2002 POSTS

JANUARY 2002

A Beautiful Portrayal and a Confused NYT Reviewer

This month, the movie A Beautiful Mind opened across the US. As I wrote in last month’s column, the film is inspired by the life of Princeton mathematician John Forbes Nash, winner of the 1994 Nobel Prize for Economics. In that column, I expressed my fear that mathematicians would complain about the historical inaccuracies and the misrepresentations the movie would inevitably contain. My fears were proved correct. Following a “Math Guy” piece I did about the film on NPR’s Weekend Edition on December 29 (right after I had seen the film in prelelease), the emails came in. The main objection was to my remark that, having been led to expect a fictional story bearing little relationship to the life of the real John Nash, I was surprised how well it captured in spirit that life—or more accurately, the understanding of Nash’s life I have from Sylvia Nasar’s excellent book of the same name, which all those who knew Nash have praised as being a good account of his life.

Well, why did I say that? First, let me begin by repeating here the statements made by film director Ron Howard and writer Akiva Goldsman that I quoted last time.

Here is what director Howard wrote:

“[The movie] captures the spirit of the journey, and I think that it is authentic in what it conveys to a large extent. Certain aspects of it are dealt with symbolically. How do you understand what goes on inside a person’s mind when under stress, when mentally ill, when operating at the highest levels of achievement. The script tries to offer some insight, but it’s impossible to be entirely accurate. Most of what is presented in the script is a kind of synthesis of many aspects of Nash’s life. I don’t think it’s outrageous.

We are using [Nash] as a figure, as a kind of symbol. We are using a lot of pivotal moments in his life and his life with Alicia as the sort of bedrock for this movie … even though we are taking licence, we are trying to deal with it in a fairly authentic way so that an audience is transported and can begin to understand. But they can’t begin to understand completely; they never could—no one could.”

According to writer Goldsman:

“[W]hat we’re doing here is not a literal representation of the life of John Nash, it’s a story inspired by the life of John Nash, so what we hope to do is evoke a kind of emotional journey that is reminiscent of the emotional journey that John and Alicia went through. In that sense, it’s true—we hope—but it’s not factual. For me, it was taking the architecture of his life, the high points, the low points, and then using that as a kind of wire frame, draping invented scenes, invented interactions in order to tell a truthful but somewhat more metaphoric story.

I think that to vet this by exposing it to historical accuracy is absurd. This movie is not about the literal moment-to-moment life of John Nash. It’s an invention … What we did is we used from his life what served the story we are trying to tell, which is why we are saying this is not a biopic. It could never bear up to that kind of scrutiny, it never wanted to, it never pretended to be a biopic. It always wanted to be a human journey, based on someone, inspired by someone’s life.”

How well did the film match up to those intentions? Pretty well, I would say. No, more than that, extremely well. I think they delivered exactly what they set out to do, and moreover, did so in great form. So what’s the problem?

Well, according to the movie review in the New York Times on December 21, “The paradox of Ron Howard’s new film, from a script by Akiva Goldsman, is that the story … is almost entirely counterfeit.”

Well, duhh! Just fancy that. A fictional story that is entirely counterfeit. Whatever will they think of next!

Why on earth anyone should expect this film to be true to some particular person’s life is beyond me. It’s a film, after all. True, it was inspired by real events and uses some real people’s names, but fictionalized novels, plays, and films involving “real” characters are hardly a new phenomenon. Does the New York Times reviewer find him (or her) self unable to enjoy Shakespeare’s “historical” plays because of all that fake dialogue?

But the NYT reviewer is not alone in wanting the truth, the whole truth, and nothing but the truth (gosh, what a dull movie that would be). As I mentioned already, following my NPR interview I’ve had a number of emails from mathematicians pointing out the historical inaccuracies and omissions from the film, and lamenting insufficient emphasis on the actual mathematics. I think the above quotes from Howard and Goldsman address the first complaint, and as for the second, I think we can gauge the overall level of interest in mathematics among the general population by the number of students who sign up for mathematics majors, which in recent years has dwindled almost to the point of vanishing. But the film is not about mathematics, and nor is this article. So, in true movie-Nash form, let’s cut out the foreplay and come straight to the point.

First, let’s be clear what this is. It’s a film. That alone is too imprecise. It’s like saying something is a book. Sure, but is it Tolstoy, Ed McBane, or D. G. Bayer’s book on Calculus of Several Variables? (Hands up all those who have seen that classic text?)

A Beautiful Mind is a Hollywood blockbuster film, with a big budget, an accomplished director, a major star (Russell Crowe plays Nash), and a strong supporting cast of leading Hollywood actors (including Jennifer Connelly as Nash’s wife Alicia, Christopher Plummer as the psychiatrist who treats him, and Ed Harris as the mysterious DoD agent Parcher).

This is a very different medium from an art film made for a select audience, or from a film biography designed to capture someone’s life on celluloid. Sure the technologies they use are all similar (though less so than one might think), but then, at the technology level, all books are more or less the same.

A Hollywood blockbuster is meant to have wide appeal. It has to tell its story in a way that is powerful, compelling, and effective on a very large screen, with a great sound system. In this case, the audience can be expected to know nothing about mathematics (and I mean nothing, other than that it’s “all about numbers”), about John Nash, or about what life was like in a leading research university in the 1950s.

Action movies aside, the main tools at the disposal of the Hollywood blockbuster director are visually powerful scenes, actors who can convey inner feelings with their eyes, their facial expression, and their body language—often when their face or just a part of their face fills the entire screen—crisper-than-life sound, and music. The genre works primarily at the level of human emotion. Any information conveyed has to be carried on the back of the emotional delivery system. The viewer has to bring the usual suspension of disbelief that goes with any form of fiction, and be prepared to be taken on the journey the director has created. (Incidentally, I have yet to see a convincing explanation of why we find watching movies or plays—watching other people pretending to be imaginary people, doing imaginary acts—an enjoyable activity for which we are willing to spend both time and money.)

The story Goldsman and Howard set out to tell is really classic: Two people fall in love and have to overcome incredible obstacles until they get to the happy ending. In this case, the man’s brilliant start to life is derailed when he succumbs to severe paranoid schizophrenia. When, in part due to the development of new drugs, his illness goes into remission, he has to come to terms with the fact that much of his own reality for the best part of his adult life has been illusory. The woman, the young graduate student whom he marries, sticks by him throughout it all to the story’s relatively happy end. (In real life, I always felt that the real hero of the real John Nash story is Alicia. Although they divorced when his illness was at its worst and she had a child to care for, she remained by him throughout his ordeal. The two remarried last summer.)

The device Howard uses to convey to the viewer the essence of the Nashes’ ordeal is to put the audience into Nash’s own predicament, inside his own mind. Much of what you see is not “real”, it’s a Nash illusion. Some of it is clearly of this nature (although some viewers seem to have missed that point, including the hapless New York Times movie reviewer, of whom more later). But other parts are not, and unless you are quick enough to spot the clues Howard puts in (and in many cases I only noticed them the second time I saw the film, when I knew what was coming later), you are surely going to be surprised at the way some of the later action unfolds.

Then, at the moment when Nash himself becomes aware of his predicament, the audience too has to start to think back, and ask, “Gee, just how much of what I’ve just seen was intended to be “real” and which parts were “Nash-hallucinations”?” Nash is not sure, and neither are we, the audience.

One of the plot lines Howard clearly intended the audience to see as taking place inside Nash’s mind right from the start were the scenes relating to Nash’s supposed defense work, particularly the scenes at the Pentagon and later with the Department of Defense, including his interactions with the mysterious Parcher. The entire tenor changes so dramatically for those scenes, from “reality drama” to over-the-top melodrama, that there should be no doubt that this is all going on inside an increasingly deranged mind. (The melodrama in those segments becomes greater as the movie progresses.) So, I don’t think I’m giving anything away to those of you who have not yet seen the movie if I base the rest of this article on just that part of the story. There are still plenty of surprises in store that Howard clearly meant to keep hidden from us until the right time.

Surprisingly, not everyone who has seen the film seems to have realized what Howard was up to. Including some who one would have thought were past masters at understanding the medium of film. (Heavens, I’m just a simple mathematician who goes to movies purely for relaxation!)

Now, it is not the normal practice of Devlin’s Angle to engage in intellectual sparring. But the self-appointed, self-important arbiters of artistic taste who grace the arts pages of the New York Times are surely fair game. So when one of them sets him (or her) self up with full grandeur and intellectual snobbery, and then falls flat on his (or her) face, making a complete fool of him (or her) self in print, the temptation is too good to pass up.

The reviewer in question is one A. O. Scott, (New York Times, December 21). Having berated the movie makers for not crediting the viewing audience with intelligence—always a dangerous ploy unless you are completely sure you have got everything right—Scott shows s/he has missed one of the main, pivotal, but hardly subtle, points of the movie. Alas, poor Scott is not alone in not realizing that the cold war scenes in the movie were all taking place in Nash’s increasingly deranged mind. Scott writes:

More than a few mathematicians and scientists at the time, including many at M.I.T., where Nash went to teach after Princeton (not, as the film has it, to conduct top-secret defense-related research), were sympathetic to Communism, and many more … were suspected of such sympathies. While Mr. Nash was not among them, he was hardly the intrepid cold warrior depicted by Mr. Howard and Mr. Goldsman.

Wake up, A.O. Scott! Didn’t you notice how the entire appearance and tenor of the film changed the moment Nash walked into the Pentagon? Maybe the movie-Nash really did go to a Pentagon meeting. That is left open. But what we see—Nash uncharacteristically well groomed and dressed in a smart, dark suit, poised and self-assured, looking people straight in the eye, without his ever-present facial twitch, and those over-the-top military men, that science-fiction-like set, with an illuminated wall display of numbers that could not have been created in the 1950s when the early part of the movie took place—that is all clearly meant to be going on inside Nash’s mind. And what about the dark, sinister Parcher, played to look the classic G-Man of cheap fifties fiction movies? Or the melodramatic car chase ending with the bad guys driving into the river? “This is not real!” director Howard was yelling. But his cries fell on deaf ears at the New York Times. (My reference to Howard yelling just then was metaphorical. Thought I’d better add that for The Times reviewers.)

Of course, there were more subtle clues. For example, whenever Nash looks out of his MIT office door, he (and we) see MPs guarding the entrance. But when we see anyone else open a door and look out at the same corridor, it’s empty—just like a typical MIT corridor in fact. Hmmm. I wonder what that can mean?

When Parcher appears on the steps of MIT’s Wheeler Lab to talk to Nash, he does just that: appears. We’ve just seen Nash leave by that door, and no one else was around, and then, presto, there is the mysterious Parcher, standing on the top step, clearly not having come through the door. (Even more subtle, Parcher’s voice initially comes from the left of the screen, where Nash is standing, whereas Parcher appears toward the right. Nash hears the voice in his mind before he creates the image, and it takes a moment for his mind to match the two.)

There are a number of other such clues that Howard has created to make sure no one misses this major artifact of the movie.

Despite laying on the clues thickly with a large trowel, I guess Howard must have been worried that some viewers would still not “get it.” Quite late in the film, Alicia (Nash’s wife) visits Nash’s two mathematical colleagues at MIT and asks them if her husband had been doing any military work. It is possible, one of them says skeptically, since MIT does after all do some military work, but they (Nash’s two closest colleagues, remember) had never done any work of that nature and they were pretty sure Nash hadn’t either. Indeed, they had been so sure something was wrong that one of them had trailed Nash on one of his self-supposed secret assignations. The implication is clear: it had all been taking place in Nash’s head.

Interestingly, the viewer is left unsure about the reality of certain parts of the film. Whether this was intentional or not on the part of the makers, I don’t know, but that hardly matters since the effect is there. For example, we are never sure whether the movie-Nash goes to MIT simply to teach and do pure, non-defense-related research, as the real Nash did. In which case, the initial discussions Nash has with his Princeton advisor about strategically important work is also part of his mental illusions. Or is that part supposed to be “real”? I found that this lingering feeling of never knowing exactly where the border between reality and hallucination lies is one of the most powerful effects of the film, and surely a strong indication of what life must be like all the time for the real John Nash.

The mathematical aspect of this movie is, of course, why an MAA Online columnist is writing about it. But the mathematics is incidental—although writer Goldsman does weave it in to the mental illness theme. It’s about two people’s struggle with the severe mental illness of one of them. Personally, I can think of no story that has greater emotional content. And arguably the best creative medium we have to convey an emotion story—an emotion-laden human journey—is the large-screen, Hollywood blockbuster movie. (The fact that 90% of movies coming out of Hollywood are pure crap is surely irrelevant. Around 90% of books are artistic junk too, though if people find them entertaining that surely gives them value.) With A Beautiful Mind, I believe Howard and Goldsman have used the medium to its best power to produce a masterpiece of a film of that genre.

But, of course, it’s up to everyone to decide for themselves. This is a personal column, not a scientific journal, and my opinions are just my opinions. I should perhaps add that I had nothing to do with the making of the film, I do not know anyone who was involved (apart from a very brief email exchange with Dave Bayer who was the math consultant for the film) and have no stake in its success or otherwise. I do have a professional stake in the overall growth of the public understanding of and appreciation for mathematics, and for that reason have been interested in how the producers would deal with the Nash story, and how the general public will receive it.

Devlin’s Angle is updated at the beginning of each month.

Mathematician Keith Devlin ( devlin@csli.stanford.edu) is the Executive Director of the Center for the Study of Language and Information at Stanford University and “The Math Guy” on NPR’s Weekend Edition. His latest book is The Math Gene: How Mathematical Thinking Evolved and Why Numbers Are Like Gossip, published by Basic Books.

FEBRUARY 2002

The Math of Online Music Trading

The High Court may have put Internet music trading company Napster out of business, but illegal swapping of music files on the web continues unabated, with companies like Audiogalaxy, Kazaa, Morpheus, and Winmx filling the void that Redwood City, California based Napster left behind.

Yet how many teenagers who clog the phone lines transferring the latest pop songs realize that the entire enterprise is built on mathematics—that what they are really downloading are streams of numbers, computed using a calculus-based technique first developed in the early nineteenth century?

The mathematics used to convert music into numbers has its origins in the work of the French mathematician Joseph Fourier. Although his interests were in heat dispersal, the technique he discovered enables mathematicians to represent any wave form as a sequence of numbers.

Fourier showed how any function x(t) of time that is periodic (i.e., repeats itself indefinitely at regular intervals of time) can be represented as the sum of an infinite series of sines and cosines. Precisely, if T is the interval at which x(t) repeats itself (so x(t+T) = x(t) for all t), and if we set f = 1/T (f is called the base frequency of x), then x(t) is equal to a constant a(0) plus all terms of the form

a(k)cos(2.pi.k.f.t) + b(k)sin(2.pi.k.f.t)

as k runs from 1 up to infinity. This infinite sum is called the Fourier expansion of the function x(t). By virtue of the Fourier series, the sequences of numbers a(k) and b(k) are uniquely associated with the function x(t). Given a reasonably nice formula for the function x(t), it is possible to compute each of the constants a(k), b(k). The mathematical process that carries out the computations of these constants is called a Fourier Transformation. (These days, mathematicians use a computational (as opposed to analytical) analog of the Fourier Transform called the Discrete Fast Fourier Transform, which produces numerical approximations as accurate as desired.)

There is, of course, no possibility of actually computing all of these numbers, since there are infinitely many of them. However, a finite number of them will suffice to give a finite sum of terms of the Fourier series that represents x(t) within any desired degree of approximation. In the case where x(t) is the wave form of a sound wave, this provides a way of coding sound as a sequence of numbers. At least, in principle it does. An obvious question is how do you go from a sound wave to a mathematical formula x(t) that represents it in a way that lets you apply Fourier’s analysis?

A sound wave consists of a ripple in the air. What makes it sound is that our ears and more generally our hearing system interpret that air wave as sound. To give a bit more detail, the motion of the air causes a skin membrane in the inner ear to vibrate, and those vibrations are converted into tiny electrical currents that flow into the brain. It is those electrical waves that the brain actually experiences as sound. In other words, when it comes down to it, sound is ultimately an electrical wave.

A microphone works in essentially the same way, converting an incoming sound wave in air into an electrical signal. If we feed that electrical signal into a loudspeaker, then the speaker recreates (a copy of) the original sound wave. But we can also do something else to that electrical wave: we can use a method known as sampling to generate a sequence of numbers. The most common procedure is called Pulse Code Modulation (PCM). This takes as input an electrical wave and measures the voltage of the signal at moments of time a small, fixed interval apart. In the case of an audio compact disk, the sampling is done 44,100 times a second. Thus, for each second of sound input, the PCM analog-to-digital converter generates 44,100 numbers, each one the measurement of the voltage at the instant it is sampled.

In the case of a compact disk, each voltage is measured to 16-bit accuracy; that is, the system can distinguish up to 65536 (= 2¹⁶) different voltages. A sample rate of 44,100 per second coupled with 16-bit voltage measurement is sufficient to encode any sound as a sequence of numbers that, when converted back into sound, the human ear cannot distinguish from the original.

However, it takes a lot of storage capacity to capture even a three minute pop song in this fashion. A typical musical compact disc carries a stereo sound signal, each sampling measures two voltages, one for each of the two stereo sound sources, and each has 16-bit capacity, so each second of audio generates 2 x 16 x 44100 = 1,411,200 bits. Hence, a minute of CD quality music requires a massive file of 10 megabytes. Given modern compact disk technology, this is fine for the recording industry and the CD industry, but would create a major problem if people started shipping CD music files around the Internet.

Anyone with a modern desktop PC is aware that there are algorithms that can compress binary data files so they require less storage space. (PK-ZIP and Stuffit are two well known examples.) When applied to a typical text file, these packages can reduce the size of the file by as much as 80%, but with CD quality PCM files the reduction is only around 10%. Algorithms specially designed to operate on PCM files have managed to achieve 60% reduction, but that is clearly nothing like enough to support Internet music swapping. The WAV compression system familiar to Microsoft Internet Explorer users is essentially a ZIP compression of a raw sampled audio wave.

The key to shipping music files over the Internet is to abandon the idea of compressing the entire digital file so that the original sampled sound wave can be reproduced exactly (so-called lossless compression) and instead deliberately discard some of the information (what is called lossy compression). The folks who develop these algorithms begin by understanding how the human hearing system works. The aim, after all, is to produce a digital waveform that, when played back through a good loudspeaker system, sounds like the original sound. Thus, anything in the original sampled sound wave that the human hearing system cannot detect may be discarded. Regardless of how good we think our musical ear might be, it turns out that there is a lot of stuff that can be thrown away without our noticing.

A significant saving of storage space is a consequence of the phenomenon known as audio masking. This is where we hear two sounds at different energy levels at nearby frequencies. What happens is that one sound obscures the other. Hence, in coding the sampled signal, the lower energy component that is masked can be ignored. In practice, this is done as follows. It appears that the way our auditory perception works, incoming sound can be divided into separate bands, arranged logarithmically, with more bands occurring at lower frequencies. Frequencies in one band do not interact with those in another band, and our auditory system can either attend to all the bands at once or can select certain bands to pay attention to, such as the bands where the frequencies of speech occur. Within a given band, however, frequencies at significantly greater volumes tend to obscure any frequencies of lower volumes, and lower frequencies will tend to mask higher ones. Within each band therefore, masked information may be discarded.

By far the most familiar form of audio encoding (lossy compression) in use today—and the base of both a huge industry and an even bigger illegal music trading network—is MP3, which is short for MPEG-3, or even more fully MPEG – level 3. MPEG is a set of industry standards created and managed by the Moving Picture Experts Group (MPEG), which is a working group of ISO/IEC (International Organization for Standardization/International Electro-technical Commission) in charge of the development of standards for coded representation of digital audio and video. Established in 1988, the MPEG group has produced MPEG-1, the standard on which Video CD and MP3 are based, MPEG-2, the standard for Digital Television set top boxes and DVDs, MPEG-4, the standard for multimedia for the fixed and mobile web, and MPEG-7, the standard for description and search of audio and visual content. The most important differences between these standards are data rate and applications. MPEG-1 has data rates on the order of 1.5 Mbit/s, MPEG-2 has 10 Mbit/s, and MPEG-4 has the lowest data rate of 64 Kbit/s.

The original MPEG standard was divided into three levels: Level 1: System aspects; Level 2: Video compression: Level 3: Audio compression. MPEG Level 3, or MP3 as it is now widely known, was developed in 1992 by the German Frauenhofer Research Institute, and is part of the MPEG-1 and MPEG-2 specifications. It achieves a spectacular compression ratio of a sampled audio wave, ranging from a factor of 8 to a factor of 12, depending on the source. This means that the 10 MB of storage capacity to encode 1 minute of hi-fi music on a compact disk is reduced to 1 MB on a computer hard drive. Not surprisingly, the MP3 process is patented, with Thompson Multimedia holding the patents in the USA and Germany, although that has not prevented a proliferation of MP3 players being available for free download on the Internet.

MP3 divides the frequency range into 32 bands. The component of the input signal (sampled wave) in each those ranges is then subjected to a modified discrete cosine transformation that separates it into a further 18 constituents, generating a total of 576 individual frequency bands. It is within those bands that redundant (i.e., masked) components are removed. The resulting signal is then compressed further by Huffmann coding, a technique familiar to computer scientists, which represents frequently occurring values by shorter codes than used for less frequently occurring values. (For instance, it would be highly wasteful to use the default 141,120 bits of the sampled wave to encode a 1/10 second silence in a song.)

With consumer electronics stores offering new MP3 players every few months, and with millions of PC owners swapping music files illegally (as well as occasionally downloading them legitimately), to say nothing of the huge numbers of musical greetings that will be zapped across Cyberspace on Valentine’s Day, the modern music industry is clearly built on mathematics as much as anything else. One wonders what Joseph Fourier would have made of today’s applications his original mathematical analysis of waves.

Devlin’s Angle is updated at the beginning of each month.

MARCH 2002

The Shrimp and the Mathematician

The next time you dip that succulent white shrimp into the picante sauce, remember that your tasty snack is one of Nature’s marvels: a single muscle—some 40% of the total weight of the creature it belongs to—designed by nature for just one purpose: the creation of massive acceleration to propel the creature out of danger with a sudden explosive thrust that, pound for pound, makes it the envy of any Olympic sprinter.

Motion—from the beating of a heart to the familiar task of getting from point A to point B—is the essence of life. Consider: A basketball player running at full speed suddenly stops, pivots on one leg, takes two steps in another direction, then launches himself high into the air to score a basket. A fish, motionless in the water one moment, catches a sudden movement in the corner of its eye and, with a barely perceptible flick of its tail, darts off rapidly into the safety of the reeds. A cockroach scuttles across the kitchen floor to escape the sudden illumination from the overhead light you just switched on. A cormorant glides silently and elegantly above the ocean until it spots a fish in the water beneath, whereupon it suddenly swoops down in a rapid dive to secure its prey.

Evolution has equipped all living creatures with a way to move—to search for food, to seek out a mate, or to escape from danger. People and ostriches walk and run on two legs, horses and dogs on four, cockroaches on six, spiders on eight; snakes slither; fish propel themselves by pushing the water sideways with their tail; birds fly by flapping their wings to create lift and forward thrust.

How do they do it? How do the creatures that inhabit the land, the sea, and the air move? You can get some idea of the difficulty of this question from the fact that, after fifty years of well-funded research into the construction of computer-controlled machines, no one has yet been able to build a robot that can walk well on two legs. In fact, the best four- or six-legged robots do not perform anything like as well as the average dog or dung-beetle. Only the invention of the whee—thousands of years ago—has enabled Man to build efficient transportation machines. When it comes to building machines that imitate the ways Nature solved the locomotion problem, we’re still in Kindergarten.

Yet all motion comes down to just two physical principles, identified by Isaac Newton 350 years ago. One is that motion results from the application of a force. (Force = mass x acceleration.) The other is that every force produces an equal and opposite reaction. The great variety of locomotive strategies that we see around us comes not from different principles of motion but from Nature’s boundless ingenuity in finding ways to apply Newton’s two physical laws. Only in recent years have scientists started to understand how Nature achieves this feat, often with enormous ingenuity. For example, in the article How Animals Move: An Integrative View, by Michael Dickinson, Claire Farley, Robert Full, M.A.R. Koehl, Rodger Kram, and Steven Lehman, published in SCIENCE 288, 7 April 2000, pp.100-106, the authors point out that the old idea of a central brain directing the actions of all the muscles involved in motion is not at all accurate. Rather, the control mechanisms that govern movement are distributed throughout the organism, in many cases embedded in the design of the individual moving parts.

Other motion is a result of self-organization, where a collection of organisms, sometimes just single cells, somehow mange to communicate with each other to coordinate their activities to produce a cohesive motion of the entire collection, as if it were a single creature. (See the recent book Self-Organization in Biological Systems, by Scott Camazine, Jean-Louis Deneubourg, Nigel Franks, James Sneyd, Guy Theraulaz, and Eric Bonabeau, published by Princeton University Press.)

In all cases, arguably the most basic question about motion is, how do the movements of single cells combine to produce the motion of the entire organism or collection of which those cells are parts? This is where mathematics can help.

To appreciate the problem, think of the problem facing a group of dancers who have to coordinate their individual movements to give a pleasing performance, a football team who must act in unison to score a touchdown, or the musicians in an orchestra who must produce a perfect symphony. Each may have its leader – the dance choreographer, the football quarterback, or the orchestra conductor – but at the most basic level it’s the communication between the individual performers that fuses their separate actions into a single, recognizable whole. So too with all movement of living creatures.

But how exactly is the communication and the resulting coordination achieved? In recent years, collaborations between biologists and mathematicians have started to provide answers, often with some surprising and tantalizing twists, as Dr Angela Stevens informed the audience in the special symposium on mathematical modeling of animal and plant movement at this year’s AAAS meeting in Boston last month. Dr. Stevens, who works at the Max Planck Institute for Mathematics in the Sciences in Leipzig, Germany, has been using mathematics to study movement of self-organizing systems of cells. Among the curious behaviors her research had uncovered is that, sometimes, just before a group of communicating individual cells achieve perfect coordination, they generate a recognizable traveling wave pattern, not unlike the ripples that move through a line of heavy traffic on the freeway. Presumably the individual cells are communicating with each other, but how exactly are they doing so, and what produces the ripple?

Turning from the very small to the very large, mathematics has also proved useful in understanding how particular tree species propagate across a geographic region. Recent work by Mark Lewis of the University of Alberta resolves a conundrum known as Reid’s Paradox: the fact that sometimes a new species of tree will spread at a rate that in botanical terms seems impossibly fast. The solution to this puzzle came not from biology but mathematics. Lewis showed how the role of chance can lead to extremely rapid plant migration.

Long recognized as a powerful tool in physics and engineering, mathematics is now finding increasing application in the biological and life sciences, often with remarkable results. The AAAS symposium in which Stevens spoke, which was organized by mathematician Hans Othmer of the University of Minnesota, gave several tantalizing glimpses of this exciting new area of scientific study—the marriage of biology and mathematics—that is helping us to understand one of the greatest of all scientific mysteries: life itself.

Other mathematics related symposia at this year’s AAAS Meeting included one on the applications of Social Choice Theory to biology and another on wave patterns and turbulence, a topic much in the news following the crash of a jet airliner shortly after takeoff from New York’s Kennedy Airport last fall.

Mathematicians whose national conference attendance is limited to the Joint Mathematics Meetings each January would be well advised to consider going along to next year’s AAAS meeting in Denver, Colorado. Here are two good reasons. First, the mathematics talks are all designed to appeal to a wide audience of scientists, science buffs, and science journalists. That means they have a different flavor from most presentations at a mathematics conference. Second, in addition to the mathematics talks, you can wander around and enjoy a veritable smorgasbord of talks on topics in science (both natural and social), science policy, and science education.

Actually, I can think of a third good reason to attend next year’s AAAS meeting: The Colorado Rockies are just an hour’s drive away from the Denver location of the conference.

NOTE: This month’s column is adapted from a promotional article I wrote under commission from the AAAS, with financial support provided by SIAM. I am grateful to Warren Page, Secretary of Section A (Mathematics) of the AAAS for encouraging me to write that initial article.

Devlin’s Angle is updated at the beginning of each month.

APRIL 2002

Mathematics and Homeland Security

During the 1950s and 60s, the United States poured millions of dollars into mathematics research as part of the national effort to fight (or at least to avoid losing) the Cold War. This came on the heels of the crucial, and successful, role mathematicians played during the Second World War.

Today the United States finds itself in a new war, an international War on Terror. Since the opening salvo in this new war was launched on the continental USA, and because the next attack could likewise take place in America, Homeland Security is a high priority in the new struggle. And once again, the US is looking to the mathematical community to assist in the conflict.

As part of the mathematical profession’s initial response, later this month (on April 26-27) the National Academies’ Board on Mathematical Sciences and their Applications (BMSA) is holding a two-day, invitational workshop on The Mathematical Sciences’ Role in Homeland Security, hosted by the National Research Council in Washington, D.C. The aim is to bring together leading experts in the various areas of mathematics that are likely to be required in fighting international terrorist organizations, with a view to setting a national research agenda to aid the country in combating this new kind of warfare.

Mixing with mathematicians from universities, industry, and national laboratories at the workshop will be senior representatives from the Defense Advanced Research Projects Agency (DARPA), the National Security Agency (NSA), the Centers for Disease Control and Prevention (CDCP), the Directorate of Defense Research and Engineering, and of course the Office of Homeland Security.

The topics to be discussed fall into five general (and overlapping) areas: Data Mining and Pattern Recognition, Detection and Epidemiology of BioTerrorist Attacks, Voice and Image Recognition, Communications and Computer Security, and Data Integration/Fusion.

Many if not all of these areas are unfamiliar to most mathematicians, and they are quite different from the kinds of mathematics that were required to fight wars in the past, hot or cold. Statistical and computational techniques figure heavily in this new kind of strategic mathematics.

Data Mining and Pattern Recognition looks for ways to discover patterns, structure, or associations in large bodies of empirical data, such as financial or travel records. Much of the early research in this area was developed for industrial and commercial purposes, for instance, by banks to detect credit card fraud, by telephone companies to spot unauthorized use of the phone system, and by supermarket chains to identify purchasing patterns. (Why do you think they have those electronically readable “store membership cards”?) The relevance of this area of research to homeland security is obvious.

Detection and Epidemiology of BioTerrorist Attacks involves a number of lines of mathematical research. The development of mathematical models of how diseases spread is perhaps one of the most well known examples – well known in part because simple scenarios form standard examples in calculus classes. In fact, within days of the September 11 attacks on the World Trade Center towers and the Pentagon, researchers at Los Alamos National Laboratories had taken a mathematical model of traffic flow they had been developing and applied it to predict the likely spread of disease following a possible bioterrorist attack. There is significant scope for further research into the mathematics of how biological and chemical agents spread.

Another area where mathematics will be important in countering a biological or chemical attack is in early detection that such an attack has in fact taken place. In the early stages, it can be hard to differentiate between a malicious attack with a dangerous weapon and a naturally occurring outbreak of a common agent. The available data is almost always noisy, creating a need for better techniques to integrate and fuse data to identify patterns, determine sources, increase confidence, and predict the spread of infectious or chemical agents, in order that the available counter agents of containment methods may be brought to bear in the most timely and efficient fashion. As the science of patterns, mathematics may turn out to be one of the main weapons in the nation’s arsenal in fighting this new kind of war.

Voice and image recognition: Today’s terrorists operate globally, maintaining contact by telephone and the Internet. Identifying the occasional key telephone conversation among the millions that take place daily can only be done (if it can be done at all) using sophisticated automation, with monitoring systems that are able to break down voices and words into digital patterns that can be scanned for keywords. This requires the development of new algorithms to monitor communications channels in real time to provide the nation’s defense authorities with early warnings of a potential threat. Similarly, methods need to be developed for the automated screening of images sent over the Internet, to look for messages embedded in pictures (steganography), a technique believed to have been used by the September 11 terrorists.

New and more sophisticated mathematical techniques for image processing and recognition will also be required to identify potential terrorists involved in suspicious activities and to improve screening at airports and other checkpoints.

Communications and Computer Security: Most mathematicians are familiar with the basics of cryptography. This, after all, is one of the areas where mathematics played a major role in the Second World War. But with secure encryption systems now widely available to security forces and terrorist alike, the focus has shifted elsewhere, to the overall integrity of communication and computer systems. A secure cryptosystem becomes worthless if an enemy can break into your computer or disrupt the network. There is thus a pressing need for taking a broad look at computers and computer networks to examine their vulnerabilities and develop ways to defend them, including early detection of an attack. New methods for analyzing Internet traffic are likely to be important in this new area of cyber warfare.

Data Integration/Fusion is the process of synthesizing information from diverse sources in order to make prudent decisions. At present there is little by way of a reliable mathematical framework to support this kind of activity. Current practitioners make largely ad hoc use of statistics, probability, decision theory, graph theory, and tools from artificial intelligence and expert systems design. The relevant parts of these disciplines need to be merged into at least a compatible toolkit, if not a coherent theory. I know first hand from my own attempts over the past twenty years to come to grips with information representation that there are enormous theoretical challenges to be overcome in order to make progress in this now crucial area.

The Washington workshop is not going to provide answers to any of the pressing questions that need to be answered. That is not the purpose. As with the war on terrorism itself, we are in the early days of what will certainly be a long haul. The workshop is intended merely to draw up a roadmap of where we want to go and how we might get there.

Much of the work that has to be done will not be “hard, elegant” mathematics of the kind that many mathematicians (myself included) view as a thing of beauty. (Although history tells us that there is a high probability that this effort will lead to such mathematics as an unintended side-effect.) Consequently, there are likely to be few public rewards or accolades for those who choose to engage in such projects. But it is work that can only be done by mathematicians. Such was the case with the part played by mathematicians in previous conflicts. Now, as then, I doubt there will be any shortage of willing volunteers.

NOTE: The April workshop is by invitation only.

Devlin’s Angle is updated at the beginning of each month.

MAY 2002

Randomness at the Airport

During the past two months, I have made 20 domestic flights on commercial airlines. For seven of those flights, that’s just over one in three, I was pulled from the line of passengers waiting to board the plane and subjected to the purportedly “random” additional body and baggage search. (Ironically, the last occasion was when I was about to board my flight back from Washington, D.C. to San Francisco, on my way home from the Board on Mathematical Sciences and their Applications two-day conference on Mathematics and Homeland Security.)

Now, I’ll tell you, with data like that, I am sure that, whatever the instructions to the airport security personnel are, those additional searches are not carried out in a random fashion. If they were, then on every flight an average of 80 people would be stopped and searched as they boarded the plane, and no plane would take off on time.

My guess (and my hope) is that the instructions are to check every passenger that fits the very obvious profile of a potential Al Qaeda hijacker, and check sufficient additional passengers to avoid any hint that passengers are being profiled.

Of course, no one is supposed to admit that this is going on, but the authorities would be stupid not to act upon what we know about terrorist risks. In the first rounds of the US War on Terror, we did not after all drop bombs on Omaha, or even many parts of Afghanistan; rather we concentrated our military efforts on those parts of Afghanistan where Al Qaeda terrorists were known or believed to be. Likewise, we should concentrate our necessarily limited airport security efforts on identifiable sections of society where the risk is higher. And that means profiling. In the case of Al Qaeda suicide terrorists—which is after all the real threat—such profiling has unavoidable racial, cultural, and religious components, and that is unfortunate. (It’s not just racial and religious, however. Terrorists fit other, behavioral profiles as well.)

Now, I don’t like the idea of profiling that has a racial or religious element any more than the next person. Thus, I regard random searches as an arguably necessary nuisance to ordinary airline passengers, a relatively small price to pay in order to avoid overt profiling. (Although I realize that by the same logic we should drop the occasional smart weapon on US cities, just to show we are not singling out any particular group.) And were those additional searches of outside-the-profile passengers truly random, I would have no problem with them. But as my own experience indicates, they are not. (True, I’m a sample of 1, but I fly a lot, so I think the data has significance.)

Now, one of two things could be going on here. First, it is possible that I do in fact fit some profile. I am male, dark-haired, fairly fit and athletic, I always travel light (hand luggage only), and living in California gives me a permanently tanned skin. To be sure, I don’t think I look remotely like any of the September 11 hijackers, nor like any other airline terrorist whose picture I have seen. But I cannot rule out the possibility that there is something about me that raises the suspicions of the security personnel. (By the way, I have tried making eye contact with the security personnel, both with and without a smile, and avoiding eye contact altogether. It does not appear to have made any difference. As we shuffle forward toward the gate, the guard inevitably walks up to me anyway.)

On balance however, I don’t think any kind of conscious profiling is going on. Rather, I think the issue is a psychological one. The security personnel at the gate do not use a random number generator to select passengers to check. Rather, they stand by the gate and approach people. Now, the last couple of occasions they have chosen me, I’ve asked them what criteria they used, and both times I was told it was random. I am quite willing to accept that they believed this to be the case. But as anyone who has taught (or taken) an undergraduate course on probability theory will know, humans simply cannot make genuinely random selections, even when choosing abstract objects such as numbers. (Most people are surprised to discover that their attempt to write down a random sequence of numbers is inevitably far from random.) And as numerous studies in sociology and psychology have shown, when it comes to selecting people “at random” from a given population, we are far worse. All kinds of subconscious influences bear on our choice.

I like to think that the reason I keep getting pulled out is that I look like the kind of person who won’t cause the security personnel any trouble. We all like to avoid trouble if we can. The security personnel are humans, just like everyone else. Indeed, the majority are immigrants (as am I), whose poor English suggests they are fairly recent immigrants, and moreover the job is a low paying one, so they surely start with a tremendous social disadvantage in having a policing role over obviously affluent, jet-setting professionals with expensive laptop computers and first-class airline tickets. It would be highly surprising if they did not, at least subconsciously, try to minimize the possibility of conflict.

But just think about this. What it means is that the security personnel are concentrating much of their efforts on the people least likely to be terrorists. That’s fine—albeit a bit unfair on us mild mannered types—if the “random” searches are purely to hide the profiling that I hope is being done. But if, in addition, there is no profiling, then the system is badly flawed.

If there is no profiling been done, then with the “random” searches concentrating on the “good guys”, the present system is just a sham, and a dangerous one at that. On the other hand, if there is profiling, and the “random” searches are just a feelgood addition to feed our admirable need to appear free from prejudice, then by not making them truly random, and allowing them to focus on the “good guys,” we miss an excellent opportunity for a genuinely useful second line of defense. Either way, the system is flawed.

I admit that it is possible to cast doubt on much of what I say. Only a systematic scientific survey, carried out using rigorous statistical techniques, could really decide the issue. But as a mathematician I do have a good understanding of randomness, which is a mathematically precise concept. What is, I am sure, clear beyond any doubt, is that those supposedly (and possibly even intendedly) “random” gate searches are anything but. As a consequence, the security benefits that truly random searches could bring are not being taken advantage of.

Moreover, if what I suggest above is correct, then a genuine terrorist could improve his or her chances of taking a weapon onto a plane by lining up just behind two or three of the “safe” passengers (like me) who are likely to be pulled out for extra screening, leaving the screeners fully occupied with harmless passengers while the terrorist walks onto the plane.

A possible solution? In Mexico, when an arriving international passenger gets to the customs control, he or she is asked to press a button, which causes a red or green light to illuminate. If it shows green, the passenger can proceed; if it shows red, a luggage search follows. Now, I dare say that the customs officers can ensure that a red light comes up. (As with security screening, it would be crazy not to make use of human expertise to spot, in this case, potential smugglers.) But I assume that the reason for having the automated system is so that any additional random searches are just that: genuinely random. A similar system could be implemented at the nation’s airports. (It could be done by software alone, with a mark on the boarding pass.)

As mathematicians know full well, randomness, when used wisely, is a powerful tool. It could be a tremendous ally in ensuing airline security. But it has to be genuine randomness. With the best will in the world, people cannot make random choices, and if we truly value airline security, we should not ask them to.

Devlin’s Angle is updated at the beginning of each month.

JUNE 2002

The words we use

I did one of my semi-regular “Math Guy” pieces on NPR recently (May 18). The topic was mathematics and Homeland Security, following the conference on that topic in Washington that I described in the April edition of this column. (Click here to listen to the segment.)

After the show aired, I received a few letters of complaint from statisticians, who lamented my use of the phrase “Bayesian mathematics” instead of the correct term “Bayesian statistics”.

In fact, my choice of words was made with some care. Although the Math Guy segments are genuine conversations, they are not entirely random. True, there is no script, other than the introduction that the show’s host, Scott Simon, reads out, and a list of possible question prepared in advance—a list he often abandons as soon as the conversation has begun. Nor do we rehearse beforehand. But without any prior planning, there would be little chance we’d be able to get some fairly sophisticated mathematical ideas across in the 7 to 10 minutes we generally have available.

The Math Guy pieces work in large part, I think, because of what happens before we ever walk into the studios. In particular, I think about how best to get the main idea across. (In a short radio segment in a magazine show aimed at a general audience, the most you can aim for is one main point, and it has to be made using everyday language the average listener is familiar with.) Since I do not know what questions Scott will ask (except in general terms), this amounts to being clear in my mind what that one main point is, asking myself what kinds of example I might have to bring in to make my point, and thinking about words to use (or words not to use) to ensure that the message really does reach the average listener.

I’ll be brutally honest. I don’t worry about how the experts will react. I figure that they are a vanishingly small proportion of our listeners, and besides, they don’t need me to inform them what’s going on in their field anyway. Here is why I made a conscious decision in advance to use the phrase “Bayesian mathematics” rather than “Bayesian statistics” in my piece on May 18.

The main message I wanted to convey was that there is a powerful method, based on a mathematical theorem proved over two hundred years ago, for combining human judgements with often massive amounts of statistical data in order to produce improved versions of those initial judgements. To the experts, that method is “Bayesian statistics.” I reasoned, however, that to most people “statistics” is not a process of getting information from numerical data (which is what professional statisticians, and I for that matter, mean by the word). Rather “statistics” is generally understood to mean tabulated numbers, pecentages, and the like. Thus to the layperson, the phrase “Bayesian statistics” will be understood as a certain collection of numbers, possibly the scoring averages of a little known baseball team called “The Bayes”. On the other hand, everyone knows that “mathematics” is a processfor handling numbers. (This is also false, but, as Walter Cronkite used to say, that’s the way it is.) Thus, I felt, using the term “Bayesian mathematics” would convey to the audience a much better general sense of what Bayesian statisticians (sic) actually do than would using the term “Bayesian statistics.”

Now, let me repeat, we’re only talking a 7 or 8 minute radio slot here, so much of the above thinking was done as I cycled across the Stanford campus to the studio. (The conversation generally takes place with Scott in the NPR studios in Washington, D.C. and me in the campus radio studio at Stanford.) I did not go out and systematically check if my reasoning was correct. I just relied on my instincts and what experience I have gleaned both in the classroom and through my efforts to raise the public awareness of mathematics. Still, a quick sample poll among two or three nonmathematical friends I carried out afterwards (yes, I know this is not statistically valid!) confirmed what I had suspected in this case. The term “Bayesian statistics” conjurs up an image of a particular set of numbers, whereas “Bayesian mathematics” suggests some sort of computational process.

So what exactly is my point? Just this. As a mathematician, I agree that correct terminology is important in mathematics and statistics—for those who practice those disciplines. In the classroom, I regularly stress the need for precision. (Although the degree to which I do this depends on the kind of course I am teaching. I am far more demanding with a class of mathematics majors than when teaching math to business students.) But what about everybody else?—by far the greater majority of the population.

The aim of any kind of teaching, surely, is to help people advance their understanding. That can only be achieved by starting with concepts and ideas the student already has and building upon them. In mathematics, more so than in any other discipline, the words we use are particularly critical, since mathematical concepts are entirely abstract, and can only be approached through the words we use to describe them. The student cannot, in general, rely on knowledge or intuition of things in the familiar, everyday world. When we mathematicians use words such as function, relation, continuous, derivative, ring, field, etc. (and, let’s not forget, “mathematics” and “statistics”), we mean something very different from what those words conjure up in the general populace.

Now, in the classroom, we have a chance, over a semester, as we check the assignments handed in to us each week, to help the students develop an understanding of what we mean by those terms. But in a one-off lecture to a general audience, or in a radio broadcast, that is not possible. We cannot get away from the fact that any word we use will conjure up, automatically, an idea in the listener’s mind. And we have virtually no chance of getting beyond that.

Giving a brief explanation of what we mean is unlikely to be successful. After all, appreciating the distinction between the mathematical meaning of the word “relation” and the everyday meaning requires an understanding of both meanings! Moreover, an aside to explain terminology diverts attention away from the main message we want to convey.

Instead, we have to find a way to tell our story in terms of the meanings our audience automatically attach to the words we use, generally the everyday meanings of those words.

In my Math Guy contributions, I have played fast and loose with terminology from physics, computer science, aerospace engineering, psychology, and probably some others areas as well, and now also with statistics. Yet the only groups who regularly complain—and it is regular—are the “math types,” those for whom correct terminology is crucially important to practitioners, and who see it as important that their students use the right words. I believe that this excessive focus on using the correct terminology, while important in the classroom (at least, some classrooms), is one of the reasons why we mathematicians have not been as successful as other sciences in explaining to the general public exactly what it is we do and why it is important.

When it comes to getting “big picture” messages across to the general population, I think it is a mistake to focus on the details. The layperson doesn’t give a hoot what the terminology is. But (to take the example that I started with) learning that there is a way of combining expert guesswork with masses of data to make predictions about possible terrorist attacks? Boy, that’s cool stuff. (I know that, because I spent the first three days of the week following my Homeland Security piece fielding phone calls from journalists who had heard the show and wanted to know what “Bayesian mathematics/statistics” was and how it played a role in Homeland Security.) Who, apart from some professional statisticians, cares about what words those professionals use when talking among themselves?

And, of course, the same point applies, to differing degrees, whenever we teach mathematics to different kinds of students.

Devlin’s Angle is updated at the beginning of each month.

JULY-AUGUST 2002

The mathematical legacy of Islam

Today, mention of the word Islam conjurs up images of fanatical terrorists flying jet airplanes full of people into buildings full of even more people, all in the name, they say, of their god. In an equally sad vein, the word Baghdad brings to mind the unscrupulous and decidedly evil dictator Saddam Hussein. Both images are as unrepresentative as they are understandable, a sad reflection on the ease with which a handful of crazed fanatics, lacking the ability or the wit to bring about change by peaceful means, can hijack not just a plane or a country but an entire cultural heritage and its associated religion. For those of us in mathematics, and by extension all scientists and engineers, the sadness is even greater. For the culture that these fanatics claim to represent when they set about trying to destroy the modern world of science and technology was in fact the cradle in which that tradition was nurtured. As mathematicians, we are all children of Islam.

Following the advent of Islam in the seventh century, Islamic forces attacked and conquered all of North Africa, most of the middle East, and even parts of Western Europe, most notably Spain. The capital of this empire, Baghdad, was established on the Tigris River. Its location made it a natural crossroads, the place where East and West could meet. Baghdad quickly became a major cultural center.

With the emergence of a new dynasty, the Abbasids, in the middle of the eighth century, the Islamic Empire started to settle down politically, and conditions emerged in which mathematics and science could be pursued. By and large, the early mathematical work done by Arabic scholars was predominantly practical, and not very deep—certainly nothing like the mathematics of the ancient Greeks a thousand years earlier. Nevertheless, the subject appears to have been viewed as important and prestigious. Early Islamic scholars imported to Baghdad books on astronomy and mathematics from India.

Early in the ninth century, the Abbasid caliphs decided to adopt a more deliberate approach to the cultural and intellectual growth of the empire. They established the House of Wisdom, a sort of ninth century academy of science, and started to gather together scholarly manuscripts in Greek and Sanskrit, together with scholars who could read and understand them. Over the following years, many important Greek and Indian mathematical books were translated and studied, leading to a new era of scientific and mathematical creativity that was to last until the 14th century.

One of the first Greek texts to be translated was Euclid’s Elements. This had a huge impact, and from then on the Arabic mathematicians adopted a very Greek approach to their mathematics, formulating theorems precisely and proving them formally in Euclid’s style. Like Greek mathematics, which was defined more by the common language in which it was written and carried out, rather than the nationality of the practitioners, Arabic mathematics was determined largely by the common use of Arabic by scholars of many nationalities, not all of them Arabic or Muslim, spread throughout the Islamic Empire.

One of the earliest and most distinguished of the Arabic mathematicians was the ninth century scholar Abu Ja’far Mohammed ibn Musa Al-Khwarizmi, who was an astronomer to the caliph at Baghdad. His name indicates that he was from the town of Khwarizm (now Khiva), on the Amu Darya river, south of the Aral Sea in what is now Uzbekistan. (Khwarizm was part of the Silk Route, a major trading pathway between Europe and the East.) Al-Khwarizmi’s full name can be translated as “Father of Ja’far, Mohammed, son of Moses, native of the town of Al-Khwarizmi”.

Al-Khwarizmi wrote several books that were to be enormously influential. In particular, his book describing how to write numbers and compute with them using the place-value decimal system that came out of India would, when translated into Latin three hundred years later, prove to be a major source for Europeans who wanted to learn the new system.

In fact, Al-Khwarizmi’s book on arithmetic with the Hindu-Arabic numbers was so important, it appears to have been translated several times. Many translations began with the phrase “dixit Algorismi” (“so says Al-Khwarizmi”), a practice that led to the adoption in medieval times of the term algorism to refer to the process of computing with the Hindu-Arabic numerals. Our modern word “algorithm” is an obvious derivation from that term.

Another of Al-Khwarizmi’s manuscripts was called Kitab al jabr w’al-muqabala, which translates roughly as “restoration and compensation”. The book is essentially an algebra text. It starts off with a discussion of quadratic equations, then goes on to some practical geometry, followed by simple linear equations, and ending with a long section on how to apply mathematics to solve inheritance problems. The Englishman Robert of Chester translated Al-Khwarizmi’s algebra book from Arabic into Latin in 1145. The part dealing with quadratic equations eventually became famous. Such was the influence of this work that the Arabic phrase al jabr in the book’s title gave rise to our modern word “algebra”.

After Al-Khwarizmi, algebra became an important part of Arabic mathematics. Arabic mathematicians learned to manipulate polynomials, to solve certain algebraic equations, and more. For modern readers, used to thinking of algebra as the manipulation of symbols, it is important to realize that the Arabic mathematicians did not use symbols at all. Everything was done in words.

One of the most famous Arabic mathematicians was ‘Umar Al-Khayammi, known in the West as Omar Khayyam, who lived approximately from 1048 to 1131. Although remembered today primarily as a poet, in his time he was also famous as a mathematician, scientist, and philosopher, doing major work in all those fields.

It was largely through translations of the Arabic texts into Latin that western Europe, freshly emerged from the Dark Ages, kick-started its mathematics in the tenth and subsequent centuries.

It was around the tenth century that “cathedral schools” sprang up in many parts of Europe,. Designed to train clerics, they concentrated on the trivium (grammar, logic, and rhetoric), with more advanced students going on to the quadrivium (arithmetic, geometry, music, and astronomy). Their creation helped spur an increased interest in mathematics. To fuel that interest, scholars turned to the ancient works preserved by the Islamic culture, many of them in Spain. For instance, Gerbert d’Aurillac (945-1003), later to be Pope Sylvester II, visited Spain to learn mathematics, then returned to France where he reorganized the cathedral school in Rhiems. He re-introduced the study of arithmetic and geometry, taught students how to use the counting board, and even used Hindu-Arabic numerals — though apparently not the full place-value system we use today.

In the centuries that followed, many European scholars spent time in Spain translating Arabic treatises on various subjects. Latin was the language of the European scholars, and thus the target language for the translations. Since few European scholar knew Arabic, however, the translation was often done in two stages, with a Jewish scholar living in Spain translating from the Arabic to some common language and the visiting scholar then translating from that language into Latin. In the same way, many ancient Greek texts, from Aristotle to Euclid, were also translated into Latin, whereupon they began to make an impact in the West.

In addition to the translations of Al-Khwarizmi’s works, of particular note was the appearance in 1202 of Fibonacci’s book Liber abaci, which described the Hindu-Arabic place-value system for representing numbers, and explained how to compute with them. Fibonacci’s treatment was so good that it arguably had more influence than any other source on the eventual acceptance of the new number system around the world, including Al-Khwarizmi’s writings that had come much earlier.

The full story of Fibonacci is a fascinating one, which I will turn to in a future column. The point I want to make now is that it was through translations of the Arabic texts that western Europe was able to develop its own mathematical traditions so rapidly, paving the way for the scientific revolution in the seventeenth century and thence to the scientific and technological world we take for granted today. Many of those Arabic texts were themselves translations of still earlier Greek works from a thousand years earlier.

Without the dedication and commitment to science of the Islamic scholars of the 9th to the 14th century, who both preserved important scientific works and pushed forward the limits of mathematical and scientific knowledge, it is not at all clear that Western Europe would have become the world leader in science and technology. And had that not been the case, it is unlikely that the United States (as we know it today) would have inherited that leadership role.

I suspect that Osama bin Laden, as an educated man from a very wealthy family, is fully aware of the crucial role played by Islam in the development of the West’s scientific tradition. I doubt that the same is true for the hordes who pour out into the streets of Iraq and Pakistan in his support, to rejoice the slaughter of men, women, and children they have never met, living in countries they have never visited. I doubt also that a sense of Islam’s ancient tradition of scientific scholarship and learning is possessed by the fanatical few who, at bin Laden’s bidding, believe that the surest way to achieve immortal greatness in the eyes of their god is to commit mass murder as a first step towards turning back the advances in science and technology that they see as so evil, and returning humankind to the Stone Age.

Ignorance, we used to say, is bliss. Maybe that was once the case, although I very much doubt it. Be that as it may, I think that the clear message of September 11 and the events that have unfolded in the months since then, is that ignorance is dangerous, leaving the gullible ignorant wide open to manipulation by unscrupulous and evil individuals. It is also, as I have tried to indicate, deeply sad.

NOTE: For further details of some of the history mentioned in this column, see the new book Math through the Ages, by William Berlinghoff and Fernando Gouvea, published by Oxton House Publishers in Maine.

Devlin’s Angle is updated at the beginning of each month.

SEPTEMBER 2002

The crazy math of airline ticket pricing

Earlier this year, I was scheduled to fly from California to Germany to give some lectures at an international conference. When I came to book my flights, almost a month before the trip, the cheapest round trip economy class tickets from San Francisco to Frankfurt on any major airline was $5,750. (That’s right, almost six thousand dollars; in economy class.) Needless to say, the conference organizers could not cover this outrageous fare, so I pulled out and stayed at home.

This month, I have to fly from San Francisco to Boston to film a TV program. The cost of the round trip is $2,350, again in economy class. A few weeks later, I fly all the way to Japan and back, half way round the globe, for a mere $795 in economy.

Clearly, the price attached to an airline ticket has got nothing to do with the length of the journey. It has everything to do with supply and demand. The airlines employ some complicated computer programs to try to meet a number of goals, the main two being to avoid flying with any seats empty and to maximize the total net revenue to the airline.

A third goal is to attract customers from other airlines and a fourth is to create customer loyalty among regular flyers. These goals are what lie behind all those promotional packages and those loyalty membership programs like United’s Mileage Plus or American Airlines’ AA Advantage. These programs create additional complexities to the pricing system. For example, as a 100,000+ miles a year member of United’s Mileage Plus, I was able to buy a round trip ticket from San Francisco to Milan for later this month for a bargain basement price of $1,000 and upgrade it to business class at no cost, but a colleague who will be traveling with me, who bought the same ticket at the same time, and who is only a regular Mileage Plus member, was unable to upgrade.

Additional rules are designed to encourage flyers to book two weeks in advance, to fly on particular days of the week or at unpopular times, to include a Saturday night stay, to take along a companion, and to use less congested airports.

Also, many airlines offer incentives to passengers who make their own bookings on the Internet. And the savvy traveler who can stay awake long enough to make an Internet booking after midnight will occasionally come across so-called “Internet specials”, super cheap deals that disappear when the rest of the nation wakes up and starts booking their airline travel.

Further complexity arises from the sale of blocks of seats by the airlines to third parties, often called “bucket shops”, who then sell them at discount prices (or “consolidated fares”).

The result of trying to meet all these different goals, in a highly volatile market, is a ticket pricing system that has grown increasingly complex over the years, to the point where on some flights, no two passengers will have paid the same price, and where it may be cheaper to fly from Boston to New York via London, England, than to take the direct shuttle flight. (Yes, this one has actually occurred.)

Faced with all this confusion, with computers constantly monitoring sales and adjusting fares as often as ten times a day, the only real option for the fare conscious air traveler is to use a Web service to try to locate the best deal. A number of such services have sprung into being in recent years, among them Expedia, Travelocity, and CheapTickets. These services trawl the Web looking for the cheapest fare for any given destination and dates of travel.

But just how well do those search engines do? Not very, is the answer. And with good reason. Airline pricing has grown so complex that it is now practically impossible to design an algorithm that will find the cheapest fare. In mathematical terms, the (idealized) problem of finding the cheapest airfare between two given locations is actually unsolvable, and even if you specify the actual route or the flights, the (idealized) problem of finding the lowest fare is NP hard, which means it could take the fastest computers billions of years to solve. This is the perhaps surprising result obtained recently by mathematician Carl de Marcken.

The story began a few years ago, when Jeremy Wertheimer, then a graduate student in computer science at MIT, and a group of fellow graduate students set out to develop an airline pricing search engine. When what looked like being a few months work failed to yield the expected results, the group began to examine the nature of the airline pricing system. Their study did lead them to develop a powerful price-searching system, which the company they formed, ITA software, markets to both Orbitz (a fare searching site owned by five of the major US airlines) and Delta Airlines, who use it to drive their search utilities. But it also led to the discovery, by research team member de Marcken, that they were chasing the end of a rainbow.

The problem was that all the different pricing rules interact in ways that not even those who designed the pricing systems could begin to fully understand. Mathematically, this made the (idealized) problem of finding an optimal fare between two given locations undecidable, which means that it is impossible to write a computer program to solve the problem. For the same reason, the more specific (idealized) problem of finding an optimal fare for a particular route, while theoretically solvable, turns out to very similar to a classical mathematical problem known as boolean satisfiability, which has long been known to be NP-complete—which means it could take the fastest computer longer than the lifetime of the universe to find the solution.

Admittedly, in order to study the airline pricing problem mathematically and obtain these results, de Marcken had to make some simplifying assumptions that don’t apply to real airline travel, such as allowing an unlimited number of destinations, flights of unlimited length, or arbitrarily long lists of rules. But the implications for real airline pricing are unavoidable.

You can get a sense of just how complicated the real situation is when you consider that with airlines offering thousands of different fares, with different sets of rules governing the different legs on each trip, if two people take a round trip together, with three flights in each direction, there can be as many as 1,000¹², or around 10³⁶, fare combinations. If you printed out a ticket for each possible fare, the pile would stretch all the way to the nearest star, Proxima Centauri, four light years way.

NOTE: This article was based on a longer story by Sara Robinson in the July/August issue of SIAM News, published by the Society for Industrial and Applied Mathematics.

Devlin’s Angle is updated at the beginning of each month.

OCTOBER 2002

The 800th birthday of the book that brought numbers to the west

Liber abaci, the book that gave numbers to the western world, is exactly 800 years old this year.

Numbers are so ubiquitous in modern life that it is easy to take them entirely for granted—to fail to notice how indispensable they are. Yet think how different life would be without them. How would we measure our height, weight, or wealth? How would we measure temperature or speed, or keep track of time or record the date? How would we pay for goods, or receive payment for our labor? How would we measure out and weigh groceries? What method would we use to “number” the pages of a book? What would take the place of telephone numbers or postal codes or street addresses? And these are just a few of the more visible uses of numbers. Beneath all of modern science, technology, medicine, business, and commerce lie oceans of numbers and mathematics.

But it wasn’t always this way. In fact, the way we write numbers today, using just the ten symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, and the methods we use to compute with them are less than two thousand years old. And for much of that period this system was unknown in the western world.

Prior to the advent of the modern way to write numbers, the most common system in use was the one invented by the Romans. The Roman numeral system, still found today in certain specialized circumstances, began with the simplest number system of all: the tally system, where you simply make a vertical mark to record each item in a collection: I, II, III, IIII, IIIII, etc.

This becomes hard to read once you have more than four or five items to count, so the Romans introduced a few additional symbols: V for five, X for ten, L for fifty, C for a hundred, and M for a thousand. For example, using this system, the number one thousand two hundred and seventy eight (1,278) can be written as MCCLXXVIII. This works out as M + C + C + L + X + X + V + I + I + I, or in modern notation 1000 + 100 + 100 + 50 + 10 + 10 + 5 + 1 + 1 + 1, which sums to 1278.

The order in which the symbols M, C, L, X, V, I are written does not matter, whereas the order in which we write the digits in the number 1278 very definitely does make a difference. In technical language, the Roman numeral system was not positional. (Later variants of the Roman system did have a positional element.) Much of the power and efficiency of our modern system comes from its positional nature.

Roman numerals are fine for recording numbers, and for doing simple additions and subtractions, which meant they were adequate, if somewhat cumbersome, for commerce and trade. But multiplication and division are not at all easy, and there is no way the Roman system could form the basis for any scientific or technical work. (Merchants and accountants used a physical abacus to do arithmetical computations.)

Then, in 1202, a young Pisan scholar called Leonardo wrote a book, Liber abaci (“The Book of Calculation”), in which he described a remarkably efficient new way to write numbers and do arithmetic that he had learned from Arab traders and scholars while traveling through North Africa. They, in turn, had picked it up from the Indians, who had developed it over many hundreds of years in the early part of the first millennium.

Leonardo was born in 1175 AD in Pisa (we assume), and died in 1250, presumably in Pisa. His full name is Leonardo Pisano (Leonardo of Pisa), but he is better known today as Fibonacci, a name that probably arose as a contraction of the Latin filius Bonacci (son of Bonacci). There is no evidence that Leonardo ever referred to himself this way. The name seems to have been given to him by later scholars. Leonardo did sometimes refer to himself as “Bigollo,” which was a Tuscan dialect term meaning traveler.

Fibonacci’s father, Guilielmo (William) Bonacci, was a Pisan merchant, who (from around 1192) held a diplomatic post in North Africa. Guilielmo was based in Bugia (later Bougie and now Bejaia), a Mediterranean port in Northeastern Barbary (now Algeria). Bugia lay at the mouth of the Wadi Soummam, near Mount Gouraya and Cape Carbon. Guilielmo’s main duties were to represent the merchants of the Republic of Pisa in their dealings with the customs. At that time, Pisan merchants traded extensively there and elsewhere. (By the end of the twelfth century, the struggle between the Papacy and the Holy Roman Empire had left many Italian cities independent republics. Some of them, most notably Genoa, Venice, and Pisa, had become major maritime traders.)

Fibonacci traveled widely in Barbary with his father, and was later sent on business trips to Egypt, Syria, Greece, Sicily, and Provence. He seems to have learned much of his mathematics in Barbary. In particular, it was there that he observed the Arab merchants using a remarkable system for writing numbers and doing arithmetic.

After Leonardo ended his travels and returned to Pisa in 1200, he wrote (in Latin) a number of mathematics books, only some of which have survived to this day. His first book, and by far the most famous, was Liber abaci. In it Fibonacci described the Hindu-Arabic numerals and the place-valued decimal system for expressing numbers that we use today, and gave detailed instructions on how to compute with them (a process that became known as algorism, which subsequently led to the modern word algorithm). Fibonacci himself always referred to the numerals as “Hindu”; later writers introduced the term “Hindu-Arabic”, and even “Arabic”.

Liber abaci was a big book. The English language translation, which has just been published, runs to 672 pages. The first chapter begins:

“These are the nine figures of the Indians: 9 8 7 6 5 4 3 2 1. With these nine figures, and with this sign 0 which in Arabic is called zephirum, any number can be written, as will be demonstrated.”

Leonardo then goes on to present a large collection of problems designed to provide exercise in using the new number system. Some of the problems were of a practical nature, aimed at merchants: problems about the price of goods, calculation of profits, and conversions between different currencies. Others were more like the word problems you find in modern algebra texts, including the famous rabbit problem that led to the number sequence that today bears his name: the Fibonacci sequence. (Many of the 175,000 hits you get when you do a web search on the name “Fibonacci” are to the Fibonacci sequence. I discussed the rabbit problem and the Fibonacci sequence in this column in March 1999. Click here.)

The book also contains a geometric explanation of the rules for solving quadratic equations, but the main focus is on the arithmetic problems.

The first edition of Liber abaci appeared in 1202. Of course, in those days, books were produced by hand. No copies of that edition are known to exist today. Fibonacci prepared a second edition in 1228, which carried a preface stating that “… new material has been added from which superfluous had been removed …”.

The earliest complete printed copy of the 1228 edition is one printed by Baldassarre Boncompagni in Rome in the period 1857-1862.

Fibonacci wrote a number of other books, three of which have, along with the 1228 edition of Liber abaci, survived to this day:

Practica geometriae, published in 1220, contained a large collection of geometry problems, based in large part on Euclid’s Elements, together with a lot of practical trigonometric problems aimed at surveyors.

Flos (“The Flower”), published in 1225, is largely devoted to algebra, and contains Fibonacci’s solutions to a series of problems posed to him in a contest organized for the emperor Frederick II.

Liber quadratorum (“The book of squares”), published in 1225, is a book on advanced algebra and number theory, and is Fibonacci’s most mathematically impressive work, revealing his substantial mathematical abilities. It deals mainly with the solution of various kinds of equations involving squares, generally with more than one variable, where the solutions have to be whole numbers – the very kind of problem that led Fermat to pose his famous problem, eventually solved by Andrew Wiles in 1994.

Among the lost works are Di minor guisa, a book on commercial arithmetic, and a commentary on Book X of Euclid’s Elements, which contained a numerical treatment of irrational numbers, which Euclid had dealt with geometrically.

The title Liber abaci is sometimes mistranslated as “book of the abacus”, but it is more accurately rendered as “book of calculation”, since, not only did it say nothing about using an abacus, it described methods that eliminated the need for such a device. It is sometimes spelt with two c’s: Liber abacci.

Liber abaci was not the first book written in Europe to describe the new numeral system. For example, the ninth century Arabic mathematician Al-Khwarizmi wrote one such exposition, which, from around 1140 onwards, several scholars translated into Latin and other western European languages. Nor did Liber abaci achieve the popularity of some later, more elementary expositions, such as De Algorismo, written by the thirteenth century English scholar John of Halifax (also known as Sacrobosco). The eleven chapters of Halifax’s text dealt with topics such as addition, subtraction, multiplication, division, square roots and cube roots. Liber abaci did, however, turn out to be the most influential exposition. The reason was that those other translations were written for, and read only by, academic scholars. Interested solely in the benefits of the system within mathematics, they did not see its significance to the commercial world. In contrast, when Leonardo wrote Liber abaci, he did so for the merchants. He took pains to explain the concepts in a way that those highly practical men could understand, presenting many examples from everyday commercial life.

Fibonacci’s expository writings made him something of a celebrity. As is still the case today when accomplished mathematicians and scientists excel at exposition, Fibonacci’s skill as a writer – his ability to reach out to the layperson – came to overshadow his very significant abilities as a mathematician, and it was only long after his death that the full range of his mathematical accomplishments was finally recognized. Today, he is regarded as the greatest number theorist during the entire 1300 year period between Diophantus in the fourth century A.D. and Fermat in the 17th century.

And yet, for all his fame then and now, we know remarkably little about Leonardo the man. We do know that he became a favorite guest of the Holy Roman emperor, Frederick II, who was a great lover of learning and scholarship, with a particular interest in mathematics and science. After Fibonacci returned to Pisa in 1200, he corresponded with some of the scholars at Frederick’s court, among them Michael Scott, the court astrologer, and Theororus, the court philosopher. It was through them that the emperor came to hear of this talented mathematician. When the court met in Pisa in 1225, another court scholar who knew Fibonacci, Dominicus Hispanus, suggested to Frederick that he invite the Pisan to come to the court to demonstrate his mathematical prowess.

The court scholar Johannes of Palermo presented Fibonacci with a number of mathematical challenges, which the latter solved. Fibonacci wrote up three of his solutions under the title Flos, which he proudly presented to Frederick.

Our knowledge of Fibonacci’s travels to Africa come from a brief passage he wrote in Liber abaci:

“When my father, who had been appointed by his country as public notary in the customs at Bugia acting for the Pisan merchants going there, was in charge, he summoned me to him while I was still a child, and having an eye to usefulness and future convenience, desired me to stay there and receive instruction in the school of accounting. There, when I had been introduced to the art of the Indians’ nine symbols through remarkable teaching, knowledge of the art very soon pleased me above all else and I came to understand it, for whatever was studied by the art in Egypt, Syria, Greece, Sicily and Provence, in all its various forms.”

After 1228, there is only one further known reference to Fibonacci. That is a decree made by the Republic of Pisa in 1240, in which a salary is awarded to “the serious and learned Master Leonardo Bigollo”. The salary was given in recognition for the services Fibonacci had given to the city in the form of advice on matters of accounting and for teaching the citizens.

Postscript

Last month, I flew to Italy to attend a conference in Rome. After the meeting was over, I spent a couple of days in Pisa, curious to learn how aware were today’s inhabitants of that beautiful city of their famous predecessor—a man who quite literally changed the course of history.

A web search before I left told me that there was a street in Pisa called the Lungarno Fibonacci as well as a statue of the man, athough sources differed as to the statue’s location, with one website even claiming that there are two statues. (There are not.)

The street was easy enough to find. It runs alongside the south bank of the Arno River at the eastern end of the city, adjacent to a delightful, if slightly run-down, little park called the Giardino Scotto, named after Leonardo’s friend Michael Scott, the astrologer whom I have mentioned already, to whom Leonardo dedicated Liber abaci.

But the statue proved harder to track down. According to some web sites, it was located in the Giardino Scotto, but that is not true. It turns out that it used to be there, but some years ago it was moved. Another web source referred to the statue as being located in a cemetery adjacent to the Piazza dei Miracoli, the beautiful, green-lawned area housing the Cathedral, the Baptistery, and the famous bell tower, the Leaning Tower of Pisa.

Now, there is a cemetery next to the Piazza dei Miracoli: a small Jewish cemetery at the north west corner of the square. But that did not seem to be a very likely location in which to find a memorial to the (presumably) Catholic Leonardo. And indeed, it is not there. So where was this statue?

I walked into the official City of Pisa Information Center at the edge of the square.

“Where is the statue of Leonardo?” I asked the woman sitting behind the desk.

“Leonardo Da Vinci?” the woman replied.

“No, Leonardo of Pisa,” I answered.

The good lady, whose English seemed impeccable, looked at me as if I were from another planet. “Leonardo Da Vinci!” she repeated firmly, stressing the words “Da Vinci,” clearly intent on correcting me.

“No, Leonardo of Pisa—Fibonacci.” I tried to be equally firm.

The information officer clearly thought she was dealing with a complete imbecile. “There is no Leonardo of Pisa,” she declared. “There is no such statue here.”

It was clearly pointless pursuing this exchange. Leaving the information office somewhat frustrated, I took a second, and more thorough look at one of the tourist information signs posted around the square.

There are, it turns out, not three but four buildings that make up the religious complex of the Piazza dei Miracoli. In addition to the Cathedral, the Baptistery, and the Bell Tower, all begun around the same time in the middle of the twelfth century, there is a fourth building, the Camposanto. Its English name, the information poster said, was Monumental Cemetery. Aha!

The Camposanto was started in 1278, after the other three buildings were essentially completed. Compared to its three sisters, the face this fourth building presents to the outside world is unremarkable. Apart from an ornately carved Gothic tabernacle that rises up above one of the two large metal doorways that open out toward the Cathedral, all the visitor sees from the Piazza is a long, low, clean white stone wall. The Camposanto keeps its more discrete beauty hidden from the outside world, facing inwards, with four cloistered walkways looking onto a long rectangular lawn.

I entered the cemetery through the left-hand door, turned left and walked around the western end. And there, facing me, at the far end in front of the eastern wall, was the imposing statue of Leonardo Fibonacci. (Perhaps it would be more accurate to describe it as a statue to Fibonacci. There is no known contemporary drawing of Leonardo, so the statue may well be a work of pure fiction.)

The statue had started out in the Camposanto. Then it had been moved to the Giardino Scotto – to save it from possible damage during the Second World War, one source told me. (If so, it was a wise move, since the Camposanto was largely destroyed in 1944, and had to be extensively renovated.) After some years, exposure to the riverside weather started to take its toll on the statue, and eventually it was taken away, restored and cleaned, and then returned to its original location, alongside Pisa’s other illustrious citizens, where it belongs. Less than fifty yards from the City of Pisa Information Bureau where the lady told me there was no such monument.

Well, the employees in today’s City of Pisa information bureau might not know much about Leonardo, but the splendid location the statue occupies indicates that someone in Pisa, at least, recognizes his importance. As well they should.

Happy 800th birthday, Liber abaci

Request

In my brief time in Pisa, I was unable to find out anything definitive about the history of the statue of Leonardo. Who carved it and when? When exactly was it moved from the Camposanto to the Giardino Scotto and why? When was it renovated and moved back? If any reader knows the answer to any of these questions, or has any other information about the statue, I’d be curious to know. In fact, my interest now piqued, I’d love to hear from anyone with knowledge of Fibonacci other than that available through a routine web search.

Devlin’s Angle is updated at the beginning of each month.

Mathematician Keith Devlin ( devlin@csli.stanford.edu) is the Executive Director of the Center for the Study of Language and Information at Stanford University and “The Math Guy” on NPR’s Weekend Edition. His latest book is The Millennium Problems: The Seven Greatest Unsolved Mathematical Puzzles of Our Time , published by Basic Books.

NOVEMBER 2002

The inaccessibility of modern mathematics

In late October, my new book The Millennium Problems: The Seven Greatest Unsolved Mathematical Puzzles of Our Time went on sale across the country, and this month sees me doing the usual round of public lectures, bookstore talks, and magazine, radio and TV interviews that these days accompany the publication of any new book the publisher thinks has even the ghost of a chance of becoming the next popular science bestseller.

Of all the books I have written for a general audience, this latest one presented by far the greatest challenge in trying to make it as accessible as possible to non-mathematicians. The seven unsolved problems I discuss—the Clay Millennium Problems—were chosen by a small, stellar, international committee of leading mathematicians appointed by the Clay Mathematics Institute, which offers a cash prize of $1 million to the first person to solve any one of the problems. The committee’s mission was to select the most difficult and most significant unsolved problems at the end of the second millennium, problems that had for many years resisted the efforts of some of the world’s greatest mathematicians to find a solution.

No one who is at all familiar with modern mathematics will be surprised to find that none of the seven problems chosen is likely to be solved by elementary methods, and even the statement of most of the problems cannot be fully understood by anyone who has not completed a mathematics major at a university.

In writing the book, I had to ignore the oft-repeated assertion that every mathematical formula you put in a book decreases the sales by 50%. (Personally, I don’t think this is literally true, but I do believe that having pages of formulas does put off a lot of potential readers.) Although my book is mostly prose, there are formulas, and some chapters have technical appendices that are little else but formulas.

Now, as I gear up for the promotional campaign, I face the same challenge again. With the book, I think I found a way to present the story of the Millennium Problems in 250 pages of text. But what can I say about the book’s contents in a twenty minutes talk in a bookstore or a ten minute interview on a radio talk show? Thinking about this made me reflect once more about the nature of modern mathematics. Put simply: Why are the Millennium Problems so hard to understand?

Imagine for a moment that Landon Clay—the wealthy mutual fund magnate who founded the Clay Institute and provided the $7 million of prize money for the seven problems—had chosen to establish his prize competition not for mathematics but for some other science, say physics, or chemistry, or biology. It surely would not have taken an entire book to explain to an interested lay audience the seven major problems in one of those disciplines. A three or four page expository article in Scientific American or 1,500 words in New Scientist would probably suffice. Indeed, when the Nobel Prizes are awarded each year, newspapers and magazines frequently manage to convey the gist of the prize-winning research in a few paragraphs. In general you can’t do that with mathematics. Mathematics is different. But how?

Part of the answer can be found in an observation first made (I believe) by the American mathematician Ronald Graham, who for most of his career was the head of mathematical research at AT&T Bell Laboratories. According to Graham, a mathematician is the only scientist who can legitimately claim: “I lie down on the couch, close my eyes, and work.”

Mathematics is almost entirely cerebral—the actual work is done not in a laboratory or an office or a factory, but in the head. Of course, that head is attached to a body which might well be in an office—or on a couch—but the mathematics itself goes on in the brain, without any direct connection to something in the physical world. This is not to imply that other scientists don’t do mental work. But in physics or chemistry or biology, the object of the scientist’s thought is generally some phenomenon in the physical world. Although you and I cannot get inside the scientist’s mind and experience her thoughts, we do live in the same world, and that provides the key connection, an initial basis for the scientist to explain her thoughts to us. Even in the case of physicists trying to understand quarks or biologists grappling with DNA, although we have no everyday experience of those objects, even a nonscientifically trained mind has no trouble thinking about them. In a deep sense, the typical artist’s renderings of quarks as clusters of colored billiard balls and DNA as a spiral staircase might well be (in fact are) “wrong,” but as mental pictures that enable us to visualize the science they work just fine.

Mathematics does not have this. Even when it is possible to draw a picture, more often than not the illustration is likely to mislead as much as it helps, which leaves the expositor having to make up with words what is lacking or misleading in the picture. But how can the nonmathematical reader understand those words, when they in turn don’t link to anything in everyday experience?

Even for the committed spectator of mathematics, this task is getting harder as the subject grows more and more abstract and the objects the mathematician discusses become further and further removed from the everyday world. Indeed, for some contemporary problems, such as the Hodge Conjecture—one of the seven Millennium Problems—we may have already reached the point where the outsider simply can’t make the connection. It’s not that the human mind requires time to come to terms with new levels of abstraction. That’s always been the case. Rather, the degree and the pace of abstraction may have finally reached a stage where only the expert can keep up.

Two and a half thousand years ago, a young follower of Pythagoras proved that the square root of 2 is not a rational number, that is, cannot be expressed as a fraction. This meant that what they took to be the numbers (the whole numbers and the fractions) were not adequate to measure the length of the hypotenuse of a right triangle with width and height both equal to 1 unit (which Pythagoras’ theorem says will have length the square root of 2). This discovery came as such a shock to the Pythagoreans that their progress in mathematics came to a virtual halt. Eventually, mathematicians found a way out of the dilemma, by changing their conception of what a number is to what we nowadays call the real numbers.

To the Greeks, numbers began with counting (the natural numbers) and in order to measure lengths you extended them to a richer system (the rational numbers) by declaring that the result of dividing one natural number by another was itself a number. The discovery that the rational numbers were not in fact adequate for measuring lengths led later mathematicians to abandon this picture, and instead declare that numbers simply are the points on a line! This was a major change, and it took two thousand years for all the details to be worked out. Only toward the end of the nineteenth century did mathematicians finally work out a rigorous theory of the real numbers. Even today, despite the simple picture of the real numbers as the points on a line, university students of mathematics always have trouble grasping the formal (and highly abstract) development of the real numbers.

Numbers less than zero presented another struggle. These days we think of negative numbers as simply the points on the number line that lie to the left of 0, but mathematicians resisted their introduction until the end of the seventeenth century. Similarly, most people have difficulty coming to terms with complex numbers—numbers that involve the square root of negativ—even though there is a simple intuitive picture of the complex numbers as the points in a two-dimensional plane.

These days, even many nonmathematicians feel comfortable using real numbers, complex numbers, and negative numbers. That is despite the fact that these are highly abstract concepts that bear little relationship with counting, the process with which numbers began some ten thousand years ago, and even though, in our everyday lives, we never encounter a concrete example of an irrational real number or a number involving the square root of -1.

Similarly in geometry, the discovery in the eighteenth century that there were other geometries besides the one that Euclid had described in his famous book Elements caused both the experts and the nonmathematicians enormous conceptual problems. Only during the nineteenth century did the idea of “non-Euclidean geometries” gain widespread acceptance. That acceptance came even though the world of our immediate, everyday experience is entirely Euclidean.

With each new conceptual leap, even mathematicians need time to come to terms with the new ideas, to accept them as part of the overall background against which they do their work. Until recently, the pace of progress in mathematics was such that, by and large, the interested observer could catch up with one new advance before the next one came along. But it has been getting steadily harder. To understand what the Riemann Hypothesis says, the first problem on the Millennium list, you need to have understood, and feel comfortable with, not only complex numbers (and their arithmetic) but also advanced calculus, and what it means to add together infinitely many (complex) numbers and to multiply together infinitely many (complex) numbers.

Now that kind of knowledge is restricted almost entirely to people who have majored in mathematics at university. Only they are in a position to see the Riemann Hypothesis as a simple statement, not significantly different from the way an average person views Pythagoras’ theorem. My task in writing my book, then, was not only to explain what the Riemann Hypothesis says but to provide all of the preliminary material as well. Clearly, I cannot do that in a ten minute radio interview!

The root of the problem is that, in most cases, the preparatory material cannot be explained in terms of everyday phenomena, the way that physicists, for example, can explain the latest, deepest, cutting-edge theory of the universe—Superstring Theory—in terms of the intuitively simple picture of tiny, vibrating loops of energy (the “strings” of the theory).

Most mathematical concepts are built up not from everyday phenomena but from earlier mathematical concepts. That means that the only route to getting even a superficial understanding of those concepts is to follow the entire chain of abstractions that leads to them. My readers will decide how well I succeed in the book. But that avenue is not available to me in a short talk.

Perhaps, then, instead of trying to describe the Millennium Problems themselves, I’ll tell my audiences why they are so hard to understand. I’ll explain that the concepts involved in the Millennium Problems are not so much inherently difficult—for they are not—as they are very, very unfamiliar. Much as the idea of complex numbers or non-Euclidean geometries would have seemed incomprehensibly strange to the ancient Greeks. Today, having grown familiar with these ideas, we can see how they grow naturally out of concepts the Greeks knew as commonplace mathematics.

Perhaps the best way to approach the Millennium Problems, I will say, is to think of the seven problems as the commonplace mathematics of the 25th century.

And maybe that will turn out to be the case.

Devlin’s Angle is updated at the beginning of each month.

Mathematician Keith Devlin ( devlin@csli.stanford.edu) is the Executive Director of the Center for the Study of Language and Information at Stanford University and “The Math Guy” on NPR’s Weekend Edition. His book The Millennium Problems: The Seven Greatest Unsolved Mathematical Puzzles of Our Time was just published by Basic Books. Much of the above discussion is taken from the introduction to that book.

DECEMBER 2002

Why are equations important?

Given the hours that mathematics teachers spend instructing students how to solve equations, it would be easy to assume that the most important thing to do with an equation is to find a solution. But that is rarely the case. Most of the equations that arise in real world contexts cannot be solved. Even if they can, it is often simpler and faster to use a computational method to find a numerical solution. The real power of equations is that they provide a very precise way to describe various features of the world. (That is why a solution to an equation can be useful, when one can be found. )

Before I find myself inundated with hundreds of angry emails from teachers who don’t want their students getting the idea that learning how to solve equations is not important, I should say that it is indeed an important exercise. But the reason is not that the student is likely to find her or himself actually solving equations—outside the math class, that is. Rather, mastering the processes required to solve equations is arguably the best way to become adept at understanding what equations tell us. This, of course, is the same reason why English teachers ask their students to write essays. Few of those students are likely to go on to become novelists or journalists, but writing essays is the best way to learn how to use written language.

One of the most dramatic illustrations of the unimportance—outside of mathematics itself—of solving equations is provided by modern physics. The fundamental theory of matter that physicists work with today is the most accurate scientific theory the world has ever known. Predictions made on the basis of the fundamental equations of matter have been experimentally verified to many places of decimals. And yet, none of those equations has been solved. You have to go back to the 1920s to find equations of matter that anyone has been able to solve.

Living as we do in a world filled with high tech gadgets that depend upon modern physics—the computer I am writing this on and the CD player that is keeping me entertained as I do being just two such—it is obvious that the lack of a solution has hardly held the physicists back, or the engineers who take modern physical theory and turn it into products. Without the precise understanding provided by the equations, the world would not have silicon chips, compact disk players, MRI medical examinations, or many of the other things we now take for granted. But none of those applications required that those equations be solved in the strict mathematical sense.

Physicists have spent the past eighty years trying to find a single framework that explains what are now believed to be the (only) four fundamental forces of nature: electromagnetism, gravity, the strong nuclear force, and the weak nuclear force. Most of the effort has been directed toward developing an extension of quantum theory of a kind that physicists call “quantum field theory” (QFT). The picture of matter that QFT has given us, which represents our best current knowledge of the nature of the material that makes up universe, is generally referred to as the “standard model of particle physics.”

Edward Witten of the Institute for Advanced Study in Princeton, New Jersey, one of the present leaders in this ongoing research, has described the current version of QFT as “a twentieth century scientific theory that uses twenty-first century mathematics.” By that, he means that much of the mathematics remains to be worked out—in other words, mathematicians have yet to solve the equations!

It may seem that Witten is being hard on the mathematicians for being so tardy, but in fact he is simply being realistic. Scientists and mathematicians have been in this position before. Much of Newton’s science depended on the methods of calculus, which he invented for the purpose, but the details of calculus were not fully worked out as a mathematical theory until two hundred years later!

A specific unsolved mathematical problem arising from QFT research was chosen as one of the seven Millennium Problems that the Clay Mathematics Institute announced in the year 2000, offering a prize of $1 million to the first person to solve each problem. (I describe these seven problems in as close to layperson’s terms as I can in my recent book The Millennium Problems: The Seven Greatest Unsolved Mathematical Puzzles of Our Time.) This particular Millennium Problem, the only one on the list that comes from modern physics, asks for a solution (under certain specified conditions) to the Yang-Mills equations (a quantum field analog of Maxwell’s equations for electromagnetism), together with a subsequent explanation, based on that solution, of the so-called “mass gap” (the conjectured, and hitherto observed, minimum level to the mass that any matter may have).

Despite its origins in physics, the problem as stated is essentially a mathematical one. Indeed, many physicists regard the problem as in large part already solved. MIT’s Frank Wilczek, one of the leading figures in QFT, and one of the initiators of quantum chromodynamics (QCD), the most comprehensive theory within the overall QFT framework, comments:

“Specifically, one has direct evidence for the existence of the basic elements of the theory [QCD]—quarks and gluons—and for the basic interactions the theory postulates. Most of the evidence is from studies of jets in high-energy processes, and comparison of their observed properties with very precise and unambiguous calculations in QCD … Another sort of evidence is from actually integrating the full equations directly, using powerful computers. This work directly addresses, and to me effectively solves, the Clay problem. We not only know that there is a mass gap, but have calculated it, and successfully compared it with reality. Of course I understand that numerical results, however convincing and well controlled, are not traditionally considered mathematical proofs.”

If physicists like Wilczek regard the Clay problem as already solved, why did the Clay Institute include it in their list of the seven most difficult and important unsolved mathematical problems at the start of the third millennium? The answer is provided by Arthur Jaffe of Harvard University, an expert in the mathematics of quantum field theory and until recently the director of the Clay Institute. He said, “The problem of Yang-Mills Theory and the Mass Gap Hypothesis was chosen as a Millennium Problem because its solution would mark the beginning of a major new area of mathematics, having deep and profound connections to our current understanding of the universe.” In other words, solving equations is an important goal within mathematics.

Jaffe’s remark does not, however, imply that a solution to the Yang-Mills Theory and Mass Gap problem, if one were discovered next week, would not have major consequences for physics. On the contrary, it would, almost certainly, lead in time to an increased understanding of matter, and from there to who-knows-what new technological gadgets to improve and enhance our world. But – and this is why Wilczek and Jaffe are not in conflict here – applications will almost certainly not start with the solution, as such, rather will come from the methods used to find that solution. As is so often the case in mathematics, in the long run, the method is likely to be more important than the answer.

Remember, all you students out there, I am not saying that solving equations is not important. It is important for several reasons. Rather, my point is that it is generally not the most important thing to do with an equation, whose real power is as a formal and precise description of our world.

We should also remind ourselves that any new mathematical result has the potential to change the world. Many have.

Devlin’s Angle is updated at the beginning of each month.

Mathematician Keith Devlin ( devlin@csli.stanford.edu) is the Executive Director of the Center for the Study of Language and Information at Stanford University and “The Math Guy” on NPR’s Weekend Edition. His book The Millennium Problems: The Seven Greatest Unsolved Mathematical Puzzles of Our Time was just published by Basic Books.