JANUARY 1998
Math + Rigor = College
According to the recent US Department of Education Report Mathematics Equals Opportunity, completion of a rigorous program of mathematics at high school pays huge dividends when it comes time to enter college, regardless of the course of study pursued at college level.
Does this provide support for the back-to-basics movement? Not at all. At least, not at all if by “basics” you mean the ability to perform mental arithmetic or do long division. The message in the DOE report is that, for most people, when it comes to high school math, it’s not what you learn that counts, it’s the mental skills you develop.
When the automobile became widely available, skill at riding a horse was replaced by skill at driving a car. Likewise, in the age of the pocket calculator and the electronic computer, computational skill is no longer necessary. We need other abilities. Training students to be a poor imitation of a $30 calculator is a waste of time for both teacher and students.
Whenever I say this kind of thing, I am invariably bombarded with accusations that I am advocating a lowering of standards. Not at all.
Using—I mean using properly—calculators and computers does not represent a reduction in skill or the need for accuracy. On the contrary, successful use of today’s computational aids requires far greater mathematical skill, and much more mathematical insight, than we old timers had to master to get our sums right.
Those who claim that use of computational aids and an increased emphasis on conceptual understanding lead to lower standards are way off target. In my experience, the only thing that leads to lower standards is a lowering of standards. And you can lower standards just as easily with an abacus as with the latest Pentium computer.
Today we live in a society that is largely shaped by mathematics—though by one of life’s paradoxes, the more important mathematics has become in our lives, the more it has disappeared into the background. You would not know it unless you looked closely, but large parts of modern communications, transport, medicine, entertainment, sport, financial trading, and even law enforcement, all make heavy use of, often sophisticated mathematics. In the industrial age, we burned fossil fuels to drive the engines of society. In the information age, the fuel we burn is mathematics. The mathematics involved is so specialized that we cannot hope to teach it in our schools. What we can—and should—do is make sure our children are prepared to acquire, quickly and efficiently, what particular math skills they require when the time comes.
The bulk of that basic skill set on which each individual can build in later life has little to do with numbers or arithmetic. The industrial age was an age of number and arithmetic. The information age is quite different. The mathematics used today is the mathematics of abstract patterns, relationships, and structures. As we continue to revise our curriculum for the high school math class of the next millennium, we have to accept the fact that the mathematics we teach today’s students will not be (at least, should not be) the same as their parents learned. But that does not make it easier or less rigorous. Quite the opposite.
In terms of arithmetic, getting the right answer to a long division problem using pencil and paper is no more important to today’s new citizen than being able to use a horse-drawn plow. On the other hand, getting the right answer to a problem using a powerful calculator or a computer can be crucially important. It’s also much harder. It’s not the same problem the student is solving, of course. That’s why the math curriculum has to change. The need for accurate, rigorous, precise logical thinking is more important to more people today than at any time in history. To try to achieve that ability by harking back to the mathematics taught a half century ago, as continues to happen in states across the nation, will surely fail with today’s students. They—and we—deserve better.
– Keith Devlin
Devlin’s Angle is updated at the beginning of each month.
Keith Devlin ( devlin@stmarys-ca.edu) is Dean of Science at Saint Mary’s College of California, in Moraga, California, and a Senior Researcher at Stanford University. A regular commentator on mathematics issues for National Public Radio, he has just finished work on a six-part television series and accompanying book called Life By the Numbers, which set out to show how mathematics is used in sports, entertainment, communications, transport, business, in fact, in all walks of life, both work and play. Hosted by movie star Danny Glover, Life By the Numbers will air nationwide on PBS in April. The book will be published at the same time.
FEBRUARY 1998
… Before This Decade Is Out …
Speaking at the Joint Mathematics Meetings in Baltimore, Maryland, in January, the Director of the US National Security Agency, USAF Lt. General Kenneth A. Minihan, called on the mathematics community to join with the NSA and other defense agencies to prepare to fight World War Four. According to Minihan, this will be the third war in which the outcome will depend in large part on the hidden role played by mathematicians.
World War II was the first major conflict in which mathematics played a major role, Minihan observed, citing how it was the breaking of the German Enigma code by British mathematicians that overcame the U-boat blockade in the North Atlantic, and the cracking of the Japanese purple code by American mathematicians that gave the Allies the entire German plan for defending the European mainland against a seaborn invasion from England.
The NSA is a direct successor of those wartime US codebreakers. Today, the agency is the largest single employer of mathematics Ph.D.s in the world, hiring between forty and sixty new math Ph.D.s each year. Their main task is to break foreign codes.
According to Minihan, the end of the Second World War was followed almost immediately by another major global conflict: The Cold War could justifiably be called World War III, Minihan suggested. Where World War II was measured in terms of battled won, he explained, World War III was measured by battles not fought—at least, not in a physical way. It was a struggle for secrets, with spies the front-line troops and mathematicians providing the technical support.
In one significant respect, major mathematics conferences are different from other scientific gatherings. The astronomers, biologists, chemists, physicists, and whatever use their big meetings to announce major new discoveries. But, because of the nature of contemporary mathematical research, it is rare for a major new result to be announced at a large mathematics meeting. These days, when a mathematician makes a major discovery, he or she first shows the result to a small number of friends, who check the proof for correctness. This can often take days, sometimes several weeks. Only after the discoverer is sure that there are no mistakes does he or she arrange to present the result at a meeting. By then, news of the breakthrough has inevitably leaked out and circled the globe via the Internet.
But the absence of new mathematical results does not mean that there is nothing of note going on. With four thousand professionally active mathematicians gathered together, the January Joint Meeting provides an opportunity to check the pulse of the US mathematical community and to look for any significant new trends.
A visit by the NSA director can be viewed as highly significant. This was a first. Admittedly, with the meeting being held in Baltimore, it was a local gig for Lt. General Minihan. But that alone would hardly justify a major address from the agency’s director. Minihan’s mind was on World War IV. This next struggle, he said, would be fought in cyberspace.
As the world of the twenty-first century becomes increasingly dependent on the world wide web and succeeding technologies, mathematicians are going to play a crucial role in ensuring that this infrastructure remains secure and intact, Minihan asserted. To succeed, it would take the collaborative effort of both the NSA’s large staff of researchers and the mathematics community at large. In an era where the communication media are public and open, only mathematics could provide the necessary security.
The importance the government currently attaches to mathematics was brought home by a second unusual visitor to the Baltimore mathematics meeting. For the first time ever, a US Secretary for Education made a major address at a Joint Meeting. Taking as his title “The State of Mathematics Education: Building a Strong Foundation for the Twenty-First Century, Secretary Richard W. Riley told a packed conference hall that it was time to bring an end to the current “math wars” in K-12 education, a battle that pits one teaching philosophy against another. Those involved in K-12 mathematics education should declare a truce and work together to find a mix of information and styles that work, Riley advised. It was important to “Recognize that different children learn in different ways and at different speeds,” he remarked.
Professional mathematicians must make the education of the next generation of mathematics teachers a major priority, Riley continued, observing that, in the present era, “The basis of essential knowledge must be mathematics.” College and university mathematicians should take a critical look at the way they taught their students—the young mathematicians who would become the high school teachers of the future, Riley advised, asking the audience to “Remember [that] teachers teach the way they themselves were taught.”
“Make the preparation of K-12 mathematics teachers a priority,” Riley urged, adding, “The message [about the importance of mathematics and the priorities in K-12 mathematics education] should come from you.” He called on university mathematicians to create new partnerships with their local communities, with museums, with high schools, and with local businesses, to help get the word out.
Another powerful call for help from the mathematical community to improve K-12 mathematics education came from a third plenary speaker at the meeting: Gail Burrill, the President of the National Council of Teachers of Mathematics. A twenty-five year veteran of the high school math class and a former recipient of a Presidential Award for Excellence in Mathematics and Science Teaching, Burrill is now at the University of Wisconsin.
After intriguing the audience with an example of the motivational mathematical problems she thought were required to inspire interest in mathematics among a greater proportion of today’s young people—yes, the solution required a calculator . . . and a whole variety of thought processes besides—Burrill cautioned that the generation coming through the K-12 system today are very different from those who teach them at school or university.
Having been born into the information age, today’s school and university students do not view calculators and computers as devices to aid thinking, Burrill observed. For them, the use of that technology forms an integral part of their thinking. “We do not know how they are thinking,” Burrill told her audience. “They are doing something fundamentally different from us.”
In order to help today’s students develop their mathematical skills, therefore, we would have to present them with math problems that have meaning for them, she said, problems that fitted in with their world view. Problems that call for a much broader range of thought processes than the more narrowly defined skill-and-drill problems of yesterday’s (and in some places today’s) math class. Math problems far more like the kind of problems that arise in real life, in fact. It was, said Burrill, a difficult challenge, one that could only be met if the professional mathematics community took a lead.
The last time the United States made mathematics a major priority was in the aftermath of Sputnik. Startled by the Soviet breakthrough in space exploration, throughout the 1960s and 1970s, the National Science Foundation pumped millions of dollars into the preparation of hundreds of mathematics and engineering Ph.D.s and the support of mathematical research. As a result of that effort, Americans not only went to the moon, they led the world into the microcomputer age.
At the same time, increasing amounts of money were allocated to improve the state of K-12 mathematics education. There the results were far less dramatic. [Now for the part where I inject my own views, a move that invariably infuriates half my readers and brings cries of support from the other half.] One can argue endlessly about the significance of the various international comparisons of the measured abilities of K-12 math students across the board —the USA generally scores poorly—and equally endlessly about the causes. But regardless of that debate, for the sake of our young people who are the citizens of tomorrow, and for the sake of the society which will depend on them when they become adults, we must ensure that the mathematics education we provide those students is both stimulating and a match for their abilities.
The next few years will show whether the three calls for action heard in Baltimore turn out to be just rhetoric or if there is more to it. If there is, then January 1998 in Baltimore could turn out to be every bit as significant as President John F. Kennedy’s 1961 promise to put a man on the moon and bring him back alive before the decade was out.
– Keith Devlin
Devlin’s Angle is updated at the beginning of each month.
Keith Devlin ( devlin@stmarys-ca.edu) is Dean of Science at Saint Mary’s College of California, in Moraga, California, and a Senior Researcher at Stanford University. He has just finished work on a book to accompany a six-part television series, Life By the Numbers, which will be broadcast nationwide on PBS in April. The book, which has the same title as the television series, will be published at the same time by John Wiley and Sons.
MARCH 1998
Forget “Back to Basics.” It’s Time for “Forward to (the New) Basics”
The report card for US secondary school math education came in at the end of February, in the form of a report based on the Third International Mathematics and Science Study (TIMSS). In tests of mathematics general knowledge and advanced mathematics, the results obtained by US 12th-graders were among the lowest of the participating countries.
Secretary of Education Richard Riley called the results “entirely unacceptable.” It is hard not to agree. The question is, what is the best response? It is tempting to look for ways to simply improve our rankings on the international league tables. But math education is not an Olympic sport. Topping the league tables might make us feel good. But as the nation that leads the world in science and technology, the question we should be asking is: How can we lead the world in math education? And since school education is supposed to be a preparation for adult life, that means: What is the best way to prepare our children for life in the twenty-first century?
The first thing to realize is that the nation needs only 3 or 4 percent of the population to be highly skilled in mathematics. Of the remainder, hardly any will ever need or make real use of any appreciable knowledge of, or skill in, mathematics. What mathematics they need and use they have probably already met by the time they are fourteen years old.
But that does not mean we should filter out the top 3 percent and then drop math from the high schools for the others. According to a Department of Education white paper, “Math Equals Opportunity,” released in fall of last year, completion of a rigorous program of mathematics at high school pays huge dividends when any student enters the work force or goes to college, regardless of what profession or course of study the individual chooses. This effect is even greater for low-income students.
Using data from several long-term studies, the report found that 83 percent of high school students who took algebra and geometry courses went on to college. That’s more than double the rate (36 percent) of students who did not take these courses.
Low-income students who took algebra and geometry were almost three times as likely to attend college as those who did not. While 71 percent of low income students who took algebra I and geometry went to college, only 27 percent who did not take those courses went on to college. By way of comparison, 94 percent of students from high-income families, and 84 percent of students from middle-income families who took algebra I and geometry in high school went on to college. Sixty percent of students from high-income families and 44 percent of students from middle-income families who did not take algebra I and geometry went to college.
The study also found that mathematics achievement depends on the courses a student takes, not the type of school the student attends. Students in public and private schools who took the same rigorous mathematics courses were equally likely to score at the highest level on the NELS 12th grade mathematics achievement test.
How do we reconcile these data with my observation about most people never using any math beyond the 8th Grade? Easily. The message is that, for most people, when it comes to high school math, it’s not what you learn that counts, it’s the mental skills you develop. That gives us considerable freedom in what we teach in the math class. Let’s use it.
Today, we live in an invisible universe of mathematics. We burned fossil fuels to power the engines of the industrial age. The fuel we burn to drive the information age is mathematics. Mathematics is used in modern communications, transport, medicine, entertainment, sport, financial trading, law enforcement, science, engineering, and many other areas of life. Most of that mathematics is so specialized that we cannot hope to teach it in our schools. What we can—and should—do is make sure our children are prepared to acquire, quickly and efficiently, what particular math skills they require if and when the time comes in later life.
Much of that basic skill set on which each individual can build in later life has little to do with numbers or arithmetic. The industrial age was an age of number and arithmetic. Today’s information age is quite different. The mathematics used now involves abstract patterns, relationships, and structures (as well as numbers). Consequently, the mathematics we teach today’s students should not be the same as their parents learned. That does not make it easier or less rigorous. Quite the opposite.
For example, using—I mean using properly—calculators and computers does not represent a reduction in skill or the need for accuracy. On the contrary, successful use of today’s computational aids requires far greater mathematical skill, and much more mathematical insight, than we old timers had to master to get our sums right.
In addition to ensuring that our students can get the right answer using modern technology, we should also try to interest them in mathematics as a human creation, developed over the centuries to improve the quality of our lives. To do that, we need to show them some of the many different ways that mathematics plays a major role in today’s society, including some of the mathematics developed during our own lifetime.
In my view, those who cry “Back to basics” have got it wrong. The call should be “Forward to (the new) basics.”
Who knows, if we answer that call, we might even produce a generation that is not math phobic or paralyzed by math anxiety.
– Keith Devlin
Devlin’s Angle is updated at the beginning of each month.
Keith Devlin ( devlin@stmarys-ca.edu) is Dean of Science at Saint Mary’s College of California, in Moraga, California, and a Senior Researcher at Stanford University. He has just finished work on a book to accompany a six-part television series, Life By the Numbers, which will be broadcast nationwide on PBS in early April. The book, which has the same title as the television series, will be published at the same time by John Wiley and Sons.
APRIL 1998
Mathematics: As Seen on TV
Dull, boring, rigid, uncreative, lifeless, irrelevant. Those are some of the adjectives I typically get from my students at the start of my regular semester-long math class for non-science majors. It could be worse: Their answers could have been in response to a request to describe me as their instructor, rather than to give me adjectives that they think apply to mathematics. But it’s still disappointing. As I encounter each year’s crop of new arts and humanities majors, I realize that C. P. Snow’s two cultures are as far apart as they were when Snow coined the term over forty years ago.
Many of the students in my class will go on to be managers, broadcasters, newspaper and magazine editors, and lawyers. Some will become local, state, or even national politicians. And a great many will become teachers. Quite frankly, I don’t mind if they can’t do much math. But I mind a lot if, when they go on to occupy influential roles in society, they are almost totally ignorant of what mathematics is and the major—though generally hidden—role it plays in present-day life.
For years, I have longed to have at my disposal the kind of glossy videotapes that my colleagues in botany, zoology, physics, and astronomy can show to their non-science majors classes. Now, at last, that is about to change. On April 8, PBS broadcasts the first espisode of the new, six-part television documentary series Life By The Numbers. Funded by the National Science Foundation and Texas Instruments, the main goal of the series, according to senior producer David Elisco, is “to counter two thousand years of bad press for mathematics.”
Though mathematics is the topic, the series does not set out to teach math, nor even to exhort people to learn math. Rather, the goal is to show just how much the world in which we live has become dominated by mathematics. It shows that mathematics provides us with a pair of eyes through which we can see what would otherwise be invisible, how it helps athletes win Olympic gold medals, how it is used in the modern movie industry, and how it provides a medium for creative artists to express the ideas produced in their imagination.
By giving educators the rights to tape the series off the air and use it freely in the classroom for up to six years, the producers and their supporters hope that teachers will use it as an educational resource. Educational use is supported by the provision of various educational supplements: activity packs, a companion book, and an interactive web site.
In order to achieve a broad educational impact, each of the series’ six episodes is split into five or six segments, having appeal to different age groups. Though middle and high school students are the main educational target, the producers went to some lengths to include material suitable for the college and university market. Not the math and science majors. Though they will probably enjoy watching the series, and will undoubtedly learn something from it, television is a poor medium for teaching advanced mathematics. For the non science majors, however, Life By The Numbers offers the college or university professor a useful resource.
To my mind, if a college or university demands of its non-science students that they take a single math course, the main goal of that course should be to create an awareness of the role played by mathematics in contemporary society. Our arts and humanities graduates should realize that we are living in a mathematical universe, a world shaped, and in large part controlled, by mathematics. They should know that, when we listen to music coming from a CD player, when we speak to each other on the telephone, when we get on a bus or fly in an airplane, when we attend a major sporting event, when we go to the movies or the theater, when we go into hospital or draw money out of the bank, when we read the weather forecast, or when we watch TV or listen to the radio, we are dependent on mathematics, much of it highly sophisticated. They should know that not only for their own enrichment and empowerment, but also because, throughout their adult lives, they will be called on to make decisions that, directly or indirectly, influence the support society gives to mathematics research and education.
The challenge facing the professor who wants to achieve this (currently lofty) goal is that most of those arts and humanities students whose somewhat apprehensive faces look back at us at the start of each semester have already formed a pretty solid impression of what mathematics is like: dull, boring, rigid, uncreative, lifeless, irrelevant—the adjectives I began with.
Though I get pilloried by many of my colleagues every time I say it, it seems to me as obvious as 1+1=2 that, in a single semester, I haven’t a ghost of a chance of changing such views by forcing my students through yet another repeat of the kinds of math classes they sat through in several years of high school.
Far better, surely, to take the class on a journey that shows them how mathematics relates to, and is used in, various aspects of everyday life. Forget the familiar “problem solving” aspect of mathematics. Show them instead that it is a powerful conceptual framework for understanding the world we live in.
For instance, to catch the attention of the English majors, I often show my students how Noam Chomsky used mathematics to “see” and describe the abstract patterns of words that we recognize as a grammatical sentence.
With tapes of series Life By The Numbers series available, I am at last able to do what my colleagues in the natural and life sciences do in their non-majors classes: I am no longer restricted to simply telling. I can show as well.
Of course, for many students—and for many faculty for that matter—the utility of mathematics does not light a spark of interest. This is where Life By The Numbers delivers far more than you might expect. In particular, the very first episode shows how, increasingly, mathematics is being used as a creative medium, akin to the artist’s canvas and paint, the sculptor’s stone or bronze, or the novelist’s words. This is an aspect of mathematics that we should emphasize far more, I believe. Though the “traditional” applications of mathematics may not catch the attention of the students and professors in the Faculty of Arts, human artistic creativity certainly does. Let’s take advantage of that interest.
For example, the very first episode opens with a profile of Doug Trumbull, the special effects wizard behind movies such as 2001: A Space Odyssey, Blade Runner, and Close Encounters of the Third Kind. These days, Trumbull spends a lot of his time developing a new kind of cinematic experience: immersion theater, where in addition to seeing a screen and hearing sound, the audience experiences motion. As Trumbull says in the first episode, in order to translate his initial idea to a theatrical reality, he relies on mathematics. It is through mathematics that the idea becomes a blueprint—a specification—for a ride.
Other segments from the first episode deal with the theory of perspective in painting, with the geometrically inspired work of the artist Tony Robbin, and with the immersive digital art of Macos Novak.
Dull, boring, rigid, uncreative, lifeless, irrelevant? Only if you find life that way.
– Keith Devlin
Devlin’s Angle is updated at the beginning of each month.
Keith Devlin ( devlin@stmarys-ca.edu) is Dean of Science at Saint Mary’s College of California, in Moraga, California, and a Senior Researcher at Stanford University. He was a lead consultant of the Life By The Numbers television series, and has written the companion book by the same title, just published by John Wiley and Sons.
MAY 1998
Is that a fact?
Facts about brain science don’t often crop up at cocktail parties. Indeed, there is only one that I know of, and that’s the well known tidbit that our brains do not grow any new cells after we reach adulthood. I’ll bet that, like me, you have known for years that, as far as the brain is concerned, it’s all down hill from the moment we get our driver’s license. Every day, another ten thousand cells die. Or is it a hundred thousand? A million? No matter, it’s a bunch, right?
Having this factoid at the ready provides a great line of escape when we find ourselves talking to someone whose name we can’t quite recall. We look sheepish and blurt out, with an embarrassed laugh, “Well, you know, my brain cells have been dying off by the millions for the past thirty years (or whatever), and the old memory isn’t what it used to be.”
Unfortunately, scientists have just discovered that this old stalwart piece of scientific knowledge isn’t true after all. In an article just published in the Proceedings of the National Academy of Sciences, Dr. Elizabeth Gould of Princeton University and her colleagues present evidence showing that our brains continue to generate new cells throughout our lives.
Well, actually, the crucial experiments were performed not on people but on marmoset monkeys, but if you’re a brain scientist, that’s close enough apparently.
The new discovery has enormous implications for future developments in medical science, raising the possibility of treatments for victims of stroke or Alzheimer’s disease, by finding ways to stimulate the brain into producing new cells to replace those killed or affected by the disease.
But what does it tell us about the nature of scientific knowledge when something that has been regarded as a scientific fact for decades, by expert and layperson alike, suddenly turns out to have been false all along? What else might follow? Will we wake up tomorrow to find that water can sometimes flow up hill, or that E is not, after all, equal to m-c-squared? Is scientific truth no more reliable than the word of a former White House aide in pursuit of a lucrative book deal?
Speaking as a scientist, the most honest answer I can give is that this kind of unexpected reversal probably won’t happen in the two cases I just cited. But I can’t be absolutely, 100 percent sure. No one can. Because science is not about determining the absolute truth. In science, there is no such thing. What the scientist does is make careful observations and take measurements as accurately as possible, and then look for an explanation that best fits those observations and measurements. If that explanation is sufficiently complete, and encompasses a variety of other observations and measurements, it is called a “scientific theory”. The theory of gravity and the theory of evolution are two of the best known examples.
Despite what you sometimes read, a scientific theory can never be proven. The best that the scientist hopes for is “confirmation”—that further experiments, observations, or measurements agree with predications made on the basis of the theory. Confirmation can make the scientist more confident that the theory is “right”. But there are no cast iron guarantees.
In contrast, just a single experiment or observation can be enough to overturn a scientific theory. Though you can never prove a theory is true, you can “easily” prove one is false. This understanding of how we obtain scientific knowledge was captured some years ago by the philosopher of science, the late Sir Karl Popper, who talked about the purpose of scientific experiment being to seek to “falsify” a theory. The more resistant a theory is to attempts to being proved wrong, the more reliable we can take it to be.
This is why, for all that scientists may occasionally find themselves with egg on their faces, as when it is discovered that human brains generate new cells just as do other parts of the body, the scientific method continues to provide us with the most reliable knowledge we can ever have. Unlike in other walks of life, “scientific truth” is not decided by an opinion poll, by a ballot, by persuasive rhetoric, by passing laws, or by force or violence. In science, “truth” is determined by the observed facts. Once Dr. Gould and her colleagues had made their crucial observations and others had confirmed their findings, the matter was regarded as settled. The old king—no new brain cells—was dead. Long live the new king. Far from diminishing our faith in science, such about face should make us more confident.
Of course, mathematics is quite different, isn’t it? In mathematics, once something has been proved, it remains true for all time. Well, yes, that’s true—up to a point. In axiomatic, pure mathematics, once a mathematician has established the truth of a statement A on the basis of the axioms, then, provided the proof is correct, A will still be true ( on the basis of those axioms) a century from now, ten centuries from now, indeed for all time. But this nice, clean picture is misleading on at least two grounds.
First, except for the simples of cases, it is never an easy matter to be certain a given proof is correct. I examined this problem in a playful mood in my December, 1996 post.
Second, though many mathematicians pursue their entire careers without any concern for the real-world origins of the systems they study or for the applications their work may lead to (and in my mind there is absolutely nothing wrong with adopting that approach), the fact is that mathematics is just one of a number of conceptual systems (or ways of thinking) that humankind has developed to help us to understand the world we live in and to improve our lives and increase our chance of survival (both our personal survival and the survival of our genes). As it happens, it is one of the most successful of those conceptual systems, and lies beneath many of the others, including all the sciences. But it is still just one among a number of conceptual frames. And when we think of mathematics as a conceptual framework to understand (or predict) the world, a result in pure mathematics, proved on the basis of certain axioms, is only as “true for the world” (and hence only as useful) as those axioms apply to—or match—the world.
One of the most obvious examples of a change in the way an axiom system was viewed occurred with geometry, of course. For two thousand years, Euclidean geometry was regarded as “the geometry of the world we live in.” When non Euclidean geometries were discovered in the nineteenth century, they were at first treated as mere curiosities invented by mathematicians. But when Einstein’s theory of relativity replaced Newton’s framework, we were forced to acknowledge that Euclidean geometry was not, after all, the geometry of the universe. It provides a good approximation that serves us well in most earthly circumstances, and several unearthly ones as well. But from a scientific standpoint, Euclid’s axioms do not provide a sound basis for proving mathematical results about the universe we inhabit. After Einstein, the old “mathematical truth” had to be discarded and replaced by a new one. The king is dead. Long live the king. I think such a change is just as good for mathematics as for other sciences.
– Keith Devlin
Devlin’s Angle is updated at the beginning of each month.
Keith Devlin ( devlin@stmarys-ca.edu) is Dean of Science at Saint Mary’s College of California, in Moraga, California, and a Senior Researcher at Stanford University. He was a lead consultant of the Life By The Numbers television series, and has written the companion book by the same title, just published by John Wiley and Sons.
JUNE 1998
The Bible Code That Wasn’t
Just over a year ago, journalist Michael Dronin’s book The Bible Code hit the bookstores—and soon after that the television screens and review pages of national newspapers—and quickly became an international bestseller. The book’s publisher, Simon & Schuster, ran full page newspaper ads proclaiming that “In all of history, few books have completely changed the way we view the world. The Bible was one. The Bible Code is another.”
Both author and publisher made a considerable amount of money—despite the fact that, as was immediately clear to both mathematicians and serious Bible scholars, the book’s main thesis was complete nonsense.
Indeed, before the year was out, Maya Bar-Hillel, Dror Bar-Natan, and Brendan McKay had shown that the same kind of “amazing revelations” Dronin claimed were hidden in The Bible could be found in the Hebrew translation of War and Peace (and almost certainly in any other work of comparable length). Bar-Hillel and Bar-Natan are mathematicians at the Hebrew University in Jerusalem, McKay is a mathematician and computer scientist at the Australian National University. In the latest issue (Spring 1998) of Chance, the magazine of the American Statistical Association, the three provide an in-depth report of the result they first announced in 1997.
In a nutshell, the “secret code” by which Dronin claimed the Bible revealed hidden messages was that of the “Equal Letter Skip,” or ELS for short. The idea is to start with some letter in the text and then skip ahead by some fixed number of letters to spell out . . . well, whatever you get. For example, if you start with the first (Hebrew equivalent of the letter) T in Genesis and skip 49 letters, the 50th letter is an O, the 50th after that is an R, and the 50th after that is an H. Wonder of wonders, you have just spelled out the word TORH, the Hebrew word for Bible (pronounced Torah). To the naive or numerically challenged, this might seem miraculous. But it’s exactly the kind of thing you would expect to happen. The person searching for the “hidden messages” gets to choose which letter to start with and how big a letter skip to use, and only the successful choice is reported. With a text as long as Genesis—78,064 Hebrew letters long—then you will almost certainly be able to find any moderately short word (say up to six letters long) by an ELS for some skip-number or other. Using a computer to search through all the possibilities, it’s child’s play to find them. For instance, the word TORH itself appears 56,769 times as an ELS for some skip number or another.
The utter triviality of the thesis that had turned Dronin into an instant media star was illustrated when, in an interview with the magazine Newsweek, Dronin stated that “When my critics find a message about the assassination of a prime minister encrypted in Moby Dick, I’ll believe them.” Dronin was referring to the claim made in his book that, using the secret Bible Code, he had been able to predict the assassination of Israeli Prime Minister Michael Rabin. Discounting the rather obvious fact that many Middle East leaders were always prime candidates for assassination, McKay took up Dronin’s challenge, and in no time at all found that, when applied to Moby Dick, Dronin’s secret code revealed not only the assassination of Rabin, but also those of Kennedy, Martin Luther King, and Trotsky.
In fact, the “amazing” Bible secrets that made the headlines were not amazing at all. They are exactly what the mathematics will predict you will find, provided you search long enough—perhaps a day or so if you have a fast computer. What posed more of a challenge to the mathematicians was a little known piece of research that had been carried out ten years earlier by another two Israeli mathematicians, Doron Witztum and Ilya Rips. They had discovered that when an ELS search was run on Genesis to look for pairings of names of prominent rabbis along with their birth or death dates (which are given as sequences of letters in Hebrew, with the first ten letters of the alphabet representing the numerals), the result was a success rate that seemed much higher than the mathematics predicted.
Subsequent examination of the Witztum and Rips work by others seemed (I stress that word seemed) to confirm their findings, and as a result, despite continued skepticism by the statistical community at large, their work was published in the respectable academic journal Statistical Science in 1994. It was this earlier publication that enabled Simon & Schuster to proclaim, on the dustcover of Dronin’s book, that “The code was broken by an Israeli mathematician, who presented the proof in a major science journal, and it has been confirmed by famous mathematicians around the world.”
In fact, I am not aware that any famous mathematicians confirmed the Witztum and Rips finding. Rather, the ever-cautious statistical community was not prepared to declare the result bogus—which is what they suspected—until they had concrete proof thereof. Now, thanks to some statistical detective work by Bar-Natan and McKay, they have that proof, and the Chance article presents it.
The problem was that, for the coding system Witztum and Rips used, Genesis did perform much better (in terms of pairing the rabbis with their dates) than any other text of comparable length. It was not the size of the ELS skip that was at issue, rather how do you select the rabbis whose names are to be looked for, which of the many ways of “naming” them do you choose (how many different ways do people refer to you, for instance?), and how do you represent their dates? When you are doing ELS letter searches, each of these makes a difference. The more freedom there is to make choices, the greater the likelihood of finding “something of interest” by an ELS search.
What made Witztum and Rips’ Genesis result tantalizing was that they provided a defensible rationale for the choices they had made. To show that Genesis was not special after all (in terms of providing an ELS coding of rabbi and date pairings), Bar-Natan and McKay had to find a coding system that conformed to the Witztum and Rips rationale but for which Genesis did not turn out to be special. In other words, even though they suspected Witztum and Rips had “cooked” their chosen rules to get the result they wanted, Bar-Natan and McKay had to play by those rules. It was like agreeing to fight an opponent using the oppontent’s preferred choice of weapons. Despite this difficulty, they were successful. They found a coding system for rabbis and dates that conformed with the Witztum and Rips rationale for which the Hebrew translation of War and Peace fared just as well as the Bible.
In their article in Statistical Science, Witztum and Rips had concluded that “the proximity of ELS’s with related meanings in the Book of Genesis is not due to chance.” Bar-Hillel, Bar-Natan and McKay end their recent Chance article by quoting that passage, and adding their own conclusion: “It must therefore be due to design. The design, however, may well be human, not devine.”
A final thought: The world we live in is indeed full of wonders—many of them genuinely “amazing”—and mathematics and statistics can help reveal those wonders to us. Moreover, whatever your religious belief (or absence thereof), there is little doubt that the Bible contains a great many “messages” that require effort to “uncover”. It is therefore a great pity that so much attention is paid to the kind of nonsense peddled in The Bible Code.
For further details, see McKay’s web site at: http://cs.anu.edu.au/~bdm/dilugim/torah.html.
– Keith Devlin
Devlin’s Angle is updated at the beginning of each month.
Keith Devlin ( devlin@stmarys-ca.edu) is Dean of Science at Saint Mary’s College of California, in Moraga, California, and a Senior Researcher at Stanford University. He was a lead consultant of the Life By The Numberstelevision series, and has written the companion book by the same title, just published by John Wiley and Sons.
JULY 1998
Buttered Toast and Other Patterns
Last month’s column took a look at the book The Bible Code, Michael Drosnin’s runaway bestseller that attached mystical and possibly religious significance to what is surely the familiar behavior of sufficiently random sequences. Specifically, by taking a suitably long text (in Drosnin’s case, the book of Genesis), choosing a starting letter, and then skipping through the text to pick up every N’th successive letter for a chosen integer N, various recognizable words will be spelled out. As I pointed out in my article—and as many others have pointed out elsewhere—there are so many choices of starting point and skip-length N available that it is inevitable that, for some choices of starting letter and skip-length, the process will lead to recognizable words, and indeed to phrases that can be regarded as having significance.
The fact, is, human beings are remarkably good at seeing patterns. Indeed, it is arguable that the ability to see patterns is our key evolutionary trick, the ability that has ensured our survival as a species in the Darwinian battle for survival. Armed with modern computers, our pattern finding ability is even greater. Hence the Bible Code.
But spotting a pattern is one thing; deciding whether or not that pattern has any significance is quite another. That was where Drosnin went astray. The patterns he saw had no significance. Sometimes, however, we perceive a pattern that is not simply a random effect. Take, for instance, the oft-cited observation that a slice of buttered toast knocked from the breakfast table will usually land buttered side down. Most of us believe we have noticed this annoying tendency of buttered toast. But what is going on?
One explanation would be that the toast lands butter-up half the time and butter-down the other half, in an entirely random fashion. According to this explanation, our perception of a pattern of butter-down behavior is a result of our paying more attention to those occasions when the toast drops butter-side down. After all, the butter-side down landing causes us to clean up the carpet, and hence makes the event more significant—and more memorable—than when it lands dry-side down. For a dry-side down landing, we either blow on the dry-side to remove any imagined dust and then eat the toast or else throw it in the bin, depending on the status of the floor—no big deal either way.
An alternative explanation would be that there is a physical law that says toast tends to fall buttered side down, and the pattern we notice is for real.
Surprisingly, the second explanation is correct. Not that there is a law of nature specifically dedicated to buttered toast. Rather, it’s a matter of the height of the typical breakfast table, coupled with the fact that when the toast is on the table, it is assuredly butter-side-up. As the toast slides across the edge of the table, with part of the toast hanging off the table, gravity causes the toast to start to rotate. Thus, when the toast falls, it is rotating. For a typical breakfast table, there is just enough fall time for the toast to complete one half a revolution before it hits the deck. The result: the toast lands butter-side-down.
This explanation can be easily checked experimentally without ruining your carpet. Just place a book on your desk, title-side-up, and push it off the edge. It is likely to land title-side-down.
Another case where our pattern-spotting ability also turns out to be reliable is our sense that, when we stand in line at the supermarket checkout, one of the lines next to us moves ahead faster than the line we have chosen. The explanation is straightforward. Assuming that the delays that slow up lines affect every checkout line randomly, and that we join a line having two neighboring lines, the probability that our line moves fastest of the three is 1/3, while the probability that one of the other two adjacent lines moves faster is 2/3, twice as high. The odds are two-to-one that you will watch an adjacent line move ahead of you!
Of course, this explanation does not work if you join one of the lines at the edge of the checkout area. In that case, there is only a 50% chance that the neighboring line will move faster than yours. Much better odds. Moreover, there is the added advantage that you are probably more likely to find yourself standing in line with another mathematician who understands elementary probability theory, and the two of you can pass the time bemoaning the slow progress of your line.
One final example where our perception of a pattern turns out have a rational explanation is the annoying tendency that when we want to use a map, the main location we are interested in lies on or near to a crease in the map, where it is difficult to make out all the details (especially in an older map), or else near the edge, where we need a second map to complete the picture. In this case, some simple geometry provides the explanation.
Given a numerical value for being “near a crease or edge”, say, within a distance from a crease or edge of one-tenth the width of the map, then for most maps—certainly for the kinds of maps designed to fit into your automobile glove compartment—the ratio of the total map area that lies “near a crease or an edge” to the overall map area will be greater than one-half. For example, for a square map with just a single crease down the middle, the ratio works out to be 0.52. For a more generous definition of “near a crease or an edge”, the ratio will in fact be much greater than one-half. Frustrated map readers might like to perform a few calculations, with different maps and different values of “nearness”. In this way, you will be able to reassure yourself that the map-makers are not out to get you. Rather, it’s a simple matter of relative areas.
– Keith Devlin
Devlin’s Angle is updated at the beginning of each month.
Keith Devlin ( devlin@stmarys-ca.edu) is Dean of Science at Saint Mary’s College of California, in Moraga, California, and a Senior Researcher at Stanford University. He was a lead consultant of the Life By The Numberstelevision series, and has written the companion book by the same title, published in April by John Wiley and Sons.
AUGUST 1998
Mathematicians and Philosophers – Chalk and Cheese?
Confession time: I occasionally read philosophy books. Like many mathematicians, for most of my career I tried to stay away from the stuff. Then, when I turned forty, I started to be plagued by “Why?” questions. Again, I know of many colleagues for whom the Big-Four-O had a similar effect. Now, having passed the even bigger Five-O, I have learned to live at peace with my interest in those nagging questions about the nature of mathematics.
My latest foray into the world of philosophy was John Haugeland’s recent tome, Having Thought: Essays in the Metaphysics of Mind. The subtitle says it all. Haugeland’s book is a fairly heavy duty philosophy book. In working my way through it, I kept being struck by the differences between philosophers and mathematicians.
Any mathematician who has ever wandered, either by accident or design, into a philosophy seminar will have found the experience somewhat bizarre. In a mathematics seminar, the presenter will write some figures, equations, graphs, and cryptic notes on a blackboard—or maybe project them onto a screen using an overhead projector—and will spend three quarters of an hour describing what those figures, equations, graphs, and notes mean. The mathematician will generally move around a lot in an animated fashion, often waving her arms in the air and pointing to something displayed on the screen. Occasionally, she will glance at her notes, scribbled hurriedly on the train or plane on the way to the meeting. An outsider observing the proceedings might get the impression that the mathematician is struggling to explain the results to herself as much as to the audience. And often that impression is not far from the truth.
At a philosophy seminar, on the other hand, the presenter stands motionless behind a podium and reads, verbatim, a pre-written, neatly-typed manuscript which he has made available to everyone in the audience. That’s right: he reads it word for word. From start to finish.
I am sure that even my philosopher friends (I had several before I wrote this article) will acknowledge that the presentation at the math seminar is much more lively than the philosophy seminar. Things switch around dramatically, however, when the presentation gives way to the question and answer session that follows. In the math seminar, chances are that no one followed much beyond the first ten minutes, so the questions—if there are any at all—are either trivial or else so far off the mark that the presenter can’t provide a coherent answer. When the philosophy presenter stops talking, however, all hell breaks loose. A casual observer would wonder why on earth the organizers had invited the presenter in the first place, given that the entire audience seems not only to disagree wildly with the views just presented, but express grave doubts that the presenter has read all the appropriate literature.
Not only do mathematicians and philosophers go about their business in very different ways, on the whole they don’t mix. Mathematicians who arguably would benefit from reading works in philosophy rarely do so—books by Daniel Dennett and John Searle excepted—and even philosophers of mathematics appear to read little contemporary work on mathematics—books by Roger Penrose excepted. As Thomas Kuhn wrote in his book The Structure of Scientific Revolutions, “. . . normal science usually holds creative philosophy at arm’s length, and probably for good reasons.” Kuhn goes on to describe those “probable good reasons”—in essence, to make progress in science it is generally better to ignore the potentially distracting meta-scientific questions that so interest the philosopher. Count mathematics in with science on that score.
I have already confessed to being a bit of a closet philosopher, in that I find many questions in philosophy deeply fascinating. My problem is always that, because of my training in mathematics, I find philosophy books so hard to read. Given the vast deluge of new research that any mathematician has to read about in order to stay abreast of his or her field, I suspect that we all develop the same approach: First read the introduction, then skip ahead to the conclusion, then glance at any figures or graphs, then rapidly guess at the main points in the author’s argument, and finally skim the main body of the text quickly to confirm or correct that initial guess. An hour at the most.
Ninety-nine percent of the time, this method works fine in mathematics. But it’s the kiss of death when you are faced with a paper or book in philosophy. There, the author has labored long and hard to craft an intricate and tightly woven argument. Every word counts, and every phrase is critical. You can’t simply cut to the chase. This is the chase, and it’s going to take time.
The main concern of the mathematics author is to get the results into print. If the results are sound, everyone will be able to figure out what is intended, even if the explanations are poor and incomplete. But the philosopher sets out to provide a defense against all anticipated attacks from all quarters. In all likelihood, the conclusion is obvious, and in any case unlikely to be surprising. The game is to supply a good supporting argument.
The mathematician establishes results by logical deduction. The philosopher constructs elaborate “thought experiments,” often involving imaginary creatures from other planets who share some—but not all—human intellectual abilities.
The mathematician writes for posterity, for the next grant, and, in his dreams if young enough, for a Fields Medal. The philosopher writes for a dozen or so (other) leading philosophers who work in the same area, against whose future criticisms the author will go to great lengths to erect a solid defense.
What then did I make of Haugeland’s Having Thought, a collection of essays written over many years, all having to do with the nature of human thought? (Incidentally, another distinction between mathematicians and philosophers is that the former usually provide their academic affiliation on the first page of any book they write, whereas philosophers—probably unduly influenced by Saul Kripke’s writings on names—often prefer to appear all but anonymous. For the record, John Haugeland is a highly respected professor of philosophy at the University of Pittsburgh. You won’t find that out from reading his book.)
Well, I found it heavy going, I’ll say that. The reason why Dennett and Searle attract so many scientist readers is that they write in a simple, direct, everyday style. But plain, everyday prose is not at all the fashion in philosophy, and Haugeland makes no such concessions for the outsider. (A chapter in Haugeland titled “Understanding Dennett and Searle” struck me as harder going than either Dennett or Searle themselves, which leads me to suspect that Haugeland is not such a pushover for those two pros as I am.)
Despite the difficult prose, however, I found the book made me think, which is surely the author’s main intention. The first chapter begins with what I found to be a persuasive and wonderfully uplifting argument explaining why cognitive psychology should be regarded as a genuine science. (The gist of the argument is to forget thinking of physics as the ideal science to which all others should aspire, and rethink what is meant by science. That clearly affects the role of mathematics in “science”. Mind you, in typical scientist’s fashion, I have just condensed 34 pages of finely crafted argument into a single sentence, so I dare say I have over- simplified Haugeland’s account.)
As I argued myself in my book Goodbye Descartes, I think that it is inevitable that, as we head into the twenty-first century, the human sciences will increasingly come in from what has hitherto been the scientific cold, both in terms of status and funding. Many of the issues Haugeland addresses are undoubtedly going to occupy a central place in the science of the next millennium. That will not only change the relationship between mathematics and science, it will influence the kinds of mathematics that are done. That’s another issue I address in Goodbye Descartes.
– Keith Devlin
NOTE: John Haugeland’s book Having Thought: Essays in the Metaphysics of Mind has just been published by Harvard University Press.
Devlin’s Angle is updated at the beginning of each month.
Keith Devlin ( devlin@stmarys-ca.edu) is Dean of Science at Saint Mary’s College of California, in Moraga, California, and a Senior Researcher at Stanford University. His book Goodbye Descartes: The End of Logic and the Search for a New Cosmology of the Mind (John Wiley, 1997) was released in paperback earlier this year.
SEPTEMBER 1998
Kepler’s Sphere Packing Problem Solved
A four hundred year mathematical problem posed by the famous astronomer Johannes Kepler has finally been solved. Mathematician Thomas Hales of the University of Michigan announced last month that after six years effort, he had proved that a guess Kepler made back in 1611 was correct.
The problem asks what is the most efficient way to pack equal-sized spheres together in a large crate. Should you pack them in identical layers, one on top of the other, with each sphere in one layer sitting right on top of the sphere directly beneath it? Or can you get more spheres into the box if you stagger the layers, the way greengrocers the world over stack oranges, so that the oranges in each higher layer sit in the hollows made by the four oranges beneath them? (The formal term for this orange-pile arrangement is a face-centered cubic lattice.) More generally, what is the most efficient packing of all?
For a small crate, the answer can depend on the actual dimensions of the crates and the spheres. But for a very large crate, you can show, as Kepler did, that the orange-pile arrangement is by far the more efficient of the two mentioned above, and indeed seems to be more efficient than any other possible arrangement.
The general problem as considered by Kepler and subsequent mathematicians is formulated not in terms of the number of spheres that can be packed together but the density of the packing, i.e., the total volume of the spheres devided by the total volume of the container into which they are packed. Since the main interest is in the pattern of the packing, rather than the shape or size of the container, the problem is further generalized by defining the density of a packing (pattern) as the limit of the densities of individual packings using that pattern for cubic crates, as the volume of the crates approaches infinity.
For example, the symmetrical packing gives you a density of pi/6 (about 0.52), the orange-pile packing gives you a density of pi/3sqrt(2) (approximately 0.74).
Kepler believed that the face-centered cubic lattice was the most efficient of all arrangements (in terms of the density of the packing arrangement), but was unable to prove this. So were countless succeeding generations of mathematicians.
Though it never achieved the general fame outside of mathematics as Fermat’s Last Theorem, proved four years ago by British mathematician Andrew Wiles 350 years after it had been posed, Kepler’s Sphere Packing Conjecture was very similar, in that it was ridiculously easy to understand the problem but seemed fiendishly difficult to solve. Moreover, both problems had subtle difficulties that led a number of mathematicians to believe they had found a solution that subsequently turned out to be false. Most recently, in 1993, a highly-respected mathematician at the University of California at Berkeley produced a complicated proof of the Kepler Conjecture which, after many months of debate, most mathematicians decided was incorrect.
Major progress on the problem was made in the 19th century, when the legendary German mathematician and physicist Karl Friedrich Gauss managed to prove that the orange-pile arrangement was the most efficient among all “lattice packings.” A lattice packing is one where the centers of the spheres are all arranged in a “lattice” (a regular three-dimensional grid—think of a three-dimensional analogue of a lattice fence).
But there are non lattice arrangements that are almost as efficient than the orange-pile, so Gauss’s result did not solve the problem completely.
The next major advance came in 1953, when a Hungarian mathematician, Laszlo Toth, managed to reduce the problem to a huge calculation involving many specific cases. This opened the door to solving the problem using a computer.
In 1994, Hales worked out a five-step strategy for solving the problem along the lines Toth had suggested. Together with his graduate student Samuel Ferguson, Hales started work on his five-step program.
By 1996, Hales was giving talks at universities describing the progress he had made, and speculating that with about two more years work, he expected to have cracked the problem.
Last week, Hales announced that he had succeeded, and posted the entire proof on the Internet. You can see the proof for yourself at Hales’ web site. But be warned: The proof involves 250 pages of text and about 3 gigabytes of computer programs and data. To follow Hales’ argument, you will have to download his programs and run them.
As Hales himself admits, with a proof this long and complex, involving as it does a great deal of computation, it will be some time before anyone can be absolutely sure it is correct. By posting everything on the world wide web, Hales has, in effect, challenged the entire mathematical community to see if they can find anything wrong. If no one can, then eventually everyone will agree that the proof is correct. At the moment, experts who have visited the web site and looked through the material think that it looks convincing.
The situation is reminiscent of the solution in 1976 of the Four Color Problem by Kenneth Appel and Wolfgang Haken. Their proof that four colors are all you need to color any map drawn on a plane, no matter how complicated (so that countries having a common border are colored differently) also involved a mixture of ordinary mathematical reasoning and masses of computation. On that occasion, it was several months before there was general agreement that the proof was correct. But that was before we had the web. With any mathematician anywhere in the world now able to get immediate access to Hales’ entire proof and all the computer programs, we can expect a consensus to emerge much quicker.
What use is the new result, you might ask? Well, the problem actually began as an applied problem. A military problem, in fact. Sir Walter Raleigh had asked a mathematician friend to give him a simple mathematical formula for calculating the number of cannonballs he had in the piles on the deck of his ship. The mathematician, Thomas Harriot, gave Raleigh his formula, but realized that he did now know whether the cannonballs were being stacked in the most efficient fashion. He passed the problem on to Kepler.
What about modern applications? Well, knowing that Kepler was correct probably has no direct applications in itself. But the whole topic of the efficient packing of spheres is a crucial part of the mathematics that lies behind the error-detecting and error-correcting codes that are widely used to store information on compact disks and to compress information for efficient transmission around the world. In today’s information society, you can’t get a more significant application than that. So who can say what might come out of any advance in our understanding of sphere packing.
And let’s not forget that, around the world, thousands of greengrocers and supermarket assistants can now rest assured that the way they arrange the oranges has now been proved mathematically to be the most efficient.
– Keith Devlin
Devlin’s Angle is updated at the beginning of each month.
Keith Devlin ( devlin@stmarys-ca.edu) is Dean of Science at Saint Mary’s College of California, in Moraga, California, and a Senior Researcher at Stanford University. Devlin describes Kepler’s sphere packing problem in his book Mathematics: The Science of Patterns, published in paperback by W. H. Freeman in 1996.
OCTOBER 1998
Why Does Back-to-School Imply Back to Math?
Last month, across the nation, parents breathed a sigh of relief as their children returned to school.
Although the schools that those children returned to will differ in a number of ways, one thing they will all have in common is the preeminent status given to English, math, and science. Why do these three subjects occupy the privileged position they do? Indeed, in most districts they are the primary performance indicators that state officials and parents use to evaluate how well a particular school is doing.
In the case of English the answer is obvious: everyone in today’s society needs to be literate and able to communicate well. But in a world where everyone can afford a pocket calculator and a great many people seem to be successful in life with little or no mathematical ability or knowledge of science, why do we place so much emphasis on math and science?
Whatever the answer, it’s an emphasis with a long history. The desirability for all citizens to have an appreciation of mathematics and an understanding of the nature of the world they live in was first expressed over two thousand years ago by Plato in The Republic.
In the USA, the importance of scientific knowledge was emphasized by such early Americans as Benjamin Franklin and Thomas Jefferson, the former founding the Philadelphia Academy of Science in 1751. But it was relatively recent that a mechanism was established to provide general mathematics and science education for all American children. It began in the early years of this century, in large part as a result of the educational theories of John Dewey, who spoke of developing in (all) students “scientific habits of the mind.” Speaking at the 1909 Symposium on the Purpose and Organization of Physics Teaching in Secondary Schools (School Science and Mathematics 9, pp.291-292), Dewey said:
“Contemporary civilization rests so largely upon applied science that no one can really understand it who does not grasp something of the scientific methods and results that underlie it; … a consideration of scientific resources and achievements from the standpoint of their application to the control of industry, transportation, communication … increases the future social efficiency of those instructed.”
Though it is plain that Dewey’s goal of “scientific habits of the mind” was never achieved on any appreciable scale, he and his followers clearly had a major impact on the K-12 educational curriculum, where mathematics and science have ever since ranked immediately after English as the most important subjects in the curriculum.
In 1989, prompted by the continuing poor performance by US middle and high school students in international comparisons, President George Bush reaffirmed Dewey’s call in his “America 2000” agenda, with the express goals:
- By the year 2000, US students will be first in the world in science and mathematics achievement.
- By the year 2000, every adult American will be literate and will possess the knowledge and skills necessary to compete in a global economy and exercise the rights and responsibilities of citizenship.
The familiar justification for mandatory math and science education is that today’s world is so heavily dependent on mathematics and science (in large part through technology) that everyone needs to be proficient in those areas. I don’t agree. That’s like saying that because our lives are so dependent on the automobile, everyone should be able to fix a car. An automobile-dependent society requires an adequate number of well-trained, skillful auto engineers and mechanics, but for most of us it’s enough to know how to drive. Similarly for math and science.
A much better justification for mandatory math and science may be found in Dewey’s phrase “scientific habits of the mind.” For the vast majority of pupils, it’s not so much the subject matter of math and science that is important as the mode of thinking that is involved: the need to collect and weigh evidence, the need to base decisions on that evidence, the ability to think logically, and a willingness to change one’s opinion on the basis of new evidence.
For example, last year the Department of Education released a white paper (the Riley report) highlighting the importance of precollege mathematics for entrance to college and success on the job market, especially for low-income students. Using data from several long-term studies, that report found that 83 percent of high school students who took algebra and geometry courses went on to college. That’s more than double the rate (36 percent) of students who did not take those courses. Low-income students who took algebra and geometry were almost three times as likely to attend college as those who did not. Moreover, students who had completed those courses did noticeably better at college than their colleagues who did not.
Note that the report did not say anything about passing those math courses, or of achieving a good grade. Simply taking the courses was what led to the benefits. What’s more, the benefits were there regardless of the subjects the students chose to pursue at college. The English, history, and art students benefited along with the math and science majors. It’s the thought process that makes the difference.
The evidence seems clear: A daily dose of mathematical and scientific thinking is as good for the mind as a daily walk or jog is for the body.
– Keith Devlin
Devlin’s Angle is updated at the beginning of each month.
Keith Devlin ( devlin@stmarys-ca.edu) is Dean of Science at Saint Mary’s College of California, in Moraga, California, and a Senior Researcher at Stanford University. His latest book The Language of Mathematics: Making the Invisible Visible, has just been published by W. H. Freeman.
NOVEMBER 1998
Math Becomes Way Cool
After years of relative obscurity, math is suddenly hip, or so it seems. Over the past two or three years, books about mathematics and mathematicians have made their ways onto the bestseller lists, television series about mathematics have been aired, radio programs have carried stories about mathematics, and newspapers and magazines have discovered that large numbers of their readers are interested in articles about mathematics.
Within the past eighteen months, math has even made its into the movies, with two feature films in which not only was the lead character a mathematician, but we even see mathematics explained on the silver screen: Good Will Hunting and Pi.
What’s going on here? Math in the movies? Is this the start of a trend? How long before we go to a film and there’s a quiz at the end?
I jest, of course. The popular association of math with quizzes is one that I have split much writer’s ink—and several hours in radio and television studios—trying to overcome. But why did the writers of Good Will Hunting and Pi make their lead characters mathematicians?
Good Will Hunting, written by Matt Damon and Ben Aflick, who starred in the film, is a Hollywood feel-good movie about the problems involved in moving from one social world to another, brought about by being born with an unusual ability. The hero, Will Hunting (played by Damon), has to have a precocious talent that can surface without any formal training and which the audience will regard as completely incomprehensible, which makes math the number one pick.
Pi, shot on a tiny budget by first-time director Darren Aronovsky in grainy black-and-white, much of it using a shaky hand-held camera, is a dark, brooding, Kafka-esque film about the human obsession to find order in the universe, especially scientific and religious order. Mathematics is the discipline above all that tries to find perfect order in the world. Hence the lead character—he’s hardly a hero in the conventional sense—is a mathematician. (Much of the universe’s order involves the mathematical constant pi, of course, which gives the movie its title.)
In each film, the director had to include sufficient mathematics to establish the characters. The mathematics shown on screen in Good Will Hunting, though correct, is deliberately chosen to be unfamiliar to the audience, and is not explained. Viewers are supposed to be baffled. In Pi, on the other hand, the mathematics shown has to connect to the audience’s own (possibly distant) memories of mathematics, and remind them that mathematics involves finding formulas that describe order in the world, formulas that often involve pi. (The formula for the area of a circle is given considerable prominence at one point.)
Math in the movies is unusual, but Good Will Hunting and Pi are not the first films to include some mathematics. In the 1980 romantic comedy It’s My Turn, for instance, star Jill Clayburgh plays a college math professor, and the film opens with her lecturing to a graduate mathematics class. She proves a well known (to mathematicians!) theorem of a branch of advanced mathematics called homological algebra. It’s all correct. The point of having Clayburgh’s character be a mathematician is simply to establish Clayburgh as a highly intelligent intellectual. Math plays no other role in the film (unless you include the eternal triangle).
Going back to 1971, Dustin Hoffman’s character in Straw Dogs was also a mathematician. Again, the mathematics shown on the blackboard at one point is the genuine article (gravitational equations on that occasion). And, as with It’s My Turn, the aim is to establish Hoffman’s credentials as an intellectual—in his case, as an expert in a field generally regarded as having nothing to do with personal violence. (The director was Sam Peckinpah, so you can guess what kind of ending that film has.)
In the 1992 movie Sneakers, starring Robert Redford, a pair of freelance spies battle foreign agents for a powerful code-breaking chip. As the film makes clear, modern security codes depend on lots of heavy duty mathematics. At one point, we see the chip’s inventor lecturing on the mathematics behind its design. Again, it looks right, as it should, given that one of the world’s leading experts on the mathematics of cryptography was an advisor on the film.
In the movie Contact (1997), based on the novel by the late Carl Sagan, star Jodi Foster makes a good job of defining prime numbers for a bunch of Washington bigwigs, and explaining why primes provide a good way to communicate with intelligent aliens. (The number pi would be better.)
The Mirror Has Two Faces (1996) shows math professor Jeff Bridges explaining the Twin Prime Conjecture (there are infinitely many pairs of primes only two apart, such as 3 and 5 or 11 and 13) to English professor Barbara Streisand.
In a little known 1995 film called Antonia’s Line, a chronicle of five generations of women, we see Antonia’s granddaughter Theresa grow from being a child prodigy to become a professional mathematician, who, with unfortunate stereotypical coldness, prefers to lecture on cohomology and read research papers on differential geometry rather than nurse her baby. (All the real-life mathematician mothers I know do both.)
Finally, let’s not forget Stand and Deliver, the excellent 1987 dramatization of the real-life story of the late Jaime Escalante, an inspired math teacher who managed to teach calculus to a class of socially deprived, Hispanic high school students in riot torn South Central Los Angeles. This is really about math teaching rather than math, though for most people (apart from mathematicians) the two are the same.
As the lead character says in Pi, mathematics is about identifying and analyzing patterns. What, if any, pattern can we see in the inclusion of mathematics in movies?
Apart from films such as Sneakers, where the plot brings in mathematics automatically, the most common attraction seems to be the very fact that, for most people, math is so incomprehensible. This immediately sets the mathematical character apart in terms of intellectual ability. The audience expects the mathematician to do clever things. The problem facing the director is to show the audience the mathematician at work in a credible way that intrigues at the same time as it baffles.
Do Good Will Hunting and Pi really indicate a new Hollywood trend, where all the leading starts will clamor to play the role of a mathematician? Will we see Clint Eastwood eveball a young student along the line of a stick of chalk and growl “Go on, prove my theorem.” Somehow, I doubt it. Looking back over all those other films involving a mathematician, I suspect that what we witnessed of late was merely the familiar bunching of random sequences. But who knows? In any event, I’ll give my two thumbs up to the idea of having more mathematician movie heroes—for the simple reason that mathematicians should not be the only group not represented in the movies.
Going back to my original question: Movies apart, why the sudden popular interest in mathematics? I’ll leave that as an exercise for the reader. I’ll give my answer in next month’s column. Let me know what you think, and I’ll try to include some views that differ from mine. (Experience tells me there will be no shortage of those!)
– Keith Devlin
Devlin’s Angle is updated at the beginning of each month.
Keith Devlin ( devlin@stmarys-ca.edu) is Dean of Science at Saint Mary’s College of California, in Moraga, California, and a Senior Researcher at Stanford University. His latest book The Language of Mathematics: Making the Invisible Visible, has just been published by W. H. Freeman.
DECEMBER 1998
1998: A Good Year for Math?
Was 1998 a good year for mathematics or a poor one? It depends on the yardstick you use to measure performance. In terms of media attention, it has probably been the best year ever (see my November column, “Math Becomes Way Cool”). But if you are looking for major new results, it has been a relatively poor year.
At about this time each year for the past four years, the American Mathematical Society has published an attractive (full color, high gloss) pamphlet titled What’s Happening in the Mathematical Sciences. Written by mathematics writer Barry Cipra, WHIMS sets out to do what its title suggests: It provides accounts of the most significant developments in mathematics over the previous twelve months, written at an accessibility level roughly the same as Scientific American.
Volume 4 of WHIMS appeared recently, and it’s as good as the previous three issues. Barry Cipra is to be praised for his excellent mathematical writing, Paul Zorn has done a magnificent job as editor, and the AMS does the mathematical community a great service by publishing this annual summary.
One thing that struck me as I read this latest issue, however, is that all the developments described occurred in previous years. Now, don’t get me wrong. I am not drawing any conclusions from this observation. For one thing, the acquisition of human knowledge proceeds at its own, often erratic pace. For another, there is more to mathematics than major breakthroughs that hit the front pages of daily newspapers.
The reason I noticed the absence of new results in the latest WHIMS was simply that 1998 has been the very year in which math has found its way onto network television, national radio, and the national press, not just occasionally but with some frequency. There were even both a movie and a deodorant called “Pi”.
Why this sudden interest? Probably a range of factors. Following my “Math Becomes Way Cool” article, a number of readers wrote to tell me their own explanation. I suspect that, as several of my correspondents suggested, the most significant factor leading to the sudden surge of interest was Andrew Wiles’ proof of Fermat’s Last Theorem.
Yes, it was over five years ago when Wiles first announced that he had found a proof, and four years since he was able to correct a major error in that first proof and produce an argument that the experts agreed was correct. But because of the media attention that Wiles’ breakthrough attracted, public awareness of mathematics as an area of research was raised to a hitherto unprecedented level. That high level of interest was sustained by the subsequent appearance of a television documentary and an associated, best selling book on Wiles’ dramatic breakthrough.
Add into the mix equally successful recent biographies of Paul Erdös and Nobel prizewinner John Nash, and you have a media environment in which math is floating around with everything else. (The thing to remember about radio and television is the degree to which they are parasitic on newspapers and magazines, and on each other. Once a topic is “in the loop,” it tends to get a lot of coverage. It’s getting into the loop in the first place that’s the problem for the media hungry publicist.)
The Contents of Barry Cipra’s WHIMS
Cipra kicks off Volume 4 of WHIMS with a discussion of game theory, taking as his peg the 1997 victory of IBM’s Deep Blue chess computer over World Chess Champion Garry Kasparov.
He follows that with a discussion of some recent work that suggests an amazing possible connection between the Riemann Hypothesis of analytic number theory and a problem in quantum physics. The connection would imply that the zeros of the zeta function can be interpreted as energy levels in the quantum version of some classically chaotic system.
After giving readers his own account of Erdös, Cipra goes on to tell us about some recent advances in the use of computers to do mathematics. First, the widespread availability of powerful computer algebra systems has led to a revival of some decidedly old-fashioned algebraic techniques in geometry.
Next, Cipra reports on the 1996 success of the theorem-proving program EQP in proving the Robbins Conjecture, a puzzle that had resisted all attempts at a solution since it was first proposed in the 1930s. The conjecture was that a particular collection of valid propositions of boolean algebra actually constitutes an axiom system for boolean algebra.
Taking the symbol + to denote logical disjunction (or) and N for negation (not), one way to axiomatize boolean algebras is by the following four axioms:
(1) p + q = q + p
(2) (p + q) + r = p + (q + r)
(3) p + N(q + N(q)) = p
(4) p + N(N(q) + N(r)) = N(N(p + q) + N(p + r))
In 1933, Edward Huntington showed that axioms (3) and (4) can be replaced by the single statement:
p = N(N(p) + q) + N(N(p) + N(q))
Huntington’s student Herbert Robbins suggested the following alternative to his adviser’s axiom:
p = N(N(p + q) + N(p + N(q)))
However, neither Robbins nor Huntington was able to prove that this alternative could be used in place of Huntington’s axiom. Nor was anyone else. The question remained unresolved until EQP found a proof. The new proof derived a known axiom from the Robbins axiom, in just short of 50,000 steps.
After computers come flour beetles, when Cipra reports on some fascinating experimental work that shows that the population growth of a controlled colony of the insects shows all the familiar signs of chaotic dynamics: period doubling bifurcations, strange attractors, and the like. (Life imitating mathematics?)
Quantum computing—still all promise, but a fascinating concept that will change the face of computing if it ever becomes a practical proposition—is the topic of a chapter titled From Wired to Weird. That in turn is followed by a report on a new method for public-key cryptography that has the mathematical potential to rival the popular RSA system.
Cipra caps off his tour with a chapter on mathematics and art and a reprint of Henri Poincare’s famous 1908 essay “Mathematical Discovery”.
What’s Happening in the Mathematical Sciences, Vol 4 is available directly from the AMS for $14 plus post and packing. To order by phone call toll-free 1-800-321-4AMS (4267) in the US and Canada, or 401-455-4000 worldwide. Ask for itemcode: HAPPENING/4. To order electronically, click here.
– Keith Devlin
Devlin’s Angle is updated at the beginning of each month.
Keith Devlin ( devlin@stmarys-ca.edu) is Dean of Science at Saint Mary’s College of California, in Moraga, California, and a Senior Researcher at Stanford University. His latest book The Language of Mathematics: Making the Invisible Visible, has just been published by W. H. Freeman.
profkeithdevlin.org 