable manner-the trajectory was unknown and the errors were
random. Much as in the case of weather forecast, it was necessary to evaluate past
16
and present conditions to predict a future situation. Wiener then developed
mathematical tools to analyze the statistical behavior of the noisy coordinates,
"filter out" the errors-or "smooth" the data-and estimate the future trajectory
of the target.44He worked out a complete mathematical solution that for the first
time clearly combined the fields of statistics and communication engineering.4 5
His novel results were published in a report of restricted circulation dubbed the
"Yellow Peril"-because of its yellow cover and its frightening mathematics.4 6
The Yellow Peril was an influential publication during wartime,47 and
Shannon read the document with interest.48 As a researcher in Bell Labs'
mathematics department under the legendaryHendrik Bode (every electrical
engineering student today knows "Bode plots" from control textbooks), he was
also working on the problem of the trajectory prediction. While engineers at the
labs developed the machine itself-the hardware-Bode's group worked on the
"software." They realized that Wiener's solution was mathematically perfect, but
not the best one to be implemented in practice; it assumed, for example, a signal
varying infinitely in time, while real trajectory measurements lasted just a few
seconds.49 To attack the problem and devise a more practical solution, the group
drew on their knowledge of communication systems. In a classified report, the
authors, Shannon among them, noted that "there is an obvious analogy 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."5 0 Electronic filters and other concepts and devices
developed for the telephone network could now be used in gunfire control
systems. In that report, they proposed a better way to remove the errors from the
noisy coordinates, including details on how to implement this solution with an
eletromechanical computer.
The solution came in the end of the war and was not used in the Bell Labs'
gunfire control systems that were sent to the battle field-and which played an
important role in the war, shooting down thousands of V-Is, the rocket-powered
bombs fired by the Nazis against targets in England.5' But the results obtained by
17
the mathematics group and also by Wiener at MITand others in the NDRC's
division on gun fire control proved important to advancethe understanding of the
means we use to represent the world-be it speech or airplane coordinates-in the
realm of electronic machines.5 2 When electrical signals flow in gunfire computers,
in the telephone network, or in any other system, we are dealing essentially with
the verysame process: transmission and manipulation of information.
NOT LONG AFTER the beginning of the war, Shannon was working also with
cryptography, another of his assignments. Bell Labs hadseveral projects on
secrecy systems, especially on speech scrambling,techniques to protect a
telephone conversation from eavesdroppers. One of these projects was possibly
the most secretive of the whole war: the development of an encrypted
radiotelephone system to connect Washington and London-known as the "X
System."
Engineers at Bell Labs had been experimenting with various methods to
scramble speech. But when they found one they thought was fairly good, someone
always figured out a way to break it. They then turned to the only system they
knew could create virtually unbreakable messages: the telegraph.53 In telegraphy,
messages are sent by the opening and closing of contacts in an electric circuit.
When the contacts are closed, an electric pulse is sent-a "dot" is a short electric
pulse and a "dash" is a longer one. When the contacts are open, no electric current
flows in the line-and a "blank space" is sent. The telegraph system, therefore,
used the presence and absence of current, or sequences of on-off pulses, to
represent all messages. Engineers knew that by combining a message with a
random sequence of on-off pulses knownjust to the senderand the receiverwhich
they called a "key"-they could obtain a perfectly secure message; this
encrypted message would also be a random sequence of on-off pulses and there
18
was no way to attack this cryptography scheme-it was virtually unbreakable.5 4
Could the same idea be applied to speech?
In the telephone system, speech is transformed into an electric signal that
varies proportionally to the air vibrations of a person's words and sounds. The
signal is analogous to the air vibrations it represents-and we call this signal
"analog." Bell Labs researchers realized it was difficult to scramble analog
signals; sometimesjust by listening carefully to an analog-encryptedconversation
it was possible to understand what was being said. They needed, therefore,
something similar to the on-off pulses of telegraphy.
The solution came with two techniques now known as "sampling" and
"quantization." The idea was to approximate a continuous signal by a series of
"steps"-as if we superimpose the continuous signal by what seems a stairway
that goes up and down following the shape of the signal. When we chose the
number of steps used in the stairway, we are "sampling" the signal (each step is a
sample). Also, we can imagine that each step in the stairway has a different height
Sromthe "ground;" when we determine these heights, we are "quantizing" the
signal. In this way, a continuous signal is transformed into a discrete sequence of
numbers. Now, this sequence of numbers could be combined to a random numeric
key using special computing operations to create an encrypted conversation.
Throughout the war, the X System was used by the high commands in the
United States and in England to work the war strategy, sure that the enemy
couldn't eavesdrop the conversation. It was one of the first digital communication
systems55, for it transmitted information with digits, while analog systems use
continuous, "analogous" signals.
The development of the X System was very secretive and Shannon had no
access to all the details--'"They were the most secret bunch of people in the
world," he recalled years later.56 He worked on a small part of the puzzle without
seeing the whole picture. Nevertheless. he played an important role in the project;
Shannon was asked to inspect the "heart" of the system: the encryption scheme.
19
His job was to verify that nothing had been overlooked and that the method used
was really unbreakable.5 7
During that time, Shannon's interactions with the other researchers were
very restricted, and many of his interlocutors couldn't discuss what they were
doing. One of these interlocutors was Alan Turing. An eminent British
mathematician, one of the world experts in cryptography and secrecy systems,58
Turing was a leading member of the team that broke the Nazi secret code
"Enigma." His computing machines at Bletchley Park, fifty miles northwest of
London,deciphered the encrypted messages and delivered Hitler's plans straight
to Prime Minister Winston Churchill's desk.
In January 1943, Turing came to Bell Labs in New York to consult on
speech scrambling. He stayed in the country for two months and had occasional
conversations with Shannon.5 9 They couldn't discuss Turing's work on the
Enigma nor their work at the laboratories. So during lunch or at teatime in the
cafeteria,they talked about things that interested both of them and that they could
discuss freely, things such as computers and the possibility of a machine
simulating the human brain.60 And what about systems capable of manipulating
and transmitting information? Working on the breaking of the Enigma,Turing
had developed a kind of measure of information very similar to the one Shannon
would develop in his 1948 paper. Shannon's unit of information was the "bit."
Turing's was the "ban."6 Did the British mathematician contribute any insight to
information theory? "He somehow didn't always believe my ideas," Shannon told
Bob Price. "He didn't believe they were in the right direction. I got a fair amount
of negative feedback."
Despite their disagreement, Shannon and Turing were working on fields
that shared many concepts and methods-but with different goals. While in
cryptography you want to protect a message from eavesdroppers, in information
theory you want to protect a message from transmission errors. In both fields you
need a measure of information and you deal with coding and decoding methods.
So during the war, while studying cryptographic techniques and consulting on
20
projects like the X System, Shannon could carry on a parallel work, the one he
started at Princeton. He could work on the problems of transmission of
information.
In 1945, he wrote a classified report titled, "A Mathematical Theory of
Cryptography," in which he used probability theory to study the subject in an
unprecedented mathematically rigorous way. To some, the paper transformed
cryptography from an art to a science.62 In this work, Shannon introduced several
concepts that would appear later in his "A Mathematical Theory of
Communication," terms such as choice, information, and uncertainty, revealing
the close connection between the two fields. In fact, in a footnote in the beginning
of that report he wrote: "It is intended to develop these results in a coherent
fashion in a forthcoming memorandum on the transmission of information. " '
Shannon was literally announcing what would be his 1948 seminal paper.
Towards the end of the text, when discussing some problems in the encryptionof
English text, he used yet another term, one that would become always associated
with his own name: "information theory." °4
ONE DAY IN 1947, Shannon went to his colleague Brockway McMillan, a
mathematician from MIT who had recently joined Bell Labs, and asked for help
with a problem. Shannon said he needed a proof for a theorem related to the
reliability of communication and sketched a diagram on a piece of paper. On the
left side, he drew a bunch of points, each below the other. He did the same thing
on the right side. He then connected some of the points: from one point on the left
departed several lines to points on the right. The resulting figure resembled a
paper fan. Shannon's explanation, however, was not very clear; he didn't express
the problem in mathematical terms so that McMillan could really understand what
had to be proved. McMillan found those ideas somewhat obscure. He couldn't
help his colleague.65
21
Shannon worked on information theory almost entirely alone. He
eventually talked about it with some colleagues, but never officially and never in
detail as he did for antiaircraft control systems or cryptography. As in most of his
works, Shannon waited until he could grasp the problem clearly in his mind to
finally write up his results. It was not different with his 1948 paper. "There were
no drafts or partial manuscripts," wrote Robert Gallager, a professor of electrical
engineering at MIT, in a recent paper on Shannon. "Remarkably, he was able to
keep the entire creation in his head."6 For Shannon, information theory was
almost a hobby, which he kept in the scarce spare time he had. During wartime,
many researchers worked ten hours a day, seven days a week.67 He worked on
information theory sometimes at night, at home, not during office hours.'
Shannon worked first at the laboratories' headquarters in Manhattan,
during the beginning of the war, and later moved to Murray Hill when the first
buildings were erected around 1941. Not long after, he got his own office on the
fourth floor of Building 2 on the east part of the campus, a sizable space with two
generous windows overlooking a green area. His door was closed most of the
time. While his colleagues got together to fly kites and play word games during
lunch, Shannon preferred to stay alone, working on his own.69
The mathematics department functioned as a consulting group, providing
expert advice to other researchers from Bell Labs or other organizations such as
the military.7 0 But its members also worked on their own projects, they could
pursue "exploratory" research that wasn't dictated by an engineering
department." Shannon very much appreciated this freedom to follow his own
interests. He liked to lock himself in his office and just think.
Shannon did interact with others. He liked to laugh and play jokes.
Sometimes he invited a colleague for a chess match in his office. The matches
attracted others, who watched by the door while Shannon, most of the time, beat
his opponent. "He wasn't a great champion, but he was quite good," says
McMillan, who occupied the office next door to Shannon's. "Most of us didn't
play more than once against him."72
22
But it was alone that Shannon created information theory. In an interview
in the late 1970s, he said his ideas were most developed around 1943 to 1945.
Why it wasn't published until 1948?"I guess laziness," Shannon said, adding that
those were busy times because of the war and information theory "was not
considered first priority work." It was cryptography that allowed him to work on
information theory. "That's a funny thing that cryptography report," he told Price
about the 1945 confidential memorandum, "because it contains a lot of
information theory which I had worked out before, during those five years
between 1940 and 1945."73 In fact, the connection between the two fields is so
straight that many believe that cryptographyoriginated information theory. This is
not true, as Shannon was thinking about the problems of communication much
before coming to Bell Labs. But the question thus remains: what led Shannon to
his novel approach to communication, to his 1948 paper, to the ideas that stunned
engineers? If it was not cryptography, what did? Price asked this question to
Shannon on that evening in 1982 and his answer was: "The real thing for me was
Hartley's paper."7 4
* * *
RALPH HARTLEY JOINED the research laboratory of Western Electric, the
manufacturing arm of AT&T, in 1913. He started working on transatlantic
wireless communication and later with telephone and telegraph systems. In 1925,
AT&T and Western Electric combined their engineering departments to form Bell
Labs, and Hartley became a member of the new organization. Three years later,
he published an important paper titled "Transmission of Information" in the Bell
System TechnicalJournal, an in-house publication.7 5
In this work, Hartley proposed to find a quantitative measure for the
transmission of information that could be used to compare various systems, such
as telegraph, telephone, and television. At that time, the very notion of
information was fuzzy and he beganthe paper clarifying what the concept meant
23
for engineers. Hartley considered it necessary to eliminate what he called the
"psychological factors" in communication. 7 6
He imagined an operator sending a
message in a telegraph cable. Opening and closing the contacts of the telegraph,
the operator sends a sequence of symbols down the line. What messages should
be considered "valid"?
We can imagine, for example, operators who speak different languages. A
message that is not intelligible to one operator could be meaningful to another. In
other words, from the engineering standpoint, the meaning of a message is not
important. Given a set of symbols such as an alphabet, any combination of
characters should be considered "valid." In fact, Hartley imagined that instead of
a human operator, the message could be generated by an automatic mechanism, a
random event such as a ball rolling into pockets that determine the opening and
closing of the telegraph contacts. This randomly generated message should be
considered as "valid" as any other; the telegraph system should be able to handle
and transmit it as well as any other message.
For Hartley, therefore, communication could be thought of as the
successive selection of symbols from a finite set of symbols. A person, for
example, mentally selects words from a vocabulary to talk to another. "By
successive selection a sequence of symbols is brought to the listener's attention,"
he wrote. "At each selection there are eliminated all of the other symbols which
might have been chosen."' Hartley observed that a symbol conveys information
because there are other possibilities to it. You can't communicate if your
vocabulary has just a single word. The more words you have, the more choices
you can make-and the more information you can communicate. Hartley
expected that by measuring this "freedom of choice" he could obtain a measure of
information.
But it couldn't be a direct measure. It couldn't simply be that as choices
rise, information rises because these two variables-the number of choices and
the amount of information-increase in different rates. Suppose a telegraph
operator has an alphabet with 26 letters. When sending a message with, say, one
24
single letter,there are 26 possible choices. For two letters,the number of choices
increases to 676 (there are 26 times 26 possible combinations for two characters).
For three letters, the number is much higher: 17,576. If we continued in this way,
the number increases extremely fast-a so-called exponential growth. Hartley
realized that, while the number of possible choices increases exponentially, there
is no such exponential increase in the information conveyed. For a
communication engineer, a telegram with 20 characters should convey twice the
information of one with 10characters.
Is there a way to make the measure of information proportional to the
number of selections? In other words, is there a way to make each further
selection add a fixed amount of information tothe total so that, instead of growing
exponentially, this total amount grows linearly? Hartley found a way to
accomplish that using the mathematical function that is the "inverse" of
exponentiation. This function is the logarithm. If you "apply" the logarithm to an
exponential curve, the curve becomes a straight line. Using the logarithm, Hartley
derived a formula that transformed the exponential growth of choices into a linear
growth of information.78 With this formula, a twenty-charactertelegram would
contain twice the information of a ten-character one-just as expected.
But so far Hartley had considered just discrete processes, such as a
telegraph operator who chooses one symbol from a set of symbols. What about
other forms of communication, such as telephone, radio, or television, in which
information is sent as continuous signals? Hartley demonstrated that the problem
is essentially the same: a continuous signal could be approximated by a series of
successive steps that follow the shape of the signal. This is exactly the idea of
"quantization" used by Bell Labs engineers in the X System. So communication
in the continuous case could also be considered a succession of choices-the
choices being the "heights" of the steps--and the amount of information could be
quantified in the same way.
Hartley had a measure of information that could in principle be used for
discrete or continuous signals. Now he could analyze what limits the transmission
25
speed of a system.Bell Labs engineers knew that theycould send images over the
"air," from one antenna to another, as in the television system. But why was it so
hard, as experiments showed, to send images over a telephone line?79 What makes
one channel so different from the other and how much information each could
transmit?
In most communication systems, information is transmitted by an
electrical signal. Suppose you are sending a signal that goes up and down
continuously, in a smoothly varying way. Imagine you are holding one end of a
string, the other end fixed, and you move your hand up and down with a constant
speed. In this case, the undulation in the string varies regularly and endlessly,
always in the same way-it is predictable. A signal like this has always the same
frequency and it conveys no information. But suppose you start to move your
hand wildly, with varied speeds. Now you are generating an undulation much
more complicated, with an arbitraryand unpredictable shape. This signal contains
not only one but several frequencies--it is made up of several components. The
more frequencies a signal contains, the more rapidly it can change-and the more
information it can convey. This range of frequencies is called bandwidth. If a
signal contains frequenciesfrom 200 to 3,200 hertz, for example, its bandwidth is
3,000 hertz. This is approximately the bandwidth required to transmit a telephone
conversation. A television transmission requires roughlytwo thousandtimes more
bandwidth, or 6 million hertz.80 That is why it was so hard to send an image over
a telephone line: the bandwidth of this channel is too "narrow." The art of
communication, therefore, is to match the message to the medium.
Very close to Hartley, another Bell Labs researcher explored similarideas.
Born in Sweden, Harry Nyquist came to the United States at age eighteen and
studied electrical engineering. In 1917, he received a PhD degree in physics from
Yale University and joined AT&T.8 ' Nyquist's seminal work on control
engineering would transform him into a legendary figure in the field.
Nyquist made important contributions to communication as well. In the
1920s, he studied the transmission of signals in the telegraph system with a
26
mathematical tool known as Fourier analysis, which decomposes a complicated
signal into a sum of simpler components.2 This allowed him to understand deeply
how these components affectthe transmission speed. In accordance to Hartley, he
also concluded that the bandwidth defines how much information you can send
over a channel. 8 But Nyquist went further in understanding the boundaries
between continuous and discrete representations. He showed that once you had
approximated a continuous signal with a series of discrete steps, you could use
these steps to reconstruct notjust a similar signal, but an identical one. The finite
bandwidth of a signal limits the amount of information it can carry. So if you had
enough "steps," the reconstructed signal would be a perfect copy of the original.
"The crucial point is that a finite amount of information implies an essentially
discrete message variable," wrote James Massey, a former professor of electrical
engineering at ETH,Zurich's Swiss Federal Institute of Technology, in 1984.
'The world of technical communications is essentially discrete or 'digital'."4
Hartley and Nyquist sought a way to measure the maximum transmission
speed of a given system. They attacked the problem not only by inspecting
electric currents with oscilloscopes and voltmeters; they used mathematics and
abstract tools in a novel approach to understand communication. Especially
important was the connection they established between discrete and continuous
signals. "Nyquist's and Hartley's notions began to resemble digital
representations, as their techniquesanalyzed the subtleties of converting discrete
pulses to and fromthe continuous world," writes David Mindell, a historian of
technology at MIT, in the book Between Human and Machine. '"These men laid
the groundwork for the theory of information that Claude Shannon would
articulate in 1948."85
Indeed, Shannon cites both in the very first paragraph of "A Mathematical
Theory of Communication." Hartley's paper, in particular, which Shannon read
while a student at Michigan, impressed him very much.8 6 He mentioned Hartley's
work in his February 1939 letter to Vannevar Bush.87 And he mentioned it also to
Herman Weyl when he arrived at Princeton.88 Shannon thought the paper was
27
very good, but it left a lot more to explore. His thinking on information theory, as
he recalled later, began with Hartley's paper.8
After Hartley's and Nyquist's work in the 1920s,communication theory
"appears to have taken a prolonged and comfortable rest," in the words of John
Pierce, a Bell Labs and an eminent communication engineer." World War H
brought important advances, but the field remained a kind of technological
archipelago, with several islands of knowledge that didn't talk to each other.
Radar, developed during the war, was still a secretive subject. The telephone
system was operated almost entirely by AT&T.At universities, communication
engineering textbooks amounted to the impressive quantity of two; students
learned radio transmissions and technologies such as AM and FM.9'
Communication had advanced significantly during wartime, but it was far from a
unified science.
* * *
SEVENTY-SEVEN PAGES, twenty-three theorems, seven appendixes with
mathematical proofs. A lengthy paper for today's standards, "A Mathematical
Theory of Communication" was published in two parts in the July and October
1948 issues of the Bell System TechnicalJournal, then widely circulated among
electrical engineers and those interested in the communication field. The paper
presented a unifying theory of communication, a mathematical theory that didn't
depend on a particular systemor technology.
Shannon imagined communication as the flow of messages along a series
of stages, which herepresented schematically with a block diagram (seefigure
below). The diagram showed a series of blocks, one connected after the other,
through which a message is transmitted. The first stage is the source, where the
message is originated. The second stage is the transmitter, which transforms-or
encodes-the original message into a form suitable for transmission. The encoded
message is then sent over the communication channel. During its way through the
28
channel, the message nmaybe affected by errors. In other words, the channel is
plagued by noise. And noise is everywhere: it is in space, where magnetic storms
can disturb a signal; inside electronic equipment, where spurious currents can
corrupt data; within an optical fiber, where energy losses degrade the light
transmitted. It is impossible to eliminate all the noise from a channel-you have
to live with it. When the encoded message leaves the channel, it reaches the
receiver, which performs the inverse operation of the transmitter; that is, it
decodes the message and delivers it to the final stage: the destination.
INFORMATION
SOURCE TRANSMITTER CHANNEL RECEIVER DESTINATION
NOISE
SOURCE
COMMUNICAION BLOCKSShannon's classic schematic diagram of a
general communication system. All practical systems can be broken
down into parts that perform the same functions as these boxes.
Shannon's block diagram became a classic concept in the communication
field. Its generality reflected the fundamental character of the theory. "All the
many practical different schemes-radio, TV, carrier pigeon--can be broken
down into parts that perform the same functions as these boxes," says David
Slepian, a colleague of Shannon in Bell Labs' mathematics department.9 With a
simple figure showing a bunch of interconnected boxes, now the main questions
of communication could be clearly formulated. How much information does a
source produce? What is the best way to encode a message in the transmitter? Is
29
there a limit to the amount of information we can send over a channel? How badly
will the noise affect the transmission? What characteristics of the receiver and the
destination are important for the communication process?
Shannon began his paper by noting that frequentlythe messages produced
by an information source have meaning.That is, they refer to things or concepts
that "make sense" to people. But from the engineering standpoint, Shannon
observed, these "semantic aspects" were not important-agreeing with Hartley's
attempt to eliminate the "psychological factors" in communication. For Shannon,
too, any message selected should beconsidered "valid." What is meaningful to a
person-a certain kind of music, a text in a foreign language-can be
meaningless to another. And a system should be able to transmit any message. So
what matters for communication engineering, Shannon said, is that a message is
selected from a set of possible messages. As he recalled later, he wondered about
various kinds of sources. What is the simplest one? Is there a fundamental
information source to which we can compare all the others? Shannon then thought
about the toss of a coin.93
Consider, for example, the case when you toss a coin and want to tell the
result to a friend. This is the simplest source of information: there are just two
outcomes-heads and tails-that are equally likely. (One might think about a coin
with heads on both sides. But you don't need to toss this coin to know the
outcome. So this coin produces no information.) So we can regard the toss of a
fair coin as having unitary information. And there are many ways you can tell the
result to your friend. You can simply shout the outcome. Or you canjump once
for heads and twice for tails. Or you can agree on more complicated schemes,
using a flashlight, smoke signals, or flags. No matter how you decide to
communicate the result, the amount of information conveyed is always the same.
We can denote each possible outcome by the digits 0 and 1-called
binary digits. When we use binary digits, we are counting in the binary system, or
what is called "base 2" (as opposed to the "base 10," or the decimal system,
which we normally use); in this case the unit of information is the bit. The story
30
goes that, one day during lunch, some Bell Labs researchers were thinking of a
better term for "binary digit." What about binit? Or maybe bigit? John Tukey, one
of the men at the table put an end to the discussion: the best and obvious choice,
he said, was "bit."4 To communicate the outcome of the toss of a coin--or any
fifty-fiftyprobability selection for that matter-we need to sendjust one bit of
information. We send either the digit 0 or the digit 1. The fact that there are two
digits and only one has to be sent is at the very basis of the concept of
"information": information can be conveyed by a digit because there exists an
alternative to it. If there isn't another possible choice-as in the case of the coin
with heads on both sides-the amount of information is zero.
For Shannon, information was a measure of uncertainty. But not in a
negative way. Uncertainty in a sense of something newsy. You don't want to read
last week's newspaper. You want today's paper because it brings things you don't
know yet, are uncertain about. Communication, therefore, was the resolving of
uncertainty. The more uncertainty, the more information needed to resolve it.
Hartley had used this same idea of a message chosen from a set of possible
messages to derive his measure of information. His formula worked fine for the
cases when the messages have all the same chance of being selected. Shannon,
however, noticed that usually choices occur in a much morecomplex way.
Communication can't be just like the toss of a coin. In written English, for
instance, some letters are used more frequentlythan others.The letter E appears
much more than z, Q, or X. And more than that, we also form words, so a
particular selection depends also on the previous ones. Take a common word like
THE. So in English there is a great chance of an E to be selected after a T and an
H. But Hartley's formula couldn't be used to reflect such situations. Actually, it
couldn't be used even for the simple toss of an unfair coin, for it couldn't deal
with the differentprobabilities for each face. Shannon realized he needed to
generalizeHartley's formula, make it mathematically morepowerful so that any
information source could be "measured." To derive a new formula, he sought a
31
new mathematical model for the source, ore that could generate complex things
like English.
Shannon realized he needed a model that could produce messages based
on complex rules of probabilities. One that could produce a selection depending
on the previous selections made. And one that should be able to generate not
some, but all possible messages of a certain source. The mathematical model that
can do all that is known as a stochastic process. The interaction of molecules of
boiling water in a pan, the fluctuations in the price of stocks, the "random walk"
of a drunk in a sidewalk-these are all examples of phenomena that can be
modeled as a stochastic process. They are essentially random events. At a certain
instant, you can't predictthe precise position of each water molecule, or the exact
price of the stock, or where the drunk will be. But stochastic processes have a
statistical behavior. And we can analyze statistics to draw certain conclusions.
Thus using a stochastic model, physicists can deduce the temperature of the
water; stock analysts can have an idea of the variation of prices; and
mathematicians can estimate how far the drunk will go.
In information theory, a stochastic process is used as a model to generate
the messages. It is a mathematical machine that runs endlessly spilling out the
messages according to probability rules. And we can define the rules. You can
start with simple rules and then introduce more and more rules-more constraints
to how the messages are generated. Shannon gave some examples of how rules
can change a stochastic process that generates English text.
You can start with the simplest model possible. Write each letter of the
alphabet in small pieces of paper and put them in a hat. Now take one piece, write
down the letter, and put the piece back in the hat. Shannon did this experiment
with 26 letters and a space and included a typical sequence in his paper:95
XFOML RXKHRJFFJUJ ZLPWCFWKCYJ FFJEYVKCQSGHYD
QPAAMKBZAACIBZLHJQD.
32
Keep doing this and, in principle, you will come up with all Shakespeare's
plays. But the way things are written, be it a grocery list or Hamlet, is not like a
bunch of monkeys typing randomly on a typewriter. Language has a statistical
structure, and you can incorporate this structure into a stochastic model. You can
create a model that takes into account that some letters appear more frequently
than others. You can also include in the model the probability for a certain letter
to be selected after, say, two letterswere previously chosen. Shannon included an
example of this case:
IN NO IST LAT WHEY CRATICT FROURE BIRS GROCID
PONDENOME OF DEMONSTURES OF THE REPTAGIN IS
REGOACTIONA OF CRE.
Rather than continue with letters, Shannon then jumped to word units. He
considered the probabilities of two words appearing togetherand generated a
sequence of this kind using words from a book:
THE HEAD AND IN FRONTAL ATTACK ON AN ENGLISH
WRITER THAT THE CHARACTER OF THIS POINT IS
THEREFORE ANOTHER METHOD FOR THE LETTERS THAT THE
TIME OF WHO EVER TOLD THE PROBLEM FOR AN
UNEXPECTED.
Shannon then concluded that a stochastic process-in particular, one
special kind of stochasticprocess known as a Markov process-could be a
satisfactory model of English text. In more general terms, he noted, a sufficiently
complex stochastic process could represent satisfactorily any discrete source.9
Once he had the model. Shannon proceeded to find a measure of
information. "Can we define a quantity which will measure, in some sense, how
much information is 'produced' by such a process, or better, at what rate
information is produced?" he wrote in the paper. Shannon deduced a formula that
33
was a generalization of Hartley's-a kind of weighted sum that takes into account
the probability of each possible selection.The formula can be applied to the
simple throw of a coin, to a message written in any language, to a ball taken from
a box with five red balls and three blue ones, or to any other probability process.
In the simplest case of the toss of a coin, Shannon's formula assumes the
following form:
H =-p logp - q logq
In this expression, H is the amount of information, measured in bits;p is
the probability for heads and q the probability for tails; log means the logarithm,
as used by Hartley. (Since we are working with the binary system, we use the
"logarithm to the base 2," which means that the "log" in the formula is actually
"log2". The logarithm of a number to the base 2 is how many times you need to
multiply 2 to get the number. For example, the logarithm of 8 to the base 2 is 3,
because you need to multiply 2 three times to get 8. Others bases can be chosen,
but in information theoly, the base used is 2, and the unit of information is the bit.)
For a fair coin, heads and tails have the same probability, thus p and q
have the same value: 50 percent. Plug these values in the formula and you get H
equals to I bit, just as expected. Now imagine you have an unfair coin, one that
has one side heavier than the other, weighted away from fifty-fifty probability.
Lets say it gives heads 70 percent of the time and tails just 30 percent. Use the
formula and H works out to be approximately0.9 bit. The amount of information
decreased. Does that make sense? Yes, because the unfair coin is more
predictable; when it lands as heads, we are less surprised, we have learned less.
So the unfair coin produces less information on average than the fair one.Toss ten
fair coins and you get 10 bits. Toss ten unfair coins and you get 9 bits.
But of course, communication is not only about tossing coins. Information
sources are things like a telephone conversation or a television broadcast. And in
most systems, information is represented with electric signals. Shannon knew
34
from Hartley and Nyquist that a continuous signal could be converted into a
discrete form and the theoretical results such as the measure of information would
still hold. He adopted the "conversion" method described by Nyquist in his papers
on telegraphy and developed it further into what is known now as the "sampling
theorem." 97 And perhaps more importantly, Shannon knew that a system that
transmitted discrete information could actually work. During the war, Bell Labs
engineers had built things such as the X System and had explored communication
schemes that mixed continuous and discrete signals.
The team working on the X System had found that the technique they
developed to transform a signal into digits was very similar to one devised and
patented years earlier by a British engineer.This method also consisted in
approximating a continuously variable signal-like the analog signal of a
telephone call-by a series of steps with different "heights" ("samples" of the
signal). But instead of sending the numeric values directly, as in the X System,
they would be converted to the base 2, that is, to numbers with just two digits: Os
and s. A "height" of 23, for example, would be transformed into the binary
number, or code, 10111. If a 1 means the presence of an electrical current and a
0 the absence of current, you can send any message using just on-off signalsuniform
and unequivocal electric pulses. This method of transforming a signal
into codes and then into pulses was called "pulsecode modulation," or just PCM.
It was widely studied at Bell Labs after the war and later implemented in various
telephone equipment." What we call now a digital communication system has its
roots in PCMand other methods to manipulate and transmit information in
discrete sequences of digits.
The terms "analog" and "digital" appeared nearly simultaneously during
the war. Both technologies evolved together as computer and communication
engineers decided which was better for the various problems they had to solve.9
Shannon realized the potential of the discrete representation-that with digital
information one could do things it was impossible with analog information.
"Before Shannon, the continuous, or analog case was considered the basic one for
35
communications with the discrete, or digital case a kind of extreme special case,"
says Massey, from ETH. "Shannon reversed this viewpoint. " ° °
In his 1948 paper, Shannon showed that one of the advantages of a digital
system was that you could choose how you represent your message with bits-or
how you encode your information. We can, for example, analyze statistical
patterns of the messages produced by a source and use this knowledgeto
compress the messages before transmission. Consider the case of English text. We
could use the same number of bits to encode each letter. We could assign, say,
00001 for A, 00010 for B, and so on. But if E appears more frequentlythan z,
why use the same number of bits? We could, for example, assign just one bit for E
and a code with more bits for Z. This same idea could be applied to all letters of
the alphabet and by using this more efficient code we could save a good deal of
transmission time. The idea of using efficient codes was not new. Samuel Morse
used short combinations of dots and dashes for the most common letters and
longer combinations for uncommon ones. While the letter E was represented by a
single dot, an X was a dash-dot-dot-dash sequence. 0'
But Shannon's insight was to take this idea even further. We could assign
codes with different lengths not just for the letters individually, but for pairs or
groups of letters (ED, for instance, is very common, while QZ very rare). And we
could do the same thing for words. Frequent words like THE would have shorter
codes while infrequent words would have longer ones. It is possible to obtain such
efficient codes because of the statistical nature of language--certain patterns
repeat themselves. Shannon called these repetitive and predictable patterns
"redundancy." Eliminating redundancy it was possible,on average, to compress
information and save transmission time.
"Two extremes of redundancy in English prose," he wrote in his 1948
paper, "are represented by Basic English and by James Joyce's book Finnegans
Wake." Basic English vocabulary, he noted, is limited to 850 words and has many
repetitive patterns-the redundancy is very high. Joyce's book is full of new
words that expand the language structure, reducing its redundancy.
36
Shannon also observed that the redundancy of a languageis related to the
existence of crossword puzzles. If the redundancy is zero, the languagehas no
patterns-there are no constraints. So any sequence of letters is a word and any
combination of them can form a crossword. If the redundancy is too high, the
language has too manyconstraints and it is difficult to form arraysof wordsthey
don't "cross" easily with each other. Shannon estimated the redundancy of
English to be roughly 50 percent, about the right amount to make possible large
crosswords (later he revised this value to 80 percent). "The redundancy of
English," he wrote in an article for the Encyclopaedia Britannica, "is also
exhibited by the fact that a great many letters can be deleted without making it
impossible for a reader to fill the gaps and determine the original meaning. For
example, in the following sentence the vowels have been deleted: MST PPL HV
LTTL DFFCLTY N RDNG THS SNTNC. "
0
But how do you know how much a message can be compressed? How do
you know you have the most efficient code? Shannon's measureof information
gives the answer. His formula "captures" the statistical structure of a sourceand
translates it into a single number given in bits. This number tells how many bits
you have to use to encode the information in the most efficient way. What is the
bestcode, say, for English text? Consider the case of letters written in pieces of
paper and selected randomly-the monkeys in the typewriters. Forthis situation,
using Shannon's formula, H equals to 4.7 bits. Since a communication system
cannot transmit a fraction of a bit, you would need 5 bits on average for each
letter. To send a text with 1,000 letters you would need 5,000 bits. Now consider
you are encoding not single letters but groups of letters that form words. In this
case, according to Shannon's calculations, H would be equal tojust one 1 bit.
That means that some words would be represented by longsequences of bits,
other words by short ones, and on average, when you count the total number of
bits and letters, it turns out you used just 1bit for each letter. To send the same
text with 1,000 letters you would need now 1,000 bits. So when you take into
37
account the statistical patterns of English you can obtain codes that represent the
same amount of text with fewerbits.
These statistical gains are important not just for text. In fact, as Shannon
recalled later, one of the motivations for his work on information theory was to
determine if television signals could be compressed."0 These signals required a
lot of bandwidth-or a "huge" channel. Compressed signals would transmit the
images faster, saving transmission time-and moneyfor the broadcasting
companies. Today we face a similar situation with other sources and channels.
Anyone who uses a modem to access the Internet knows that transferring large
images, high-quality music, or video might take a longtime. And it would take
much longer if it weren't for compressing codes based on Shannon's ideas.
Multimedia files usually are huge, but like text, they also contain redundancy and
other patterns that can be removed.
It was an efficient coding technique that made possible the recent music
frenzy on the Internet, when millions of people suddenlystarted sharing their
favorite songs with a few mouse clicks. The so-called MP3 audio format can
transform a large audio file into a much smaller one. Usually, a ten to one
compression rate can be achieved.1 4 Take a high-quality audio file with, say, 50
megabytes, like the ones stored in music CDs. Over a modem connection that
most people have at home, the transmission would take a seemingly endless two
hours. Compressing the file with MP3, you get a 5-megabyte file that might be
transmitted in about ten minutes. With coding, the message could match the
medium.
BUT REDUNDANCYIS not alwaysundesirable. In somecases, according to
Shannon's theory, you want to add redundancy to a message. The reason is that
these repetitive patterns can "protect" the message from errors. Before Shannon,
engineers thought that, to reduce communication errors it was necessary to
38
increase the power of the transmission. When you make your signal stronger, it is
more difficult for noise to affect the communication. But doing so has a high cost:
it demands more energy, which means larger batteries or other sources of
electricity. Another way they considered to deal with errors was to send the
message repeatedly. If you send the letter Athree times and because of an error
one of the;mturns into, say, a B, you would still be able to get the right message:
with a kind of "majority vote" you still could conclude that an A was transmitted.
If instead of three you send the same letter five,ten, or one thousand times you
can improve the reliabilityof the communication even more. But notice what is
happening: now you need to send the same letter several times while you could be
sending other letters. In other words, you are squandering your precious
transmission time. The transmission rate is reduced by three, five, or one thousand
times when you repeat the messages. To make the error rate go to zero, the
repetition must increase indefinitely, which in turn makes the transmission rate go
to zero-you are forever sending the same letter.
Shannon showed that the idea of adding redundancy was right, but the
way engineers were doing it wasn't. He showed there was a much more efficient
way to encode a message to have not just good, but perfect communication. He
proved mathematicallythat using the proper codes data could be transmitted
virtually free from errors. While the methods to compress information form what
is called "sourcecoding," this part of Shannon's theory dealing with errorcorrecting
methods is known as "channel coding."
Imagine that you want to transmit four letters, A, B, C, and D. You decide
to represent A as 00, B as 01, C as 10, and D as 1-the simplest binary code for
four letters. But then you learn that the channel is "noisy." When you send a
stream of bits over this channel, unpredictable errors affect the transmission,
"flipping" the bits-a 1 arrives as a 0 or vice-versa. You might send an A (00),
but in the channel the second bit is flipped, and what you get is 01. But this is a
B. In other words: you got a transmission error. Is there a way to overcome this
kind of problem?
39
Before Shannon, engineers recurred to the repetition method. You can
decide, say, to send each letter three times repeatedly. To transmit B you send
010101. In this case, if one bit is flipped you don't get a wrong letter: you can
correct the error. For example, if the first bit is flipped you receive 110101,
which is closer to 010101 than any of the other three possibilities (000000,
101010, and 111111). What if more than one bit if flipped? Then there is no
escape: you will get errors. This is, therefore, a one-error correcting code. It
requires a total of six bits to transmit two "useful" bits (the encoded letters),
which represents a transmission rate of two to six, or 33 percent.
Can we do better? Coding theory gives the answer: yes. The solution is to
encode each letter with specially constructed sequences of bits. Suppose now that
you represent each letter withfive bits instead of six: you represent A as 00000,
B as 00111, C as 11100, and D as 11011 (each sequence of zeroes and ones is
called a codeword). Again, this is a one-error correcting code. If one bit is flipped,
you still can get the right letter. But note that you have a more efficient code than
before. Now you use five-bit codewords to transmit each letter. So the
transmission rate is two to five, or 40 percent. In this case, you introduced the
redundancy in a more intelligent way. Code designers study the noise in a channel
to create very efficient codes that make erroneous situations very unlikely. They
know they can't always win-but they know how to win most of the time.
In coding theory, the field that deals with source and channel coding, there
is a measure of how similar two codewords are: the number of bits you have to
flip to change a codeword into another is called Hamming distance, after Richard
Hamming, a pioneer in the field and also a Bell Labs researcher. The Hamming
distance for A and B in this last case is three, because you need to flip three bits of
an A to make it a B-and vice versa. A Hamming distance of three is usually
considered a good distance; that means the codewords have a good number of
distinct bits. A basic rule that coding engineers learn is that, codes with a short
Hamming distance can be messed up with each other. In systems like CD players
and hard drives, special codes are used to expand the Hamming distance so that
40
several errors can be detected and corrected. That is why you can scratch a CD
and still get crystal-clear music.The coding schemes embedded in this equipment
create complicated sequencesof bits, usually adding "error-check bits" used to fix
corrupted data.
Another importait-and surprising-concept in Shannon's paper was that
of channel capacity. Shannon showed that every communication medium has an
upper limit for transmitting data, a quantity given in bits per second. If you try to
send data above this threshold, you would necessarily get errors in the
transmission. Towards the end of the paper, Shannon related bandwidth and error
rate in an equation that gives the capacity of a channel. He put together the ideas
developed by Hartley and Nyquist into a single mathematical expression that
could measure the transmission limit of a telephone wire, a wirelessconnection,
or an optical fiber. Shannon gave engineers an ultimate measure of how well a
communication system could work. "Once you know you're close to the capacity,
you know you're doing a goodjob," said MITprofessor of electrical engineering
Hermann Haus, a leading researcherin optical communication. "So you don't
waste money trying to improve your system-you've reached the limits."'05
Shannon's informationtheory showed that coding provided the perfect
means to overcome error-a result that surprised many experienced
communication engineers. "People were thinking of a better way to communicate
in the presence of noise," MIT professor Robert Fano says. "But the notion that it
was possible to eliminate completely the effects of noise was totally unknown,
unthinkable."' Now engineers learned that, with coding, they could have errorfree
communication without reducing the transmission speed nor increasing the
power of the transmitter. If you had a channel with capacity of, say, 1 million bits
per second, you could have an error rate as low as desired and still use the channel
at 1 million bits per second-all you needed was the right code. Shannon called
this result the "fundamental theorem" for a discrete channel with noise.
To prove the theorem he used schematic representations of the relations
between transmitted and received messages-the "fan diagrams" (seefigure
41
below). A message sent over a noisy channel can arrive at the receiver altered. But
if this set of altered messages refers to one--and only one--of the original
messages, there will be no confusion in the receiving end. Roughly speaking, the
code used has to guarantee that the fan diagrams will not overlap.
POSSIBLE
MESSAGES
TOBESENT
O
POSSIBLE
MESSAGESTO
BERECEIVED
0
A MESSAGETRANSMITTED
OVERA NOISYCHANNEL...
...M,'GHTBE RECEIVED
ALTEREDDUETOERRORS
* S
*
FANDIAGRAMThe figure shows a typical "fan diagram," as this kind of
schematic drawing was known to MITgraduate students in the 1950s.The
points on the left represent the messages to be sent. Due to errors in the channel,
a transmitted message can arrive altered (points on theright connected by lines).
Using a few laws of probability theory, Shannon elegantly proved his
fundamental theorem and demonstrated that such codes must exist. But he didn't
showed how to obtain them-a task that would keepthousands of coding theorists
and communication engineers busy for several decades. Moreover,the possibility
of error-free transmission up to the channel capacity had an intrinsic "side effect."
The tradeoff is that the more errors you want tocorrect and the closer you get to
the capacity, the more complicated is the encoding of the information. That is, the
codewords become longer and longer. And you need to analyze extremely huge
chunks of data to find statistical patterns that optimize these codes. So you spend
42
a longtime encoding and decoding very long sequences of bits, which translates
into a delay that in practice might be unacceptable. You still send your data at 1
million bits per second-but you had to sit and wait for the messages transmitted
to be reconstructed.
Engineers had always sought a balance between transmission speed and
errorrate. Shannon's theory gave them full control of theseparameters. An
optimum transmission could be achieved in two clear steps. First, removing the
unnecessaryredundancy of a message, the unnecessary bits. And then adding the
right kind of redundancy-"error-correction bits"-to make the transmission
error-proof. Shannon showed that, with the proper source andchannel coding, it
waspossible to construct the perfect message for each communication system.
These ideas were so surprising and counterintuitive that ETHprofessor James
Massey compared Shannon's information theory to the Copernican revolution.
Copernicus's astronomical theory showed that the earth wasn't the center of the
universe, but was one of the many planets orbiting the sun. Shannon's also
introduced radically new concepts that turned the communication world upside
down. And despite his theory being clearly superior in scientific terms, it was
longand bitterly resisted by the advocates of the old school of thought.'7
WHEN SHANNON DEDUCED his formula for the measure of information, he
noticedsomething interesting. The combination of the logarithmic function and
probabilities had appeared in other fields before. His formula was very similar to
those used in statistical mechanics and thermodynamics to measure what is called
entropy. In a wider sense, entropy is a measure of disorder. The higher the
entropy, the greater the randomness--or "shuffleness"--of a system. A wellshuffled
deck of cards has more entropy than a deck in which, say, the cards are
grouped by suit. The concept of entropy was proposed by the German physicist
RudolfClausius in the late 19thcentury and furtherdeveloped by Ludwig
43
Boltzmann andWillard Gibbs. Shannon realized his formula was similar to those
obtained by Boltzmann and Gibbs.'0 That was no coincidence; in a sense, his
measure of information was also a measure of disorder. When we take a card
from a well-shuffled deck we are more uncertain about the outcome than when we
take a card from an ordered deck. So the shuffleddeck has more entropy and
produces more information when cards are taken. For Shannon, the connection
seemed clear: he called his measure of information "entropy."°
But a connection between entropy and information had been established
much before Shannon's work appeared. In 1929, in a paper titled "On the
Decrease of Entropy in aThermodynamic System by the Intervention of
Intelligent Beings," the physicist Leo Szilard analyzed a problem known as the
"Maxwell's demon." The demon was an imaginary being created by physicist
James Clerk Maxwell in 1867to contradict the second law of thermnnodynamics.
This law states that entropy never decreases. In fact, in most everyday
processes-as when an ice cube melts into liquid water--entropy always
increases. The liquid water is more disorganized, has more entropy than the ice
cube. British astronomer Arthur Eddington considered that the second law defines
the "arrow of time." It is such a fundamental law that, for him, it held "the
supreme position among the laws of Nature.""° In fact, the philosophical
implicationsof the second lawwere vast; every physicist had his own
interpretation.For William Thomson, better known as LordKelvin, the second
law was a confirmation of the biblical view of the universe's impermanence, for
the inexorable increase of the entropy meant a degradation of the usable energy."'
Maxwell imagined a box filled with gas and divided in the middle by a
partition with a small gate. The demon was a kind of molecular-sized intelligent
gatekeeper who monitored the molecules in the two sides of the box. In the
beginning, the gas was at the same temperature in both sides. Although the
average speed of the gas molecules determines this temperature,the molecules
don't have all the same speed; some move faster than the averagespeed, others
slower. The demon's goal was to have the faster molecules in one side and the
44
slower ones in the other. To accomplish that, according to Maxwell, he had just to
observe the molecules approachingthe gate and select which had to go to either
side. By simply opening and closing the small gate-and without adding energy
to the molecules-the demoncould make one side of the box hot and the other
cold. That was like getting boiling water and cubes of ice just by separating the
faster and slower molecules from a bowl withwarm water. Maxwell's demon
could put the gas into a more organized state-thus making the entropy decrease.
The second law of thermodynamics was threatened by a mere imaginary being.
Szilard was the first to understand how to get around the problem. He
realized that the process of observing the molecules necessarily required an
energy exchange with the system. To measure the speed of the molecules, the
being had in some way to interact with them; one way to accomplish this
measurement, physicists suggested later, was to send rays of photons that hit the
molecules andbounced back, giving the being certain information about the
molecules' speed.These measurements accounted fcr the apparent decrease of
entropy, whereas the overall entropy was actually increasing-the second law
reigned again. Szilard's explanation of the interaction between the demon and the
molecules in terms of an acquisition of information established a direct link
between thermodynamics' entropy and what we understand today as information
theory's entropy."'
But when Shannon wrote his paper in 1948, he wasn't aware of Szilard's
work."l 3 So why did he decide to call his measure of information "entropy"?
According to physicist Myron Tribus, Shanno., said to him in a private
conversation in 1961 that the suggestion came from John von Neumann. The
Princeton mathematician suggested to Shannon to adopt the term because most
people didn't know what entropy really was, and if he used the term in an
argument he would will every time.' 14
Von Neumann himself had explored the
concept of information in quantum mechanics in the early 1930s"5 , but Shannon
told Bob Price he didn't recall the conversation with Tribus. "Now, I'm not sure
where I got that idea," Shannon said to Price. "I think somebody had told me that.
45
But anyway,I'm quite sure that it didn't happen between von Neumann and me."
It seems that the connection occurred to Shannon later in his development of
information theory. In his 1945 cryptography report, Shannon regarded the
logarithmic formula as a measure of uncertainty-not entropy. But later, the
section titled "Choice, Information and Uncertainty" in that report became
"Choice, Uncertainty and Entropy" in the 1948 paper.
In that year, a few months after Shannon's paper appeared, Norbert
Wiener published his influential book Cybernetics:or Control and
Communication in the Animal and the Machine. In that work Wiener presented a
formula for the measure of information almost identical to Shannon's and also
observed its connection to the thermodynamics concept of entropy."6 At MIT, not
long after the war, Wiener used to storm into a colleague's office-a huge cigar in
one hand-burble his latesttheory and leave without further explanations. In one
of these occasions, Wiener entered Robert Fano's office and declared,
"Information is entropy.""' Shannon, while a student at MIT,took a course with
Wiener on Fourier theory, but was not veryclose to him in research or in personal
terms."8 But they did exchange some correspondence and in October 1948
Shannon wrote Wiener saying he had read Cyberneticsand found it "full of new
and interesting ideas," adding that, "It was interesting to note how closely your
work has been parallelingmine in a number of directions." In the end of the letter,
Shannon wrote: "I would be interested in any comments you might have on my
paper 'A Mathematical Theory of Communication' ... The second installment,
dealing with the continuous case, is closely related to someof your work, and
contains several referencesto your papers.""9 Wiener answered the letter. He
thanked Shannon for his interest, said he also valued his colleague's work in the
field, and commented on some other issues. About Shannon's paper specifically?
Not a word.' 2 0
"The question of separating what is Wiener's contribution and what is
Shannon's raises a lot of debate," says Fano. For him, Wiener-especially with
his work on the aircraft trajectory prediction-was the first to address the question
46
of noise and also to point out that information was in a sense statistical. But it was
Shannon, Fano says, who really realized the discrete nature of information-the
necessary step to arrive at the crucial idea of coding. So while Wiener's work
dealt with filtering noise out of a received signal, Shannon's dealt with
overcoming noise in a signal transmission.2 ' Despite their similar ideas, no
collaboration ever took place, especially in the case of information and entropy'
"I think actually Szilard was thinking of this, and he talked to von Neumann about
it, and von Neumann may have talked to Wiener about it," Shannon recalled years
later. "But none of these people actually talked to me about it before I was
thinking of this by myself."'2 3
What explains the connection between information and entropy appearing
in the works of Szilard, Wiener, Shannon, and others?'2 4 What is this mysterious
measure that can be applied to boiling water and ice cubes, the toss of a coin, and
language? Some think Shannon's quantity was badly named; thermodynamics'
entropy and Shannon's entropy are two different things.'25 Others think that
entropy is just a kind of information."6 And Shannon himself thought there is a
deep, underlying connection between the two.'27 "Some scientists believe," he
wrote in the late 1960s, "that a proper statement of the second lawof
thermodynamics requires a term relating to information."'28 That would mean that
Shannon's theory is more than just a theory of communication that tells how
engineers should transmit data. Indeed, the results were so profound that, in 1983,
the great Russian mathematician Andrey N. Kolmogorov wrote that "information
theory must precede probability theory, and not be based on it."' 29 Shannon built
his work over probability theory andwhat he found was even more fundamental
than probability theory itself. Should it alsoprecede one of the most supreme laws
of Nature, the second law of thermodynamics?
'"The discussion about entropy and information can get very
philosophical," says Gerhard Kramer, a esearcher in the mathematics department
at Bell Labs.'30 Like Maxwell's demon, this question has puzzled engineers,
mathematicians, physicists, and philosophers. "What is information?" asked MIT
47
professor Seth Lloyd, who teaches the course "Information and Entropy" together
with professor Paul Penfield. "It is knowledge that you can pass back andforth,"
said one of the freshmen (yes, the course is taught for freshmen). "It is the
description of stuff," said another. "Encoding for meaning," guessed a third.''
One of the goals of the course,conceived by Penfield three years ago, is to study
the second law of thermodynamics as a kind of information processing.
"Information and entropy are related in an intimate way," said Penfield to the
students. 'They are the same thing but in differentcontexts. " '32 The coursebegins
with the concepts of bits and codes and advances to the study of varioussituations
in which information can be manipulated, stored, and transmitted, such as in
communication systems and in quantum computers-the use of quantum
phenomena to process information.
So information can exist in a varietyof situations. But is itjust an abstract
entity or has it a physical reality of its own? Information is entropy or entropy is
information? Is there a minimum amount of energy to store a bit? And to erase
one? Scientists are looking for the answers and their exploratory ideas take
several directions.33 In a paper published in 2002, Lloyd calculated the amount of
information that the Universe can store: 109°-or ten followed by ninety zerosbits.'34"Is
the Universe a computer? It is certainly not a digital computer running
Linux or Windows," he concluded in the end. "But the Universe certainly does
represent and process quantifiable amounts of information in a systematic
fashion."35Ed Fredkin, a former MITprofessor and a friend of Shannon, believes
that the laws of nature at its most fundamental level result from a sort of digital
information processing--what he calls "digital mechanics," a substrate for
quantum mechanics.'- And Princeton physicist Freeman Dyson wonders whether
life is analog or digital. "We don't yet know the answer to this question," he
wrote in an essay on the subject. "If we are partly analog, the downloading of a
human consciousness into a digital computer may involve a certain loss of our
finer feelings and qualities. I certainly haveno desire to try the experiment
myself." 1 3 7
48
* * *
LETTERS CAME FROM all parts of the country. And also from Canada, England,
France, and Japan. They came from universities-MIT, Harvard, Princeton,
Pennsylvania, Johns Hopkins-companies' laboratories-Westinghouse, RCA,
General Electric-and governmental and military institutions-Navy's Bureau of
Ordnance, Los Alamos Laboratory, BrookhavenNational Laboratory.Shannon,
then at age 32, felt the reaction to his 1948 paper in his mailbox. Not only
engineers and mathematicians were writing, but also economists, psychologists,
sociologists, linguists. Some posed questions; others shared ideas. And most
asked for copies of his stunning paper. Just a month after the firstpart was
published, a researcher from a company's engineering department wrote asking
for "half dozen" copies. Yet another wondered if Shannon could send him the
unpublished portion.'3
Many papers in the Bell System TechnicalJournal discussed technologies
that are now outdated. Shannon's paper, on the contrary, brought ideas that are
notjust useful today-they are the basis of modern communication theory. He
formulated information theory in generalterms, detaching his concepts from the
specific systems of the time and thus transcendingthe inevitable obsolescence of
technology. "I have on occasion skimmed through other journals of 1947to form
a basis for the calibration of Shannon's work," wrote Robert Lucky,a former Bell
Labs director of research. "What kind of world was it then? What I found was a
lot of forgettable hopping, chirping, and flying engineering things that suffered
extinction in the survival of the fittest in the decades that followed.... No
museum would ever be interested in them."'39
But as with most revolutionary theories, those that force scientists and
researchers to rethink what they considered true and fundamental, Shannon's
ideas took some time to become fully accepted. "When the paper first appeared,
few people really understood it," says David Slepian. "The old school of
49
engineers thought it was nonsense." 4° In 1949, the mathematician Joseph Doob
wrote a review of Shannon's paper that caused-and still causes-a lot of debate.
"The discussion is suggestive throughout, rather than mathematical," he wrote at
one point in his review, "and it is not always clear that the author's mathematical
intentions are honorable." Doob, a great mathematician soon to become an
authority in stochastic processes, argued that the paper wasn't mathematics, that
Shannon's proofs weren't rigorous enough.
In fact, the paper did have holes. Certain steps and assumptions were not
clear. Some theorems and proofs weren't perfect. But this very approach to the
problem of communication was perhaps the main reason why Shannon's work
was so successful. Shannon was a mathematician and an electrical engineer. He
always worked in the intersection of both fields. He was employed in the
mathematics department of a telephone company. He was writing a theoretical
paper for an engineeringjournal. For these reasons, he had practical
considerations in mind. He had to avoid certain mathematical formalisms and
move ahead in his theory. "The occasional liberties taken with limiting processes
in the present analysis," he wrote halfway in the paper, "can be justified in all
cases of practical interest.""4 ' Those who understoodthe essence of Shannon's
theory-and its implications-quickly recognized the need for a balance between
formalism and pragmatism. Shannon had showed the way; from there, they had to
work the rest of the results. The next step was to refine the proofs to make sure
the theorems were valid, that the engineers weren't stepping in muddy territory,
that they could trust the theory to build what really matters: the applications.
Not long after the paper came out, some began to work on the math.
"When I read the paper I finally understood the problem Shannon was trying to
explain," says Brockway McMillan, to whom Shannon had shown a sketch of the
fan diagrams, an important element in the formulation of information theory. A
few years later, McMillan wrote a paper explaining the theory to statisticians and
putting its results into more rigorous terms.142Finally mathematicians began to
50
understand and accept Shannon's ideas. "I gave a lot of talks about that paper,"
says McMillan.'4 3
The science of communication was finally becoming a unified field, with
its diverse disciplines brought together by integrating forces such as Wiener's
theory on prediction and filtering of noisy signals, Shannon's information theory,
and other works appearing at that time in the United States, England, Germany,
and the Soviet Union. '4 A symposium organized in London in September 1950
brought together, according to one attendee, "an intriguing mixture of
mathematicians, statisticians, physicists, biologists, physiologists, psychologists
and communication engineers.""'45
Groups of information theorists formed in several parts of the world, but
the leading place was MIT. In the spring of 1951, Robert Fano started teaching
course 6.574, "Transmission of Information," a seminar on information theory for
graduate students.As part of the course, he challenged his graduate students to
come up with better proofs for Shannon's theorems. To his delight, the proofs did
come. A student figured out the most efficient method to represent information
with zeroes and ones. Another extended McMillan's formulation and proved
definitely that Shannon's idea of virtually error-free communication up to the
channel capacity was right. And in fact, there wasn't a single theorem flawed.
"We should say that after fifty years," Stanford University professor Thomas
Cover wrote in 1998, "it is clear that Shannon was correct in each of his assertions
and that his proofs, some of which might be considered outlines, could eventually
be filled out along the lines of his arguments."'
In the mid-1960s, after nearly two decades since Shannon's foundational
paper was published, most theoretical results were exhaustively explored, and all
proofs were, well, proved. People had been working for a while on the
development of codes and satisfactorily ones were available. And it seemed they
were good enough for any application. At that point some began to say that
information theory as a field was dead. There were no more problems to solve.
51
Not everybody agreed with that of course. Microchips were on their way,
some noted, and they would certainly open new possibilities for information
theory. At MIT the order was to attack practical problems. "Their advice was,
'Don't work on theory, go to the applications'," says MIT electrical engineering
professor David Forney, then a graduatestudent.'47 But it wasn't easy.Computers
were still the size of refrigerators and communication was more analog than ever.
Informationtheorists had the codes to compress information and correct errorsthe
"software"-but they didn't have the hardware to execute those instructions.
Building any system with advanced coding schemes would require new
electronics-which would certainly cost lots of money. Not surprisingly,the first
complete application of Shannon's ideas was part of a government project.
In 1958, following the Soviet Union's launch of Sputnik, the first artificial
satellite put in orbit, the U.S. government created NASA.The space race
accelerated throughout the years and several spacecraft and probes were
developed. In 1968 NASA sent to space the solar-orbiting spacecraft Pioneer 9,
designed to study the flow of the solar wind and the high-energy particle streams
coming from the sun. The 144-pound spacecraft was the first to carry a digital
system for encoding and transmitting information. Shannon's theory said that the
better the codes you use, the more noise you can overcome. In space missions,
that translated into an invaluable result: spacecraftscould go farther. In fact, with
Pioneer 9's digital coding scheme, the spacecraft could travel 40 percent farther
and still be "heard" on earth. To do the same thing using more transmission
power, NASAwould have to spend millions of dollars in better transmitters and
receivers. 1 4 8
Deep space communication andcoding, according to James Massey, was
"a marriage made in heaven."'49 "It is no exaggeration," Massey wrote in a paper,
"to say that the Pioneer 9 mission provided communications engineers with the
first incontrovertible demonstration of the practical utility of channelcoding
techniques and thereby paved the way for the successful application of coding to
many other channels." Pioneer 9 circled the sun twenty two times, covering
52
eleven billion miles, and sent 4.25 billion bits of scientific data back to earth
during its operational lifetime.' ° Succeedingspacecrafts and probes carried even
more sophisticated codes and could transmitnot only scientific data but also clear
pictures of Mars, Saturn, and Jupiter. One of those error-correcting codes is the
same used now in CD players.
In the 1970s, microchips and computers began to be mass-produced. The
hardware to run information theory's coding schemes was finally available. "With
Shannon's remarkable theorems telling communicationsengineers what ultimate
goals to strive for, and integrated circuits providing ever-improving hardware to
realize these goals, the incredible digital communications revolution has
occurred," Solomon Golomb, a professor at University of Southern California,
wrote recently.'5 ' Incidentally, the microelectronicrevolution started veryclose to
Shannon, also at Bell Labs, when researchers Walter Brattain, John Bardeen, and
William Shockley invented the transistor in 1947. One day Shannon was chatting
with Shockley and saw on his desk a small plastic object with three wires
extending from it. "This was my firstglimpse of a transistor, quite possibly the
greatest invention of the 20th century," Shannon recalled later.'5 2 A tiny electronic
device to amplify and switch an electric current, the transistor was much better
than the clumsy vacuum tubes. Several of them can command a complex flow of
bits in a circuit. And the more transistors, the more bits can be manipulated. In
1958 the first integrated circuit-or simply microchip-was invented. Since then,
every few years long more powerful microchips are produced. Today's chips
contain thousands or millions of transistors. According to an empirical trend
observed in the semiconductor industry-known as Moore's Law, after Gordon
Moore, who worked with Shockley and later became one of the founders of
Intels 3 -the number of transistors that can be packed in a chip doubles every
eighteen months. Microchips can now execute the complex codes required by
information theory.
The search for better codes that can approach the channel capacity has
kept engineers busy since Shannon's paper appeared in 1948. But in a conference
53
in Geneva in 1993, French coding theorists surprised their colleagues when they
presented a code--dubbed the turbo code-that couldget extremely close to the
Shannon limit-closer than anyone had gotten before. At first, veterans in the
coding field were skeptical, but the results turned out to becorrect, and the turbo
codes revolutionized coding for reliable communications. "Other researchers have
refined these results andhave been able to construct simpler codes that approach
capacity basically as closely as desired," says Kramer, from Bell Labs. "In the
future, you will probably be using such codes on a daily basis if you have a cell
phone."'
From multi-million dollar space probes to cell phones and CDplayers that
many people can afford, Shannon's legacy is nearly everywhere today. His
information theory is invisibly embedded in the hardware and software that make
our lives easier and more comfortable. We don't see the error-correction schemes
embedded in our hard drives, modems, and computer networks. Or the coding
technology that makes mobile phones work. Or the compression methods that
shrink sound, images, and video sent over the Internet. Or the encryption schemes
that make online shopping secure. These and other technologies are based on the
laws of communication Shannon established 55 years ago.Today's digital
machines are fueled by bits, which made possible a simpler and cheaper
representation of information. In digital systems, all sorts of data are transformed
into two-value elements-the mathematical bit becomes aphysical bit. In a CD
bits are stored with tiny pits and bumps in the plastic surface. In an optical fiber,
bits are transmitted with pulses of light and "blank spaces" (absence of light). In
certain chips, bits flow as zero- and five-volt signals. With bits we can
manipulate,copy, and communicate information in an unprecedented way. "I
think it can truly be said that Claude Shannon laid the cornerstone for the field of
digital communication," wrote communication engineer Robert Kahn in an article
in 2001. "He provided the goals which generations aspired to attain. He provided
the formalism that shaped the way an entire field thought about their discipline,
54
and his insights forever altered the landscape and the language of
communications."15
* * *
JANUARY 1949.WARREN Weaver, the man who had directed the National
Defense Research Committee's gun fire control division, wrote Shannon saying
he had recently met Chester Barnard, he new president of the Rockefeller
Foundation, and described to him what was information theory. Barnard became
veryinterested and asked Weaver to write something less mathematicalfor him.
"This turned out to be a real job," Weaver wrote, referring to a text in which he
tried to explain information theory in more accessible terms and discuss its
possible applications. Weaver asked for Shannon's feedbackon that piece and
wondered about getting it published. "Having written this out, and assuming that
it does not turn out to be too horribly inaccurate, I have tentatively considered
attempting a rewriting, briefer and more popular in form, which I might submit
somewhere for publication-probably the Scientific American," he wrote. "But I
couldn't possibly do that unless you are entirely willing to have me do so.
Comments?" Weaver's article appeared in Scientific American in July. And in that
same year, his introductory piece and Shannon's original paper were published in
book form. Information theory wasn't just technical reading anymore. It was
available in bookstores.
Throughout the years, Shannon's paper influenced-and continues to
influence-scientists, engineers, and many other individuals.A graduate student
named his golden retriever "Shannon." An undergraduate at Cornell captivated by
Shannon's paper changed his major to mathematics. Anotherbegan referring to
himself as "third generation Shannon" as he took the legacy on and started
passing out copies of Shannon's paper.'" Indeed, somesay there is no better
introduction to the field than Shannon's paper itself. "Since I am an information
theorist, Shannon gave me my life's work," wrote Robert McEliece, an eminent
55
figure in the field. "I first read parts of his 1948 paper in 1966, and I reread it
every year, with undiminished wonder. I'm sure I get an IQ boost every time."'"s
Published in book form with a subtle-but significant-change in the title,
Shannon's work now wasn't just "A Mathematical Theory...," but "The
Mathematical Theory of Communication." Soon, concepts, terms, and ideas of
information theory began to beemployed in fields other than engineering and
mathematics. In the early 1950s, scientists studied the human ear and eye as
information channels and calculated their supposed "capacities." The ear? Ten
thousand bits per second. The eye? Four million bits per second.'8 A person
speaking? About thirty bits per second.'" A group of psychologists measured a
person's reaction time to various amounts of information. The subject sat before a
number of lights with associated buttons. The lights went on according to some
patterns-conveying different amounts of information-and the subject had to
push the corresponding buttons as quickly as possible. For one bit, the reaction
time was 0.35 second. For two bits, 0.5 second. For three bits, 0. 65 second. Every
bit added increased the reaction time in 0.15 second.The experiment showed that
the reaction time increases linearly with an increase inthe amount of information
conveyed by the lights.'6" "These results," Shannon wrote in the Britannica
article, "suggest that under certain conditions the human being, in manipulating
information, may adopt codes and methods akin those used in information
theory." 6"' Could the understanding of how machines process information help us
to finallyunderstand how we human beings process information?
Another field that adopted information theory ideas was linguistics. The
concept of a stochastic process as an information sourcethat could produce
languagewas a matter of a lot of debate. If a mathematical model could generate
language,and if this model was implemented into a computer, could a machine
speak? In a paper published in 1956 in the Transactionsof InformationTheory, a
young linguistics professor recently admitted to MIT argued with vehemenceand
math as well-that Markov processes couldn't generate language.For Noam
Chomsky, certain concepts from information theory could be useful for linguists,
56
but language had a grammatical pattern that couldn't be represented by such
processes. 1
As the yearspassed, Shannon's ideas continued to disseminate to other
fields-economics, biology, even art and music. "Many were honest attempts to
apply the new exciting ideas to fields in need of help," wrote David Slepian in
1998. "Time has shown that for the most part only mirages were seen."" In fact,
information theory didn't have a major impact on any of these fields. But its
concepts proved valuable anyway.Today, linguists use the notion of entropy and
Shannon's fundamental theoryto determine the number of words of a language
and their lengths.' 4 Geneticists and molecular biologists use information theory to
study the genetic code and to investigate things such as why sex is a winning
"evolutionary strategy" for manyspecies.'6 And investors and market analysts
use information theory to study the behavior of the stock market and optimize a
portfolio of stocks.'
But the incursions of information theory into fields other than
communication as it happened inthe 1950s and 1960sdidn't please its inventor.
"Informationtheory has, in the last few years, become something of a scientific
bandwagon," Shannon wrote in aneditorial for the Instituteof Radio Engineers
information theory journal in 1956.167"Our fellow scientists in many different
fields, attracted by the fanfare and by the new avenues opened to scientific
analysis, are using these ideas in their own problems. Applications are being made
to biology, psychology, linguistics,fundamental physics, economics, the theory of
organization, and many others." Shannon called for moderation, especially
outside the engineering domains. More than that, within the field itself, Shannon
called for more rigorand diligence: "Research rather than exposition is the
keynote, and our critical thresholds should be raised. Authors should submit only
their best efforts, and these only after careful criticism by themselves and their
colleagues. A few first rate research papers are preferable to a large number that
are poorly conceived or half-finished.The latter are not credit to their writers and
57
a waster of time to their readers." It seems Shannon followed that philosophyjust
"firstrate research papers"--all his life.
CLAUDE SHANNON NEVER liked to write. He even avoided writing.'" At home,
he used to keep unanswered correspondence in a box in which he wrote: "Letters
I've procrastinated too long on." e1 Shannon left few records at Bell Labs as well.
He simply seems not to have kept paper records. On joining Bell Labs every
researcher receives a lab notebook. Apparently, he didn't turn his in to the
files."'" At the Institute for Advanced Study in Princeton, where he started
working on information theory, he left no records either."'
Shannon was always looking for a new problem. But once he found the
solution, he didn't care about writing it down or getting it published. Many times
he started and soon abandoned a draft, putting it in a file on his attic. "After I had
found answers, it was alwayspainful to publish, which is where you get the
acclaim. Many things I have done and never written up at all. Too lazy, I guess. I
have got a file upstairs of unfinished papers!" he said in an interview with Omni
magazine in 1987. "But that's true of most of the good scientists I know. Just
knowing for ourselves is probablyour main motivation."'7
And that was how Shannon worked. He always kept most things to
himself."3 He was a quiet, introspective person. Not shy, not someone who tries
to hide in a corner, avoiding contact. Shannon was very warm and cheerful and
enjoyed talking to other people. Among his family and friends he used to laugh
heartily in a good conversation. He loved puzzles, pranks, and gadgets. He could
ride a unicycle and juggle at the same time. He was tall and thin--"a frail man,"
he once described himself. He had a long face and gray eyes with a very narrow,
light gray rim around the iris.
Shannon was just self sufficient, content to be alone immersed in his
solitary research. For those who didn't know him, he seemed serious, sometimes
58
cold. But despite his fame, hetreated his colleagues and students as his equal.1 7
"We all revered Shannon like a god," said Hermann Haus, who once gave a talk
at MIT in which Shannon was in the audience. "I was just so impressed, he was
very kind and asked leading questions. In fact, one of those questions led to an
entire new chapter in a book I was writing."" 5
In 1948, Shannon met Mary Elizabeth Moore, a mathematician at Bell
Labs working in John Pierce's group. On March 27, 1949 they got married and
went to live first in New York and later in New Jersey. Like Shannon, Betty was
bright, sharp, and had a great sense of humor."A perfect couple," some say.'7
Among the benefits of marriage was one solutionto Shannon's problems. He
didn't have trouble memorizing numbers, formulas, theorems, and mathematical
constants. That was his alphabet.'7 But to overcome the pain of writing, Shannon
found he could dictate his papers to Betty, who would write down everythingon
paper.' 7 The manuscript was then sent to a pool of secretaries at Bell Labs who
typed everything (a "noisy channel" of its own, as errors appeared here and there,
as when in a paper on coding, the phrase "signaling alphabet" became "signaling
elephant"'7 9). And that was notjust a dictating-writing process. Betty, as a
mathematician and a Bell Labs scientist in her own right, would engage in the
elaboration of Shannon's ideas. Many of Shannon's papers from 1949 until the
end of his life were created this way.80
After the publication of the 1948 paper, Shannon continued contributing to
the field he had created-but not for too long. As information theory grew in
importance, Shannon became more andmore distant and uninterested,
culminating in his 1956 "bandwagon" editorial. In that same year, after fifteen
years at Bell Labs, Shannon decided to leave the laboratories. He received offers
from a number of universities to teach.'8 ' But he chose the place he knew could
offer the freedom he had always sought: MIT.
Shannon went back to MIT first as a visiting professor, and as a faculty
member in 1958.At about that time, professor Robert Fano was teaching his
information theory seminar. Shannon then began teaching an advanced topics
59
course on information theory for graduate students. He would come to class and
talk about research problems he was working on-how to build reliable machines
with unreliable components; how to maximize a portfolio of stocks;'" how to
build an unbreakable cryptography system. He was a good lecturer, but not the
kind who tries to entertain his students. He would put a problem in the blackboard
and look at it for a while. Then he would put down something else or grab a piece
of paper andcheck a passage on it. "I have the indelible recollection of a person
who could reason in theorem-sized steps, the logic of which were never easy for
other to grasp," wrote Robert Kahn, who took Shannon's course in the 1960s.'
Sometimes Shannon could be very halting, and he didn't solicit much interaction;
but he was alwaysopen to questions. "For someproblems, he had good results.
For others, he made no progress beyond formulating the problem," says David
Forney, who attended the seminar in the spring of 1964. "It was great for graduate
students looking for a thesis topic."'84
Shannon's lecture style revealed much of his own research habits: he
chose topics according to his curiosity and was always trying to simplify the
problems, often using simple games or toy examples. "He was a different model
of scientist," says Robert Gallager, from MIT."He wasn't a scholar who wants to
know everything in a field. He liked puzzles, he liked to solve complicated things
making them simple."'" Also, Shannon always sought broad, fundamental results
instead of technical details-a single question from him could lead to an entire
chapter in a book. "His way of attacking a problem and seeing how hethought
and addressed research were strong influences on my own research," says
Leonard Kleinrock, aprofessor of computer science at the University of
California at Los Angeles. Because of that influence, Kleinrock says he always
looked for the global behavior, for the underlying principles, for the extreme and
simple cases, for intuitive understanding. "Shannon was the best," he says.'
Information theory had had its roots at Bell Labs, but now MIT became,
according to an observer, the "mother church" of the field.1 87 From the groups of
information theorists at Bell Labs and MIT came significant contributions to
60
communication, computer science,and artificial intelligence. Students graduated
from MIT and started teaching information theory at places such as Cornell,
Berkeley, and Notre Dame. Others went to companies. Gallager and Massey
became consultants to Codex, which produced many coding systems for the
military and where Forney created the first commercially successfulmodems (the
company was later acquired by Motorola). Former students Andrew Viterbi and
Irwin Jacobs started Qualcomm and developed wireless technologies widely used
in the cell phone system nowadays. Jacob Ziv, an Israeli electrical engineer who
graduated from MIT in 1962, invented a method of "squishing" bits that
transformed the field of data compression and which is widely used now in
computers and other digital systems. And in 1963, Robert Fano became the head
of MIT's Project MAC,an initiative to share computing resources that later
became the famed Laboratoryfor Computer Science.'"
During those bright years of information theory at MIT, Fano says that at a
certain point Shannon was asked to be an advisor to students. But he reacted
saying, "I can't be an advisor. I can't give advice to anybody. I don't feel the right
to advise." In fact, Shannon didn't have many advisees, but the few he advised
made significant contributions in several fields. In the early 1950s, while still at
Bell Labs, Shannon supervised John McCarthy and Marvin Minsky during
summer jobs at the laboratories." Both went on to be pioneers in the field of
artificial intelligence, an area in which Shannon also made fundamental
contributions. While at Bell Labs, he published an influential paner about
programming a machine to play chess-the first to speculate about such
possibility.'90 At about the same time, using telephone relays, he built a mouse
that could find its way through a maze (actually, a mechanism under the maze
guided the mouse with magnets). "Since the drive mechanisms and relay
computing circuit were all under the maze floor," Shannon recalled later, "some
of my persnickety friends complained that the mouse was not solving the maze,
but the maze was solving the mouse."' 9 At MIT, in the 1960s, he advised Leonard
Kleinrock, who would become an Internet pioneer, and also Ivan Sutherland,
61
whose master's thesis opened the field of computer graphics.' In the late 1960s,
another student, Edward Thorp teamed with Shannon to develop a cigarettepack
sized computer to predictroulette. Thorp, Shannon, and their wives even wentto
Las Vegas to test the device, which worked as expected, except for the fact that a
minor hardware problem preventedany serious betting.'93
At MIT and elsewhere, Shannon gained the status of celebrity, and for his
work on information theory he received innumerable honors. In 1966 he was
awarded the U.S. National Medal of Science. (Russian information theorists tried
to elect him to the Soviet Academy of Sciences, but the proposal didn't find
enough support.)' In 1985 he was awarded the Gold Medal of the Audio
Engineering Society; the plaque hereceived read simply, "For contributions that
made digital audio possible."'95That same year, he won Japan's Kyoto Prize,
known as the Nobel for mathematical achievements. In his discourse, Shannon
showed his classical block diagram of communication and said: "Incidentally,a
communication system is not unlike what is happening right here. I am the source
and you are the receiver. The translator is the transmitter who is applying a
complicated operation to my Americanmessage to make it suitable for Japanese
ears. This transformation is difficult enough with straight factual material, but
becomes vastly more difficult withjokes and double entendres. I could not resist
the temptation to include a number of these to put the translator on his mettle."'g
Not long after receiving the Kyoto Prize, Shannon received a letter from
Sweden. Some good news from the Nobel Committee? Not, actually. "Dear
Professor Claude Shannon," the letter began. "I'm a collector of autographs of
persons who really have done something positive for their people and the future
of mankind." The person wonderedif Shannon could send him an autographed
photo.
The prizes and honors, however, didn't interest Shannon very much. Nor
did his celebrity status. His colleagues say he hated giving speeches. But in 1985,
for some reason, Shannon decided to go to the international symposium on
information theory-something he didn't do for years-held in Brighton,
62
England. His presence caused turmoil and the audience lined up for autographs.
"It was as if Newton had showed up at a physics conference," recalled Robert
McEliece, the chairman of the symposium.'9 7
By the early 1980s,direct and indirect applications of information theory
could be found in several fields-from deep-space probes in the far reaches of the
solar systemto modems that transmitted data over a telephone line. Shannon's
theory was known among electrical engineers, mathematicians, computer
scientists, physicists, linguists, psychologists,and Wall Street investors. But by
that point, Shannon had lost most of his interest in information theory. He rarely
could be found in his office at MIT.
Shannon became an emeritus professor in 1978 when he was already
practically retired. He spent his time at home on his own things. He liked to play
with his kids-Andy, Bob, and Peggy-and work on gadgets. He usually
concentrated on one thing at a time. He could stay up all night building a machine
and get up the next day and immediately get back to it.'" But then he would start
working on another thing, and then move to another, and another.'9 In Christmas
of 1951, Betty gave him a unicycle. He loved it. Soon he was riding it in the
corridors of Bell Labs-now a legendarymoment in the culture of the labs-and
also building at home unicycles of different shapes and sizes. "Well, let me putit
this way," Shannon told Bob Price, "that my mind wanders around and I will
conceive of different things day and night, as Betty will attest-like the science
fiction writer or something like that. I'm thinking what if it were like this, or what
is this problem, or is there an interesting problem of this type? And I'm not caring
whether somebody's working on that or whether Washington would care one way
or the other, or anything of that sort. Ijust like to solve a problem. And I work on
these all the time." The kind of problems Shannon liked involved a physical, an
engineering situation for which he could sort of tailor make a mathematics. He did
that for switching circuits, genetics, cryptography,communication, chess-playing
machines, and juggling. 2"
63
At home he kept a large room to store his stuff-the "toy room," he and
Betty called it. The largest wall was devoted to his diplomas, plaques, and prizes.
Another wall was filed by posters of Darwin, Einstein, Newton, Hilbert-and a
Smurf. Shelves were packed with collections of Swiss knives--one with one
hundred blades-hats, chess sets, musical instruments,juggling balls, rings, clubs,
and Rubik's cubes. He tried to build a machine to solve a cube-but as with many
of his theories and inventions, he never finished it. But he wrote a song:
Ta! Ra! Ra! Boom De ay!
Cu-bies Indisarray?
First twist them that-a-way,
Then turn them this-a-way.
Respect your cube and keep it clean.
Lube your cube with Vaseline.
Beware the dreaded cubist's thumb,
The callused hand and fingers numb.
No borrower nor lender be.
Rude fold might switch two tabs on thee,
The most unkindest switch of all,
Into insolubility.
In-sol-u-bility.
The cruelest place to be.
However you persist
Solutions don't exist.20'
The mouse-maze system he built at Bell Labs, the roulette prediction
portable computer, andother incredible machinesShannon built were also there,
including a robotic puppet in the form of comedian W. C. Fields-his favorite
gadget.2" Made of Erector set pieces, it could bounce juggle three steel balls over
the head of a drum, making a sonorous "thunk" with each hit.
64
Juggling fascinated Shannon, and he spent a lot of time practicing-and
also figuring out mathematical laws relating the time the objects stay in the air
and the height they had to be thrown. One Sunday in the mid-1980s, members of
the MIT Juggling Club were practicing in the field just across the street from the
Institute's main entrance when a gray haired man stopped by and asked, "Can I
measure your juggling?" "He wanted to know how long a ball stays in the air and
in the juggler's hand," says Arthur Lewbel, the founder of the club and now a
professor of economics at Boston College. "He made some measurements that
day using a stopwatch and later came for more measurements." Lewbel says that
they had no idea that the cheerful gray haired man was Claude Shannon, a
professor emeritus at MIT, a tinkerer who loved to build all sorts of gadgets, and a
man who made a lot of money in the stock market investing in high-tech
companies started by friends, such as Teledyne, Hewlett-Packard, and Codex.203
Not long after, members of the club found themselves in his living room, in a
large house in Winchester, in the suburbs of Boston, to see his machines, watch
juggling videos, and eat pizza. "Shannon was the first to apply mathematics to
juggling," says Lewbel. After his juggling measurements, Shannon came up with
ajuggling theorem and wrote an article for Scientific American. The magazine
asked for revisions, which Shannon never did, and the article was never
published.
In 1993 Robert Fano met Shannon at a friend's memorial. "I asked him
something about the past, nothing technical or mathematical," says Fano, "and
Claude answered just 'I don't remember."' "2 In the early 1990s, Shannon noticed
some memory lapses. Sometimes he couldn't drive back home. Later he wouldn't
recognize his own writings. And then his friends. It was a long battle against
Alzheimer's disease. "The last time I saw Claude, Alzheimer's disease had gotten
the upper hand. As sad as it is to see anyone's light slowly fade, it is an especially
cruel fate to be suffered by a genius," wrote Lewbel in an article on Shannon in
late 2001. "He vaguely remembered Ijuggled, and cheerfully showed me the
juggling displays in his toy room, as if for the first time. And despite the loss of
65
memory andreason,he wasevery bit as warm, friendly, and cheerful asthe first
time I met him." Shannon died in February 24, 2001, at age 84.
TODAY, THE MOVEMENT in the corridors of Bell Labs at Murray Hill is not so
intense as in the golden years of the post-war. But the cafeteria is still the place
wherescientists and researchers get together to chat, solve problems, and
exchange ideas with their colleagues. Bell Labs is now the research and
development arm of telecom company LucentTechnologies. The old buildings in
the Murray Hill campus are now filledwith high-techequipment being tested and
developed as the labs seek to remain a leading place for innovative technology.
The transistor was invented here. And so were the orbiting communication
satellite, the solar battery cell, the Unix operating system, and many other
significant inventions.' Over the years Bell Labs has collected six Nobel prizes
and over 30,000 patents.206
The members of the mathematics department don't fly kites or play word
games during lunch as in the 1940s. Now they have their own games, puzzles, and
jokes. "The other day I dialed an 1-888number to get some information on
passports," says a Russian researcher. "But I think I missed one digit and I
reached an erotic line!" Says one of his colleagues: "The number has a short
Hamming distance." Among this new generation of information theorists,
Shannon's legacyis enormous. Posters and pictures of him hang in the offices.
Papers on information theory and editorials such as Shannon's "The Bandwagon"
fill the boards on the corridors. One researcher reads Shannon's papers when she
is depressed. "Those who knew Shannon say he wasn't showy. He wasn't, say, a
Dick Feynman," says Kramer, a young researcher in the department who never
met Shannon but keeps a venerable posterof him in his office. "He must have
been captivating in his own way."207
66
At the main building in Murray Hill, on the right side of the entrance
lobby, a tall hall illuminated by a skylight, there stands a ten-foot tall bust of
Alexander Graham Bell, the inventor of the telephone and after whom Bell Labs
was named. Bell's figure, with opulent moustache and sideburns, gazes into the
horizon, while its pedestal bears the following quote: "Leave the beaten track
occasionally and dive into the woods. You will be certain to find something you
have never seen before." On the opposite side of the hall fromBell's bust, another
bust stands by a column. Claude Shannon's bust is not as tall as Bell's, but it
shares a privileged location on the hall. His figure has a thoughtful look, his head
turned a little downwards, his hand holding his chin. On Shannon's there is no
quotation,just the message:
H =- p logp - q logq
67
ACKNOWLEDGEMENTS
Several times throughout the course of this work, I met Robert Kanigel in his office on the fourth
floor of the Humanities building,where he receives his students sittingin a comfy easy upholstered
chair and offering his guests a hard wooden slatted chair. I wonder what his true intentions were;
perhaps he wanted to create an uneasy atmosphere, and provoke, inspire, push his students.
Whether that was true or not, it worked. I am indebted to Rob for his help, advice, support-for
provoking, inspiring, and pushing me during my year at MIT. He helped me to choose the topic of
this article, shape its structure, refine and polish the text. I thank him for his help, patience, and
constant cheerfulness in all-easy and hard-moments.
The faculty of the science writing program surpassed all expectations and provided
memorable moments of unique writing and editing wisdom during our "mega seminars." Iwant to
thank B. D. Colen, Alan Lightman, and Boyce Rensberger for an incredible year at MIT. Thanks
go also to Jim Paradis, head of the writing program, for all his support-and a great Thanksgiving
dinner at his home. Sarah Merrow went beyond her duties as the program administrator to help me
in all sorts of ways, becoming a good friend even before I arrived in Cambridge. Sarah helped to
make the "MIT experience" a very enjoyable and agreeable one.
At MIT I also had the opportunity to take great courses on the history of science and
technology. I wish to thank David Kaiser, David Mindell, and Rosalind Williams for their effort
and dedication to the art of teaching. Additional thanks go to Paul Penfield, who let meaudit the
interesting freshmen course on information and entropy he teaches with Seth Lloyd. (What is
information anyway?)
I am thankful to all who contributed to this work explaining to me the toughest parts of Shannon's
theory, submitting to interviews, providing invaluable material, and referring me to various
important sources. I am specially grateful to Betty Shannon, who kindly received me at her home
in Winchester for an evening of wonderful stories and a tour in the famed "toy room." For their
guidance and encouragement, I am indebted to Robert Fano, Robert Gallager, and Robert Price.
For sharing their recollections and providing invaluable help, specialthanks are due to David
Forney, the late Hermann Haus, Amos Joel, Leonard Kleinrock, Arthur Lewbel, James Massey,
Brockway McMillan, Joel Moses, Ronald Rivest, and David Slepian.
At Bell Labsin Murray Hill, I am thankful to Gerhard Kramer, James Mazo, and Debasis
Mitra. Gerhard proved to be not only an excellent host and "tour guide" at the labs, but also an
excellent teacher, patientlyexplaining to me the 'fundamental theorem," turbo codes, one-time pad
cryptography, and many other tough things.
Shannon didn't like to write. And he didn't like to give interviews neither. But in the few
occasions he agreed, the material revealed great insight on his personality, interests, and genius.
Friedrich-Wilhelm Hagemeyer interviewed Shannon and other leading figures in the field of
communication in the late 1970s. Bob Price interviewed Shannon in the early 1980s. I would like
to express my appreciation for their efforts in trying to obtain the message straight fromthe source.
Price kindly shared his transcript and the original audio recording and also helped me to find and
follow other leads. Hagemeyerreferred me to the mostrelevant parts in his comprehensive PhD
dissertation on information theory. I owe a great debt of gratitude to Hermann Rotermund, who
transformed Hagemeyer's tape-recorded analog interviews into digital MP3 files and mailed them
to me on a CD-Shannon's voice signals encoded in bits.
The three "Bobs"-Fano, Gallager, and Price-and also David Slepian and Gerhard Kramer
read the manuscript and provided invaluable comments and suggestions, helping me to "debug" the
original text and fix errors here and there.
Librarians and archivists provided invaluable help in several ways; my thanks to Leonard
Bruno and Joseph K. Brooks of the Manuscript Division at the Library of Congress, Nora Murphy
of the MIT Archives, Jenny O'Neill of the MIT Museum, the staff of the MIT libraries, and Mary
Ann Hoffman of the IEEEHistory Center.
Coming to MIT was certainly a unique experience. It wouldn't be possible without the training and
education I received at theUniversity of Sao Paulo, in Brazil. Special thanks go to Jose Jaime da
Cruz, Roseli Lopes, Ricardo PaulinoMarques, Henrique Del Nero, Felipe Pait, Paulo Sergio
Pereira da Silva, and Marcelo Zuffo. Also in Brazil, I want to express my gratitude to Helio
Gurovitz, who taught meeverything I knew about journalism before coming to MIT.
My classmates in MIT's first science writing class also helped to make this a memorable
year; for their friendship, I wish to thank Timothy Haynes, Matthew Hutson, Karen MacArthur,
Maywa Montenegro, Sorcha McDonagh, and Anna Lee Strachan. Thanks go also to other friends at
MIT and elsewhere: Paulo Blikstein, Pedro Bolle, Patrick Kann,Charles Low, Sidney Nakahodo,
Fabio Rabbani, Lidia Reboucas, Matthew Traum, and the Big Head AdventureTeam-Tadeu
Azevedo, Daniel Garcia, AndreKupfer, Humberto Marin, Marcus Paula, and Daniel Pezeta.
My family, despite being thousands of miles far to the south, was always present, providing
comfort, confidence, and inspiration. I am infinitely thankful to my father, Joao, my first editor; my
mother, Clarice, my most dedicated reader; and my little sister Maya, always a source of wise
thoughts and words.
69
NOTES
Robert Price, communication engineer, interviewby author, Lexington, Mass., April 10,2003. On July 28,
1982, Price conducted a lengthy interview with Shannon, recorded on tape. Price was interested in the origins
of information theory and more specifically in the history of a communication technology known as spread
spectrum. An edited version of Price's interview was published in 1984 (F. W. Ellersick, ed., "A
Conversation with Claude Shannon: One Man's Approach To Problem Solving," IEEE Communications
Magazine, May 1984,p. 123.)The transcript of the interviewis available, under copyright, at the IEEE
Archives in New Jersey.
2 Mary Elizabeth (Betty) Shannon, interview by author, Winchester, Mass., February 25,2003; Robert Fano,
professor of electrical engineering and computer science,MIT, interview by author, Cambridge, Mass.,
December 19,2002; Hermann Haus (deceased), professor of electrical engineering and computer science,
MIT, interview by author, Cambridge, Mass., December 20, 2002; David Slepian, information theorist.
former Bell Labs researcher, interview by author, Murray Hill, New Jersey, March 7, 2003; Arthur Lewbel,
professor of economics, Boston College, interview by author, Chestnut Hill, Mass., January 30, 2003; Robert
Gallager, professor of electrical engineering and computer science, MIT, interview by author, Cambridge,
Mass., January 15,2003.
3 For other epiphanies in science see, for example, Paul Strathern, Mendeleyev's Dream: The Quest For the
Elements, St. Martin's Press, 2001; David A.Mindell, Between Human and Machine: Feedback, Control.
and Computing before Cybernetics. Chapter 4, "Opening Black's Box," and Chapter 9, "Analog's Finest
Hour," The Johns Hopkins University Press, 2002.
4John Horgan, "Profile: Claude E. Shannon," IEEE Spectrum, April 1992, p. 74. In fact, Shannon seemed to
always evade this kind of question-see, for instance, Ellersick,op. cit. Also, in 1977,Friedrich-Wilhelm
Hagemeyer, a German graduate student in history of science, came to the United States and interviewed
several leadiag scientists and information theorists, including Shannon. His thesis was published in 1979 and
never translated into English (F. W. Hagemeyer."Die Entstehungvon Informationskonzepten in der
Nachrichtentecknik," PhD dissertation, Free University of Berlin, Germany 1979). The aut, r obtained
several of the tape recordings of the Hagemeyer's interviews. In his interview with Hagemeyer, Shannonalso
avoided tracing the origins of information theory back to a single moment or idea.
5 In 1953, just five years after Shannon's paper came out, Fortune magazine peblished an elaborate article on
information theory. The theory, wrote the author. "seems to bear some of the same hallmarks of greatness" of
quantum theory, but is "still almost unknown to the general public" (Francis Bello, "The Information
Theory," Fortune, December 1953, pp. 136-141 and 149-157).
6"A Mathematical Theory of Communication" was published in two parts in the Bell System Technical
Journal, Vol. 27, July 1948, pp. 379-423 and October 1948, pp. 623-656. For an introduction to information
theory, see Claude E. Shannon and Warren Weaver, The Mathematical Theory of Communication, University
of Illinois Press, 1949;John R. Pierce, An Introduction to Information Theory: Symbols, Signals. and Noise,
New York: Dover, 1961;and Robert W. Lucky, Silicon Dreams: Information, Man. and Machine, New
York: St. Martin's Press, 1989.
7 See, for example, Sergio Verdi, "Fifty Years of Shannon Theory." IEEE Transactions on Information
Theory, Vol.44, No. 6, October 1998,p. 2057; Amos Joel. "Telecommunications and the IEEE
Communications Society," IEEE Communications Magazine, 50th Anniversary Commemorative Issue,May
2002.
8 M. Mitchell Waldrop, The Dream Machine: J. C. R. Lickliderand the Revolution That Made Computing
Personal, Penguin Books, 2001, p. 81. (An article in the September 1952issue of Scientific American said
about the term: "It is almost certain that 'bit' will become common parlance in the field of information, as
'horsepower' is in the motor field.")
9 Gallager, interview by author. For more on the reaction to Shannon's paper, see: David Slepian,
"Information Theory in the Fifties," IEEE Transactionson Information Theory, Vol. IT-19,No.2, March
1973;James L. Massey, "Information Theory: The Copernican System of Communications," IEEE
Communications Magazine, Vol. 22, No. 12,December 1984,p. 28; John R. Pierce. "The Early Daysof
Information Theory," IEEE Transactions on Information heory. Vol. IT-19, No. 1,January 1973.
10
Colin Cherry, interview by F. W. Hagemeyer, London, England, September 6,1976 (audio recording
obtained by author).
i Solomon W. Golomb et al., "Claude Elwood Shannon (1916-2001)," Notices of the American
Mathematical Society, Vol. 49, No. 1,p. 11.
1 2
Brockway McMillan, interview by Hagermneyer,Whippany, New Jersey, February 8, 1977 (audio recording
obtained by author).
13 Fano, interview by author.
14No wonder Scientific American called the paper "the Magna Carta of the information age" (John Horgan,
"Claude E. Shannon: Unicyclist, Juggler and Fatherof Information Theory," Scientific American, January
1990).
5
s Horgan, "Profile: Claude E. Shannon," p. 72. A similar story wastold by Shannon to Arthur Lewbel
(Arthur Lewbel, "A Juggler's Tribute to Claude Shannon." IEEE Information Theory Society Newsletter,
December 2001, pp.9-12).
i Claude Shannon, interview by F. W. Hagemeyer,Winchester, Mass., February 28, 1977 (audio recording
obtained by author).
7 Shannon, interview by Hagemeyer.
18N. J. A. Sloane and Aaron D.Wyner, ed., ClaudeElwoodShannon:CollectedPapers,John Wiley & Sons,
1993,p. xi ("Biography of Claude Elwood Shannon'").This book is a comprehensivecollection of most of
the papers written by Shannon.
19 Shannon,interview by Hagemeyer.
20 Claude Shannon, Kyoto Prize speech draft, 1985,Claude Elwood Shannon Papers, Manuscript Division,
Libraryof Congress, Washington, D.C.
21 Sloane and Wyner, op. cit., p. xi.
22Claude Elwood Shannon,A SymbolicAnalysis of Relay and Switching Circuits, S.M. Thesis, MIT, 1940,
14.
Letter from Charles S. Rich, secretary of the American Instituteof Electricai Engineers, to Vannevar Bush
(May 4, 1938), Claude Elwood Shannon Papers, Manuscript Division, Library of Congress, Washington,
D.C.
24 AmosJoel, telephone engineer, former Bell Labs researcher, telephone interview by author, May 30, 2003.
25Sloane and Wyner, op. cit., p. xii.
26 Letter from R. H. Smith to Karl Compton (April 11, 1939), President Letters, MIT Archives.
27
Letter from Karl Compton to R. H. Smith (April 13, 1939), President Letters, MIT Archives.
28 Shannon, interview by Hagemeyer.
29Sloane and Wyner, op. cit., pp.455-456.
3 0
Ibid.
31 Letter from Frank Avdelotte, director of the Institutefor AdvancedStudy at Princeton, to Shannon (April
11, 1940),Claude Elwood Shannon Papers, ManuscriptDivision, Libraryof Congress, Washington, D.C.
32Shannon,interview by Hagemeyer; Shannon wouldsay the samething in his interview with Price.
33Shannon, interview by Price.
3 Shannon, interview by Hagemeyer.
s Ibid.
36 Ibid.
Brockway McMillan, mathematician, former Bell Labs researcher, telephone interview, June 3, 2003.
38Shannon, interview by Price.
39Mindell, op. cit., p. 187
40Ibid., pp. 186-187
41Ibid., p 258
42Pesi R. Masani, Norbert Wiener: 1894-1964, Birkhauser Verlag, 1990,p. 39.
4
3
Wiener has two autobiographies: Ex-Prodigy (MIT Press, 1953) and I Am a Mathematician: The LaterLife
of a Prodigy (Doubleday & Company, 1956).
Pierce, An Introduction to Information Theory, pp. 41-42.
4
5 Norbert Wiener, Extrapolation, Interpolation, and Smoothing of Stationary Time Series, Technology Press
and JohnWiley & Sons, 1949,pp. 1-10.
71
46David Jerison and Daniel Stroock. "Norbert Wiener," Notices of the American Mathematical Society, Vol.
42, No. 4, April 1995, p. 438.
47
Mindell, op. cit., p. 280.
48Shannon, interview by Hagemeyer.
49 Mindell, op. cit., p. 280.
50R. B. Blackman, H. W. Bode, and C. E. Shannon, "Data Smoothing and Prediction in Fire-Control
Systems," Summary Technical Report of the National DefenseResearch Committee, Division 7, Gunfire
Control, Washington, D. C., 1946,p. 80 (Chapter 7: General Formulation of the Data-Smoothing Problem).
51 Mindell, op. cit., pp. 254-258.
52 For a comprehensive account of the emergence of electrical signals as a way to represent the world in
control and communication systems throughout the two world wars, see Mindell, op. cit.
53 M. D. Fagen, ed., A History of Engineering and Science in the Bell System: National Service in War and
Peace (1925-1975), Bell Telephone Laboratories, 1978, p. 301.
4The cryptography scheme used in the X System was known as the "Vernam cipher" or "one-time pad" (in
the case when used keys were discarded). The one-timepad is notonly practically unbreakable, but
theoretically unbreakable as well. There are no patterns in the messages to makepossible a "statistical
attack." A "brute force" attack doesn't work either because for each key the enemy tries, he gets a different
deciphered message. In the case, say, of a fifteen-character message. the enemy will generate all possible
fifteen-charactertext strings when trying the keys; he might obtain "ATTACK TOMORROW," or "NO
ATTACKTODAY," or any other fifteen-charactertext, from gibberish to pieces of Shakespeare's plays. The
enemy has no way to decide which of the messages is the "right" one. For a complete description of this
method,see: David Kahn, The Codebrakers: TheStory of Secret Writing, NewYork: Macmillan Company,
1967, pp. 394-403.
55 Fagen,op. cit., p. 297; William R. Bennett,"Secret Telephony as a Historical Example of SpreadSpectrum
Communication," IEEE Transactions on Communication, Vol. COM-3 1, No. 1,January 1983, p.
98.
5 Shannon, interview by Hagemeyer.
57 Fagen, op. cit., p. 317.
s8 Andrew Hodges, Alan Turing: The Enigma, New York: Simon and Schuster, 1983, pp. 242-255.
59Shannon, interview by Hagemeyer.
oIbid.
61 Hodges, op. cit., p. 250.
62 Robert G. Gallager, "Claude E. Shannon: A Retrospective on His Life, Work, and Impact," IEEE
Transactions on Information Theory, Vol. 47, No. 7, November 2001, p. 2682.
Claude Shannon, "A Mathematical Theory of Cryptography," Technical Memorandum, September 1,
1945. MIT Archives, p. 7.
CAIbid., p. 74. It seems this was one of the first occurrences of the expression-an expression Shannon would
repeat exactly once in his 1948 paper (p. 393 in the original). It is also worth to mentionthat the term was
used before Shannon's paper came out. On March 24, 1948,during the Institute of Radio Engineers National
Conventionat the Hotel Commodore in New York, Shannon gave a talk titled "Information Theory." The
short abstract forthe talk read, "Limitations on the transmissionof information imposed by bandwidth, time,
and signal-to-noise ratio" (Proceedings of the IRE, 1948,p. 366). In 1949,Shannon's talk appeared inprint
as "Communication in the Presence of Noise" (Reprinted in Sloane and Wyner, op. cit., pp. 160-172).
6 McMillan, interview by author.
66 Gallager, op. cit., p. 2682.
67 Betty Shannon, interview by author.
68Shannon, interview by Hagemeyer; Shannon, interview by Price; Betty Shannon, interview by author.
69 Slepian,interview by author; McMillan, interviewby author.
70 McMillan,interview by author.
71S. Millman, ed., A History of Engineering and Science in tireBell System: Communication Sciences
(19 2
5-1980), AT&T Bell Laboratories, 1984, p. 3.
McMillan, interview by author.
73 Shannon, interview by Price.
72
74 Ibid.; Shannon also mentioned Hartley's paper in his interview with Hagemeyer and in his Kyoto Prize
spegechdraft.
R. V. L. Hartley, "Transmission of Information," Bell System Technical Journal, July 1928, pp. 535-563.
OIbid., p. 536.
7
Ibid., p. 536.
Ibid., p. 540.
79 Millman, op. cit., pp. 152-160.
So John R. Pierce, Signals: The Telephone and Beyond, San Francisco: W. H. Freeman and Company, 1981,
D
4 1 .
HendrikW. Bode, "Obituary Statement: Harry Nyquist," IEEE Transactionson Automatic Control, Vol.
AC-22, No. 6, December 1977.
82 In the 1920s, Harry Nyquist published two important papers on the transmissionof signals in telegraph
systems, bothof themcited in Shannon's 1948 paper: "CertainFactors AffectingTelegraph Speed," Bell
System Technical Journal, April 1924,pp. 324-346, and "Certain Topics in Telegraph Transmission
Theory," Transactionsof the A.I.E.E., Vol.47, April 1928,pp. 617-644 (reprinted in: Proceedings of the
IEEE, Vol. 90, No. 2, February 2002, pp. 280-305).
83Nyquistdeveloped a formula to measure the transmission speed of a telegraph system that also used the
logarithmic function.
84Massey, op. cit., p. 27.
6 Mindell, op. cit., pp. 134-135.
Shannon,Kyoto Prize speech draft, op. cit.
87 Sloane and Wyner, op. cit., pp. 455 and 456.
Shannon, interview by Price.
90Shannon,interview by Hagemeyer.
90 Pierce, An Introduction to Information Theory, p. 40.
91 Fano, interview by author.
9 David Slepian, personal communication with author, June 15,2003.
93 Shannon, interview by Price.
94 Waldrop, op. cit., p. 81.
95Shannon and Weaver,op. cit., pp. 43 and 44.
9Ibid., p. 44.
For technical and historical details on the sampling theorem,see HansDieter LUke,'The Origins of the
Sampling Theorem," IEEE Communications Magazine, April 1999.
98Millman, op. cit., pp. 399-418.
99 Mindell, op. cit., p. 318; for examples of digital computers, see, for example, H. H. Goldstine and Adele
Goldstine, 'The Electronic Numerical Integrator and Computer (ENIAC)," IEEEAnnals of the History of
Computing, Vol. 18,No. 1, 1996,pp. 10-16; M. M. Irvine, "Early Digital Computers at Bell Telephone
Laboratories," IEEE Annals of the History of Computing,July-September 2001,pp. 22-42.
1o0James Massey, information theorist, former professor of electrical engineering at ETH, Zurich, personal
communication, June 3,2003.
101 This is the code according to the International Morse Code. In the American Morse Code, an "X" is a dotdash-dot-dot
sequence.
102TheEncyclopaedia Britannica article is reprinted in Sloane and Wyner, op. cit., pp. 212-220.
103 Shannon,interview by Hagemeyer; also Bello, op. cit., p. 140.
104 Fraunhofer Institute for Integrated Circuits, "Audio & MultimediaMPEG Audio Layer-3," URL:
http://www.iis.fraunhofer.de/amm/techinf/layer3/
105 Haus, interview by author.
106 Fano, interview by author.
107 Massey, op. cit., p. 27.
103 Shannon,interview by Hagemeyer.
109 Shannon and Weaver,op. cit., p. 51.
1 1
Arthur S. Eddington, The Nature of the Physical World, Macmillan, 1948,p.74.
73
1l Anson Rabinbach, The Human Motor: Energy, Fatigue, and the Origins of Modernism, Universityof
California Press, 1992.p. 47.
112
For a comprehensive discussionabout Maxwell's demon, entropy, and information see Harvey S. Leff
and Andrew F. Rex, ed., Maxwell's Demon: Entropy, information, Computing, Princeton University Press,
1990.
,13 Shannon, interview by Hagemeyer.
114Raphael D. Levine and Myron Tribus, ed., The Maximum Entropy Formalism (A Conference Held at the
MassachusettsInstitute of Technology on May 2-4, 1978), MIT Press, pp. 2 and 3.
1a Shannon and Weaver, op. cit., p. 3.
11 Norbert Wiener, Cybernetics: Or Control and Communication in the Animal and the Machine, MIT Press
and John Wiley & Sons, 1961,pp. 62-64.
117Fano, interview by author. Fanoalso mentions Wiener's statement on information and entropy in the
preface of his book, Transmissionof Information: A Statistical Theoryof Communications, MIT Press, 1961,
P.vii.
Shannon, interview by Hagemeyer. Wiener says the same about Shannon in I Am a Mathematician, op.
cit., p. 179.
119Letter from Shannon to Wiener (October 13, 1948),Wiener Papers, MIT Archives.
120Letter from Wiener to Shannon (no date), Wiener Papers, MIT Archives.
121 Shannon cited and acknowledged Wiener's work several times in his 1948 paper. In a footnote, he
observed: "Communicationtheory isheavily indebted to Wiener for much of its basic philosophy and theory.
His classic NDRC report, The Interpolation, Extrapolation and Smoothing of Stationary Time Series (Wiley,
1949), contains the first clear-cut formulation of communication theory as a statistical problem, the study of
operations on time series. This work, although chiefly concerned with the linearprediction and filtering
problem, is an important collateral reference in connection withthe present paper. We may also refer here to
Wiener's Cybernetics (Wiley, 1948),dealing with the general problems of communication and control." In
the acknowledgements, Shannon wrote:"Credit should also be given to Professor N. Wiener, whoseelegant
solution of the problems of filtering and prediction of stationary ensembles has considerably influenced the
writer's thinking in this field." And Wiener himself claimed part of the credit for the creation of information
theory. "Information theory has been identified in the public mind to denote the theory of information by bits,
as developed by Claude E. Shannon and myself," Wiener wrote in 1956 (Norbert Wiener, "Editorial: What Is
Information Theory?" IEEE Transactions on Information Theory, Vol. 2, Issue 2, Jun 1956,pp. 48).
122Fano, interview by author.
123Shannon, interview by Hagemeyer.
124For a short story on how the concepts of information and entropy appeared in the works of Wiener,
Shannon, and others, see JeremyCampbell, Grammatical Man: Information, Entropy. Language,and Life,
Simon and Schuster, 1982.And Wiener and Shannon weren't alone; others were publishing works with
similar ideas. See, forexample, Dennis Gabor, "Theory of Communication,"J. IEE (London), Vol. 93, 1946,
pp. 429-457; William G. Tuller, "Theoretical Limitations on the Rate of Transmission of Information," PhD
dissertation, MIT, June 1948.
125See, for example, Thomas Schneider's web site on information theory and molecular biology, URL:
http://www.lecb.ncifcrf.gov/-toms/pitfalls.html#shannonentropyis_a_misnomer
126Paul Penfield, Jr., Information and Entropy, Course Notes, MIT, version 1.0.2, January 30, 2003, URL:
http://www-mtl.mit.edu/Courses/6.050/2003/
127Shannon's Encyclopaedia Britannica article (Sloane and Wyner,op. cit., p. 213).
128Ibid.,p.215.
129Thomas M. Cover et al., "Kolmogorov's Contributions to InformationTheory and Algorithmic
Complexity," The Annals of Probability, Vol. 17,No. 3,July 1989,p. 840.
130Gerhard Kramer, Bell Labs researcher, interview by author, Murray Hill, New Jersey, February 21, 2003.
31 "Information and Entropy," MIT course, lecture on February 6, 2003.
132Ibid., February 4, 2003.
33For more on information in physical systems see Neil Gershenfeld, The Physics of Information
Technology, Cambridge University Press, 2000.
74
'34 Seth Lloyd, "Computational Capacity of the Universe," Physical Review Letters, Vol. 88, No. 23, June
10, 2002, p. 237901-1.
138Ibid., pp. 237901-3 and -4.
13 Ed Fredkin, Digital Philosophy website, URL: http: //www.digitalphilosophy.org/
1
3 7
Freernan Dyson, essay on EDGE website, URL:
http://www.edge.org/3rdculture/dyson_ad/dyson_ad_index. html.Also, lecture at MIT,
October 17,2002.
138 Letter from William Altar, electronics and nuclear physics department, Westinghouse Electric Corp., to
Shannon (August 20, 1948),Claude Elwood Shannon Papers, Manuscript Division, Library of Congress,
Washington, D.C.
139Robert W. Lucky, "When Giants Walked the Earth," IEEE Spectrum, April 2001, URL:
http://www.spectrum.ieee.org/WEBONLY/resource/aprOl/refl.html
140 Slepian, interview by author.
141 Shannon and Weaver, op. cit., p. 81.
1
42Brockway McMillan,"The Basic Theorems of Information Theory," The Annals of Mathematical
Statistics, Vol. 24, No. 2. June, 1953,pp. 196-219.
143 McMillan, interview by author.
44For historical accounts of the development of the concept of information in communication and
computing, see WilliamAspray, 'The Scientific Conceptualization of Information: A Survey," IEEEAnnals
of the History of Computing, Vol. 7, No. 2, April 1985,pp. 117-140; E. Colin Cherry, "A History of the
Theory of Information," IEEE Transactions on Information Theory, Vol. 1,Issue 1,February 1953,pp. 22-
43; Hagemeyer, PhD dissertation,op. cit.; R. C. Olby et al., ed., Companion to the History of Modern
Science, Routledge, 1990,pp. 537-553 (Chapter 34: Cybernetics and InformationTheory).
145Willis Jackson, ed., Communication Theory, Papers read at a Symposium on "Applications of
Communication Theory" held at the Institution of Electrical Engineers, London. September 22nd-26th 1952,
New York: Academic Press, 1953, p. x.
14 Golomb et al., op. cit., p. 11.
147David Forney, professor of electrical engineering, MIT, interview by author, Cambridge, Mass., Jan 14,
2003.
148J. Hagenauer, ed., Advanced Methodsfor Satellite and Deep Space Communications, Lecture Notesin
Control and InformationSciences 182, Heidelberg and New York: Springer, 1992,pp. 1-17.
149 Ibid.; also, Daniel J. Costello, Jr. et al., "Applications of Error-Control Coding," IEEE Transactions on
Information Theory, Vol.44, No. 6, October 1998,p. 2533.
Pioneer 9, Space Calendar, NASA Jet Propulsion Laboratory, URL:
http://www.jpl.nasa.gov/calendar/pioneer9.html
151Golomb, op. cit., p. 10.
152Shannon, Kyoto Prize speech draft, op. cit.
153Michael Riordan and LillianHoddeson, Crystal Fire: The Invention of the Transistorand the Birth of the
Information Age, W. W. Norton, 1997, p. 239.
1997.
154Kramer, interview by author.
55Kahn, op. cit., p. 20.
156 According to comments posted on the web site "Remembering Claude Shannon," Project Echo, Center
for History and New Media, George Mason University, URL:
http://chnm.gmu.edu/tools/surveys/responses/80/
157 Robert J. McEliece, Project Echo's website, op. cit.. March 12.2001.
158Homer Jacobson, 'The Informational Capacity of the Human Eye," Science, New Series. Vol. 113, No.
2933, March 16, 1951, pp. 292 and 293.
159John R. Pierce, An Introduction to Information Theor?, p. 234.
16' Ibid., pp. 230 and 231.
161Encyclopaedia Britannica article (Sloane and Wyner, op. cit., p. 219).
~2Noam Chomsky, 'Three models for the description of language," IRE Transactions on Information
Theory, Vol. 2, No. 3, September 1956, p. 115.
75
l63 Slepian, op. cit., p. 145.
1" See, for example, Martin A. Nowak and Natalia L. Komarova, "Towards an evolutionary theory of
language," Trendsin Cognitive Sciences, Vol. 5, No. 7,July 2001, pp. 288-295; Martin A. Nowaket al., "An
error limit for the evolution of language," Proceedings of the Royal Society of London, Series B, Vol. 266,
1999,pp. 2131-2136; Joshua B. Plotkin and Martin A. Nowak, "Language Evolution and Information
Theory," Journal of Theoretical Biology, Vol. 205, 2000, pp. 147-159.
166See, forexample, Thomas Schneider's website on information theory and molecular biology, URL:
http: //www. lecb.ncifcrf. gov/-toms/; David MacKay, Information Theory, Inference, and Learning
Algorithms, Cambridge University Press, 2003, URL:
http: / /www.inference.phy.cam.ac .uk/mackay/ i tprnn/book. html (Chapter19:WhyhaveSex?
Information Acquisition and Evolution).
166See, for example, Thomas M. Cover and Joy A. Thomas, Elements of Information Theory, John Wiley&
Sons, 1991(Chapter 15:Information Theory and the Stock Market).
167 Claude E. Shannon, 'The Bandwagon," IRE Transactionson Information Theory, March 1956,p. 3.
16eShannon, interview by Hagemeyer; Betty Shannon,interview by author; Fano, interview by author.
169 M. Mitchell Waldrop, 'Claude Shannon: ReluctantFather of the Digital Age," Technology Review,
July/August, 2001, p. 71; Slepian, interview by author.
170Sheldon Hochheiser, historian, AT&T Archives, personal communication, February 19, 2003.
171 Shannon, interview by Price.
172 Anthony Liversidge, "Interview: Claude Shannon. Fatherof the Electronic Information Age," Omni,Vol.
9, No. 11, August 1987, p. 66.
173Betty Shannon, interview by author.
174 Hermann Haus, Project Echo's website,op. cit., July 30, 2001.
175Haus, interview by author.
17 6
Lewbel, interview by author.
17' Betty Shannon, interview by author.
178 Ibid.
19 Slepian, interviewby author.
mo Betty Shannon, interview by author.
181Shannon,interview by Hagemeyer.
182 "Information Theory," Seminar Notes, March 1956,Shannon Papers, MIT Archives.
183Robert E. Kahn, "A Tribute to Claude E Shannon (1916-2001)," IEEE Communications Magazine, July
2001, p. 18.
84
Forney, interview by author.
'85 Gallager, interview by author.
186Leonard Kleinrock, professor of computer science, University of California at Los Angeles, personal
communication, June 5, 2003.
187 Andrew Viterbi, interview by David Morton, Oral History, IEEE History Center, October 29, 1999, URL:
http://www.ieee.org/organizations/history_center/oral_histories
transcripts/viterbi.html
188J. A. N Lee, J. McCarthy, and J. C. R. Licklider, "The beginnings at MIT," IEEE Annals of the History of
Computing, Vol. 14, Issue 1, 1992. pp. 18-30; MIT Laboratory for Computer Science, Timeline, URL:
http:/ / timeline.1cs.mit.edu/; Institute Archives & Special Collections, MIT History, "History of the
Laboratory for Computer Science (LCS)," URL:
http://libraries.mit.edu/archives/mithistory/
histories-off ices/lcs.html.For other accountson the origins and early days of computingand
networking, see Howard Rheingold, Toolsfor Though: TiheHistory and Future of Mind-Expanding
Technology, MIT Press, 2000 (a revised version of the original 1985 issue), or on the web, URL:
http: //www. rheingold.com/texts/tft/. Waldrop, op. cit.
189Shannon, interview by Hagemeyer.
190Claude Shannon, "Programming a Computer for Playing Chess," Philosophical Magazine, Series 7. Vol.
41, No. 314, March 1950,pp. 256-275 (reprinted in Sloane and Wyner, op. cit, pp. 637-656).
191Shannon, Kyoto Prize speech draft, op. cit.
76
192 Both cite Shannon in the acknowledgements of their theses: Leonard Kleinrock, "Message Delay in
Communication Nets with Storage," PhD dissertation, MIT, 1962, p. iv; Ivan Sutherland, "Sketchpad, A
Man-Machine Graphical Communication System," PhDdissertation, MIT, 1963,p. 4.
1
9 Edward O. Thorp, "The Invention of the First Wearable Computer,"°'
IEEE 2nd International Symposium
on Wearable Computers, 1998.
'94Boris Tsybakov, Project Echo's website,op. cit., June 20, 2001.
195 Kees Immink, Project Echo's website, op. cit., July 24 2001.
16 Shannon, Kyoto Prize speech draft, op. cit.
'97Horgan, "Profile: Claude E. Shannon," p. 75. In the symposium,McEliece introduced Shannon as "one of
the greatest scientificminds of our time." Shannon reacted saying,after the applause, "This is...ridiculous!"
(Anthony Ephremides, "Historian's Column," IEEEInformation Theory Society Newsletter, Vol. 51, No. 1,
March 2001, p. 1).
196Betty Shannon, interview by author.
1 Ibid.
2o Shannon, interviewby Hagemeyer.
201 The song included several footnotes. In the first one, Shannon wrote: "We are with Elliot and will freely
use footnotes to clarify and amplify our meaning." He wasreferring to T. S. Elliot's 'The Waste Land."
Published in 1922with a "wealth of footnotes," Shannonwrote, itcaused "considerablecommotion among
the critics."
20, Horgan, "Profile: Claude E. Shannon," p. 73.
203Lewbel, interview by author; See also Lewbel, op. cit., pp. 9-12.
ZX Fano, interviewby author.
05 For different accounts on Bell Labs' history, inventions, and culture, see Fagen, op.cit.; Millman, op. cit.;
Prescott C. Mabon, Mission Communications: The Story of Bell Laboratories, Bell Telephone Laboratories,
1975; Jeremy Bernstein, Three Degrees Above Zero: Bell Labs in the Information Age, Charles Scribner's
Sons, 1984;Narain Gehani, Bell Labs: Life in the Crown Jewel, Silicon Press, 2003.
20 BellLabsAwards,URL:http: //www.bell-labs. com/about/awards. html
Bell Labs Press Release, "Bell Labs receives its 30,000th patent," March 10,2003, URL:
http://www.lucent.com/press/0303/030310.bla.html
207 Kramer, interview by author. Feynman, by the way, would have run into Shannon had he not turned down
a job offer from Bell Labs in the spring of 1941 (JamesGleick, Genius: The Lifeand Science of Richard
Feynman, New York: Vintage Books, 1993. p. 137). One wonders what the two of them might have done
together.
77