From 1997 onwards, the original files I sent in to the MAA were all intact on my computer. So the posts published here will be as they first appeared apart from editorial changes made by the MAA Online editor prior to publication. And from the February post onwards. And from the February post onwards, the files I submitted were in HTML, so the formatting will be as when it appeared. (I learned Web coding specifically for that purpose.) Within a few years, what I produced would be referred to as “vanilla html”, and just a few years later much of the html coding would be generated automatically from regular text entry.
JANUARY 1997
Why 2001 Won’t be 2001
January 12, 1997, is the birthday of HAL, the mission-control computer on the Jupiter-bound spaceship Discovery in Arthur C. Clarke’s celebrated science fiction novel 2001: A Space Odyssey. According to the book, HAL was commissioned at Champagne-Urbana on January 12, 1997. In Stanley Kubrick’s movie version, the date of HAL’s birth was inexplicably changed to January 12, 1992. In any event, whether HAL is just about to be born or preparing to celebrate its fifth birthday, with the year 2001 practically upon us, it’s natural to ask how correct Clarke and Kubrick’s vision of the future has turned out to be.
Thirty years ago when the film was made, director Kubrick endowed HAL with capabilities computer scientists thought would be achieved by the end of the century. With a name that, despite Clarke’s claim to the contrary, some observers suggested was a simple derivation of IBM (just go back one letter of the alphabet), HAL was, many believed, science fiction-shortly-to-become-fact.
In the movie, a team of new millennium space explorers set off on a long journey of discovery to Jupiter. To conserve energy, the team members spend most of the time in a state of hibernation, their life-support systems being monitored and maintained by the on-board computer HAL. Though HAL controls the entire spaceship, it is supposed to be under the ultimate control of the ship’s commander, Dave, with whom it communicates in a soothingly soft, but emotionless male voice (actually that of actor Douglass Rain). But once the vessel is well away from Earth, HAL shows that it has developed what can only be called a “mind of its own.”
Having figured out that the best way to achieve the mission for which it has been programmed is to dispose of its human baggage (expensive to maintain and sometimes irrational in their actions), HAL kills off the hibernating crew members, and then sets about trying to eliminate its two conscious passengers. It manages to maneuver one crew member outside the spacecraft and sends him spinning into outer space with no chance of return. Commander Dave is able to save himself only by entering the heart of the computer and manually removing its memory cells.
Man triumphs over machine—but only just. It’s a good story. (There’s a lot more to it than just described.) But how realistic is the behavior of HAL?
We don’t yet have computers capable of genuinely independent thought, nor do we have computers we can converse with using ordinary language. True, there have been admirable advances in systems that can perform useful control functions requiring decision making, and there are working systems that recognize and produce speech. But they are all highly restricted in their scope. You get some idea of what is and is not possible when you consider that it has taken AT&T over thirty years of intensive research and development to produce a system that can recognize the three words “yes”, “no”, and “collect” with an acceptable level of reliability for a range of accents and tones. Despite the oft-repeated claims that “the real thing” is just around the corner, the plain fact is that we are not even close to building computers that can reproduce human capabilities in thinking and using language. And according to an increasing number of experts, we never will.
Despite the present view, at the time 2001 was made, there was no shortage of expert opinion claiming that the days of HAL (“HALcyon days,” perhaps?) were indeed just a few years off. The first such prediction was made by the mathematician and computer pioneer Alan Turing. In his celebrated article “Computing Machinery and Intelligence”, written in 1950, Turing claimed, “I believe that at the end of the century the use of words and general educated opinion will have altered so much that one will be able to speak of machines thinking without expecting to be contradicted.” Though the last part of Turing”s claim seems to have come true, that is a popular response to years of hype rather than a reflection of the far less glamorous reality. There is now plenty of evidence, from psychology, sociology, and from linguistics, to indicate that the original ambitious goals of machine intelligence is not achievable, at least when those machines are electronic computers, no matter how big or fast they get.
So how did the belief in intelligent machines ever arise? Ever since the first modern computers were built in the late 1940s, it was obvious that they could do some things that had previously required an “intelligent mind.” For example, by 1956, a group at Los Alamos National Laboratory had programmed a computer to play a poor but legal game of chess. That same year, Allen Newell, Clifford Shaw, and Herbert Simon of the RAND Corporation produced a computer program called The Logic Theorist, which could—and did—prove some simple theorems in mathematics.
The success of The Logic Theorist immediately attracted a number of other mathematicians and computer scientists to the possibility of machine intelligence. Mathematician John McCarthy organized what he called a “two month ten-man study of artificial intelligence” at Dartmouth College in New Hampshire, thereby coining the phrase “artificial intelligence”, or AI for short. Among the participants at the Dartmouth program were Newell and Simon, Minsky, and McCarthy himself. The following year, Newell and Simon produced the General Problem Solver, a computer program that could solve the kinds of logic puzzles you find in newspaper puzzle columns and in the puzzle magazines sold at airports and railway stations. The AI bandwagon was on the road and gathering speed.
As is often the case, the mathematics on which the new developments were based had been developed many years earlier. Attempts to write down mathematical rules of human thought go back to the ancient Greeks, notably Aristotle and Zeno of Citium. But the really big breakthrough came in 1847, when an English mathematician called George Boole published a book called An Investigation of the Laws of Thought. In this book, Boole showed how to apply ordinary algebra to human thought processes, writing down algebraic equation in which the unknowns denoted not numbers but human thoughts.
For Boole, solving an equation was equivalent to deducing a conclusion from a number of given premises. With some minor modifications, Boole’s nineteenth century algebra of thought lies beneath the electronic computer and is the driving force behind AI. Another direct descendent of Boole’s work was the dramatic revolution in linguistics set in motion by MIT linguist Noam Chomsky in the early 1950s.
Chomsky showed how to use techniques of mathematics to describe and analyze the grammatical structure of ordinary languages such as English, virtually overnight transforming linguistics from a branch of anthropology into a mathematical science. At the same time that researchers were starting to seriously entertain the possibility of machines that think, Chomsky opened up (it seemed) the possibility of machines that could understand and speak our everyday language. The race was on to turn the theories into practice.
Unfortunately (some would say fortunately), after some initial successes, progress slowed to a crawl. The result was hardly a failure in scientific terms. For one thing, we do have some useful systems, and they are getting better all the time. The most significant outcome, however, has been an increased understanding of the human mind: how unlike a machine it is and how un-mechanical human language use is.
One reason why computers cannot act intelligently is that logic alone does not produce intelligent behavior. As neuroscientist Antonio Damasio pointed out in his 1994 book Descartes’ Error, you need emotions as well. That’s right, emotions. While Damasio acknowledges that allowing the emotions to interfere with our reasoning can lead to irrational behavior, he presents evidence to show that a complete absence of emotion can likewise lead to irrational behavior. His evidence comes from case studies of patients for whom brain damage—either by physical accident, stroke, or disease—has impaired their emotions but has left intact their ability to perform “logical reasoning”, as verified using standard tests of logical reasoning skill. Take away the emotions and the result is a person who, while able to conduct an intelligent conversation and score highly on standard IQ tests, is not at all rational in his or her behavior. Such people often act in ways highly detrimental to their own well being. So much for western science’s idea of a “coolly rational person” who reasons in a manner unaffected by emotions.
As Damasio’s evidence indicates, truly emotionless thought leads to behavior that by anyone else’s standards is quite clearly irrational. And as linguist Steven Pinker explained in his 1994 book The Language Instinct, language too is perhaps best explained in biological terms. Our facility for language, says Pinker, should be thought of as an organ, along with the heart, the pancreas, the liver, and so forth. Some organs process blood, others process food. The language organ processes language. Think of language use as an instinctive, organic process, not a learned, computational one, says Pinker.
So, while no one would deny that work in AI and computational linguistics has led to some very useful computer systems, the really fundamental lessons that were learned were not about computers but about ourselves. The research was successful in terms not of engineering but of understanding what it is to be human. Though Kubrick got it dead wrong in terms of what computers would be able to do by 1997, he was right on the mark in terms of what we ultimately discover as a result of our science. 2001 shows the entire evolution of mankind, starting from the very beginnings of our ancestors Homo erectus and taking us through the age of enlightenment into the present era of science, technology, and space exploration, and on into the then-anticipated future of routine interplanetary travel.
Looking ahead forty years to the start of the new millennium, Kubrick had no doubt where it was all leading. In the much discussed—and much misunderstood—surrealistic ending to the movie, Kubrick’s sole surviving interplanetary traveler reached the end of mankind’s quest for scientific knowledge, only to be confronted with the greatest mystery of all: Himself. In acquiring knowledge and understanding, in developing our technology, and in setting out on our exploration of our world and the universe, said Kubrick, scientists were simply preparing the way for a far more challenging journey into a second unknown: the exploration of ourselves.
The approaching new millennium sees Mankind about to pursue that new journey of discovery. Far from taking away our humanity, as many feared, attempts to get computers to think and to handle language have instead led to a greater understanding of who and what we are. For today’s scientist, inner space is the final frontier, a frontier made accessible in part by attempts to build a real-world HAL.
Happy birthday, HAL!
– Keith Devlin
Dr. Devlin is a Senior Researcher at Stanford University’s Center for the Study of Language and Information (CSLI). He is the Dean of Science and a Professor of Mathematics at Saint Mary’s College of California, in Moraga, California, and a Consulting Research Professor in the Department of Information Science at the University of Pittsburgh. He is the author of sixteen books and over sixty research articles. His books include Mathematics: The Science of Patterns, a Scientific American Library book published by W. H. Freeman in 1994, and the award winning Logic and Information, published by Cambridge University Press in 1991. This celebration of the birth of HAL, the computer in the book and film 2001, is abridged from the book Goodbye Descartes: The End of Logic and the Search for a New Cosmology of Mind, to be published by John Wiley and Sons in late January, 1997.
FEBRUARY 1997
Eskimo Pi
As everyone knows, in the last century, the State of Indiana passed a law legislating the value of pi to be 3.
Well, perhaps it’s a bit of an exaggeration to say that everyone knows it, but most people do. But that alone is very odd, since it is not true.
If you go and search the records of the Indiana House of Representatives, what you will find is that there was a bill passed that related to pi, House Bill No. 246 to be precise. What this 1897 bill did was offer the State of Indiana an angle trisection, a cube duplication, and a circle quadrature. Obviously, a misguided amateur mathematician had managed to persuade a legislator that he had something of value to offer.
Though it passed in the House, the bill had less success when it went to the State Senate. Admittedly it is unlikely that any of the good senators of Indiana was able to evaluate the claims made in the bill, but they nevertheless had the good sense to consider it not a proper matter for the law. They postponed any action on the matter, and so it has remained ever since.
The full Indiana pi story (yes, there’s more to it) is recounted by Underwood Dudley in his excellent book Mathematical Cranks, published by the MAA in 1992.
The odd thing about the whole affair is not that some midwestern crackpot deluded himself into thinking he had made a great mathematical discovery, nor that a bunch of busy lawmakers saw little harm in accepting what seemed like a free gift from one of their citizens. No, what is odd is the way the “legislating pi” story not only became established, but persevered in the face of all the facts. I have no great hopes that this article will be any more successful in stopping the myth any more than did the appearance of Dudley’s book.
At least we can console ourselves that we are not alone. Just about every discipline has its collection of enduring myths. For instance, what about that well-known observation that the Eskimos have hundreds of words for snow?
The standard explanation for this durable linguistic factoid is that snow plays an important part in the life of an Eskimo, so they need lots of ways to describe it. Both the observation and more expansive versions of the explanation can be found in numerous books and articles on linguistics. Unfortunately, for all its ubiquity, the observation is false, as is the familiar explanation.
The whole sorry saga was released to an eager world a few years ago in a book called The Great Eskimo Vocabulary Hoax, written by the linguist Geoffrey Pullum, of the University of California at Santa Cruz. Pullum based his account on some fine detective work by the anthropologist Laura Martin.
According to Martin, as recounted by Pullum, the story has its origins in an observation of the founder of American linguistics, Ralph Boas (1858–1942). In 1911, in the introduction to his book The Handbook of North American Indians, Boas mentioned, in passing, that Eskimos have four distinct words for snow, aput for snow lying on the ground, gana for falling snow, piqsirpoq for drifting snow, and qimuqsuq for a snow drift, whereas English can only make similar distinctions by means of phrases involving the one word ‘snow’ (such as ‘drifting snow’, ‘falling snow’, etc.).
And there things remained—no flurry of snow stories, if you will forgive the pun—until 1940, when Boas’s observation was picked up by a gentleman called Benjamin Lee Whorf, who included it in a popular article “Science and linguistics”, which was published in the MIT magazine Technology Review.
Whorf (1897–1941) had earned a degree in chemical engineering at MIT, became a fire prevention inspector with an insurance company in Hartford, Connecticut, and pursued linguistics as a hobby.
According to Whorf’s Technology Review article, “We have the same word for falling snow, snow on the ground, snow packed hard like ice, slushy snow, wind-driven flying snow—whatever the situation may be. To an Eskimo, this all-inclusive word would be almost unthinkable; he would say that falling snow, slushy snow, and so on, are sensuously and operationally different, different things to contend with; he uses different words for them and for other kinds of snow.”
If you count up Whorf’s examples, with the vague reference to “and other kinds of snow” counting conservatively as two, you arrive at a figure of seven Eskimo words for snow. That is already three up on Boas. And as Pullum points out, numeric inflation is not the only problem with this passage. It simply is not the case the English has only the one word for all those kinds of snow. What about the perfectly respectable English words ‘slush’, ‘sleet’, and ‘blizzard’? To which one could arguably add the words ‘flurry’ and ‘dusting’ and the skiers’ words ‘powder’ and ‘pack’.
With Whorf’s article released to the world at large, over the years the story spread far and wide. And as the story spread, so the number of words grew: ten, twenty, fifty, reaching as many as a hundred in a New York Times editorial on February 9, 1984. As Martin says, “We are prepared to believe almost anything about such an unfamiliar and peculiar group [as the Eskimos].” (What Martin would say about the good people of Indiana I’m not sure.)
So what is the real answer? How many words for snow do the Eskimos have? Well, such a question cannot possibly have a cut-and-dried answer. Which peoples are you going to count as ‘Eskimos’? There are a number of tribes that you might or might not include. What exactly do you mean by ‘a word’? If you allow derivation words, then English has a lot of words for snow: ‘snowfall’, ‘snowbank’, ‘snowdrift’, ‘snowstorm’, ‘snowflake’, and so on. However, if you decide to take a fairly hard line and ask for word roots, not derivations from those roots, then the answer varies between two and a dozen, depending on which peoples you count as Eskimos. That’s right, the most you can reasonably claim is about twelve Eskimo words for snow. Not many more than in English in fact.
– Keith Devlin
Devlin’s Angle is updated at the beginning of each month.
Keith Devlin (devlin@stmarys-ca.edu) is the editor of FOCUS, the news magazine of the MAA. He is the Dean of Science at Saint Mary’s College of California and the author of Mathematics: The Science of Patterns, published by W. H. Freeman in 1994 and Goodbye, Descartes: The End of Logic and the Search for a New Cosmology of the Mind, published by John Wiley in 1997.
MARCH 1997
How to Get Into the Newspapers
With Mathematics Awareness Week coming up soon, many a university mathematician’s fancy will be turning to thoughts of getting a story into the local newspaper. How is this done?
Well, there are no magic secrets, at least as far as I know. Over the years I’ve achieved some degree of success, and in this column I’m going to note down a few reflections. But I don’t claim any particular expertise in the matter. If you don’t agree with something I say, well that’s fine. If you think I have missed something out, please drop me an e-line and let me know—I am always looking for new ideas. My comments are directed at university mathematicians. That is the only perspective I am familiar with.
1. The first thing to ask yourself is what is it you are trying to say. Can you summarize your main point in one or two sentences in a way that a non mathematician can appreciate? If you can’t, you are probably not going to get very far. Before you approach your local newspaper, try out your one or two sentence summary on colleagues outside the sciences. If they can’t grasp your point immediately, what chance do you have with an editor (who in turn will be an “easier sell” than the average newspaper reader you are trying to reach).
2. Second question. Why should the newspaper want to print it? Alternatively, why should people want to read it? These are not the same question as “Why do I want to write it?” or “Why do I think people should want to read it?”, let alone the not uncommon “Why do I think it would be good for people to read it?” Remember, newspapers are not in the business of education, and people don’t buy newspapers to be educated. The purpose of newspapers is in part to inform, in part to entertain. Your article should have either news value or general interest value. Ways of giving news value to an article are to link it to an upcoming or recent local event, the announcement of a scientific discovery made locally (there aren’t many of those), or a multiple-of-five anniversary of some important event. For instance, the recent, mythical birthday of HAL, the computer in the film 2001, was covered by most national newspapers, and I managed to get a “Where is HAL now?” piece into my own local Sunday newspaper.
Be aware that news can be created. You just need an appropriate event. My own institution recently managed to get two major stories on one of our faculty: the first when he made a discovery (event 1); the second two years later when the research article he wrote was published (event 2). (Local prof gets published in distinguished learned journal was the angle second time around.) The content of the newspaper report was virtually the same, what changed was the new angle—the reason for publication.
3. Remember that you are not writing for the New York Times. (If you are, good luck to you. With JPBM Communications Award winning journalist Gina Kolata covering mathematics for them, I have never even tried.) Take a good look at the kinds of article that appear regularly in your newspaper. Assuming the editor knows what his or her readers are interested in (and the chances are huge that the editor has a better handle on that than you do), this initial exercise should give you a good idea of the kind of issues, and the level of presentation, that stand some chance of finding their way into print. And don’t forget to check the length of the articles that the newspaper typically publishes. It is probably quite low—between 400 and 700 words. Brevity will have to be your motto. Even if the editor likes what you have written, if it is too long, it will be cut before publication. It’s better to make the cut yourself, than run the risk of your carefully crafted theme being lost when someone else hits the delete key.
4. Your audience (to say nothing of the editor you have to get past first) is not interested in mathematics. Yes, I know that’s a shame, and wouldn’t it be wonderful if everyone in the world shared our passion for x, y, and z, but the sad fact of the matter is that they don’t. What are they interested in? Other people, mostly, and things that affect their own lives; sport; TV personalities; housing prices; the cost of living; Madonna; O.J. Simpson; possibly a few other things. So that’s where you have to start. Those are the pegs onto which you have to hang your article. Education is another of those “human interest” entryways, but remember that for the vast majority of people, mathematics education means elementary arithmetic, and will their kids be able to do their sums and get the right answer as well as those oh-so-smart Japanese children we keep reading about. (Notice that: we keep reading about them! Get it?)
5. Not only is your audience uninterested in mathematics, they know virtually nothing about it. Incidentally, if you can’t utter that last sentence without a trace of condescension or condemnation, you might as well give up now. The reader you are trying to reach has been getting on just fine without your help so far; why on earth should they voluntarily read through an article that makes them feel ignorant or inferior?
6. Make sure you really understand what it means to say your reader does not understand mathematics. Really not understand. If your university has a “pre-college” level mathematics class (don’t even think of calling it a “remedial class”), sit in on it for a few sessions. Then remind yourself that these are the people who have gotten to university, and who have a motivation to learn mathematics (even if it’s not the motivation you would like to see). Most of your readers did not get that far in their education, and what’s more it was a long time ago that they were last in a classroom. So forget all thought of formulas or assuming any concepts beyond whole numbers and triangles.
Avoid any temptation to “be precise.” You are not trying to get past the editor of Annals of Mathematics. The editor of your local paper cares not one whit about “convergent power series.” You’ll do well if you get the phrase “infinitely long polynomial” past him. (Sic. You might also find that your conscious efforts to use gender-neutral language are all in vain, though things seem to be improving on that front these days.)
7. Okay, so you have figured out what you want to say, why people might be interested in it, and how to say it. The next step is to take your draft article to the person on your campus who deals with public relations. He or she is probably a former newspaper reporter, and knows the business. At this stage, you are no longer the expert; your PR person is. Remember, he or she is on your side; the newspaper editor you are trying to get to accept your article is not. So listen to the advice you receive. If you hear the words “Ah, here is your real story!” your reaction should be one of absolute glee. You have just made it to first base. Chances are it is not the story you wanted to tell, or the one you thought you were telling. But, hey, now you know you have something that people might want to read. That’s already a major victory. Make the changes your advisor suggests. Don’t argue; make them! Remind yourself yet again that, just as you are the expert on mathematics, your PR person is the expert at getting stories into the press.
8. How to submit? One way is to let your campus PR person take over from now on. He or she has the contacts, and can ensure that your words are at least glanced at by someone in the newspaper’s editorial office. If you do have to do it yourself, send in your article to the appropriate editor with a brief cover letter, and say you will call in a few days to see if they want your help to make any changes. (This is just an excuse for you to give them a nudge.) Give your home and office phone numbers so they can contact you if necessary.
9. How to improve your chances? One way I have used that has been very successful is to find something that is going to be covered—or just has been covered—by the national press, and give a local slant to the story. (Remember those local newspapers that published stories under headlines like “Local couple killed at sea” on the day the Titanic went down.) In your cover letter, you can say something to the effect that “The Washington Post (or whatever) just published an article on X, and I wondered if you would be interested in the following local angle on the story.” (That’s how I managed to get my HAL piece into print. I told the editor that I was sure most of the national dailies would cover the event. The only local angle about my version was that I wrote it. But that was enough. The newspaper even printed a file photo of me alongside the article, rather than a still from the movie, in order to play up the local angle.)
Finally, let’s suppose all your efforts have paid off, and your words appear in print. That is not the end of the process—-unless you want this to be the only time you (and possibly any other mathematician within a fifty mile radius) get an article into the newspaper. There is one more thing you should do, and something you should not do.
10. What you should do is call or write to the editor thanking them for their help in getting the word out, and expressing your hope to work with them again at a future date. Start to build up a relationship. You never know, the next time there is a major news story with a mathematical angle, you might find your editor calling you up for a quote or, even better, an op-ed piece.
11. What you should not do is complain about the way the copy editor mangled your story and completely obscured or misrepresented your main point. This is almost inevitable; newspapers are put together at great speed by people who are expert at cutting text to fit a particularly sized slot, but who are dealing with topics of which they have little or no knowledge. Your article is just one of many that pass through their computer. (Newspapers never show you the final copy of your article, by the way, even if you send it to them well in advance of publication. Don’t even suggest it. It’s not the way they work—and if you have gotten this far in the process, you are well inside their territory, where their rules prevail.)
If you want to see your words appear exactly as they came off your printer, don’t send your article to a newspaper. Post it on your Web home page where you (and probably you alone) can read your words to your heart’s content. Remember, your main aim was (or should have been) to get your local paper to say something—anything—about mathematics. Those few readers who can see there is something wrong with the mathematics as printed will have no problem making the appropriate mental correction. For the others—the vast majority of your readers—why worry? You did not—or should not have—set out to teach your readers anything. Heavens above, the students who sit in our classes for three or four hours each week have trouble remembering what we tell them several times over. The person who glances over your article as they eat their breakfast cereal is unlikely to remember any mathematical content by the time they have emptied their bowl, let alone when they are fighting the traffic to get to work. The most you can—and should—hope for is that a little bell has rung briefly in their minds, saying “Mathematics.” You have added your epsilon increment to that far-off Nirvana where everyone appreciates that mathematics is an important and relevant part of contemporary life.
So don’t complain. Don’t even hint that there was anything wrong. The odds are, if a story with any mathematical content has gotten into the paper, the editor decided to “take a chance.” Be grateful. If the result of taking that chance is that the editor is made to feel incompetent, ignorant, or just plain stupid, then the next time a mathematician comes along, the decision will more than likely go the other way. Remember who is doing the asking here. If you don’t like the rules, play another game.
Well, my fifteen minutes are up. If you don’t like being lectured at this way, but for some perverse reason have continued to the bitter end, you can always decide to stop reading “Devlin’s Angle” in the future, just as people can skip over your newspaper masterpiece. Writing a column for commited fellow professionals is not the same as writing a newspaper article, but columnists too can easily find that if they don’t follow the rules, the outlet dries up. One of the rules about writing a column is don’t lecture to your readers. On this occasion I have deliberately broken that rule. Direct your complaints to the MAA’s Director of Publications. He has the ultimate editorial control. He can fire me.
– Keith Devlin
Devlin’s Angle is updated at the beginning of each month.
Keith Devlin ( devlin@stmarys-ca.edu) is the editor of FOCUS, the news magazine of the MAA. For many years, he wrote a twice-monthly column in The Guardian newspaper in England. A selection of those articles was published in book form by the MAA in 1994, under the title All The Math That’s Fit to Print.
APRIL 1997
When mathematics has to be theater – A tale with a moral
Several years ago, I promised myself I would never write another textbook. Not that I was losing interest in mathematics or in writing. Rather, it was getting to the point where there were so many good textbooks around, I could no longer see any place where I could make a significant new contribution. At least, that is how I saw—and see—the situation for “specialist” textbooks in my area, aimed at mathematics majors. When it comes to introductory textbooks such as pre-calculus and calculus, I see a different state of affairs.
These days, calculus textbooks come in two kinds: hard covered, called “traditional”, and soft covered, called “reform”. The former favor cover designs featuring intellectually inspiring items such as violins; the latter prefer a more sporty image, such as a hot air balloon. Of course, I jest. The two kinds of textbook really do differ, and I know from experience that writing a good textbook is no easy matter. Within each of the two calculus camps, however, the traditional and the reform, the contents of one textbook differ little from any other. The last time I was involved in making a departmental textbook adoption decision, the final choice between maybe half a dozen texts was made almost entirely on the basis of design: which textbook did we think the students would find the most attractive and the least intimidating?
So when, eighteen months ago, I was approached by a publisher and asked to write a calculus textbook, I had no hesitation in saying “no.” Even though this was not to be a paper textbook but an electronic text on CD ROM. The publisher persisted. Interactive texts are not at all like traditional texts, she insisted. An interactive text is not—or rather should not be—just a book transferred onto a disk and read on a computer screen.
“Why not approach one of the established calculus textbook writers,” I retorted, even going so far as to name several whose work I admired. “No, I think it is important to have someone who has not written a calculus textbook before,” the publisher replied. “We want an author whose first attempt to write a calculus text is to write it as an interactive text. Done properly, an interactive text should be very different from a book. The entire presentation is different. You have to re-think everything from the ground up.”
I was still not convinced, but I was certainly intrigued. As readers of my recent book Goodbye, Descartes will know, my research interests for the past several years have been in language, thought, and human cognition. Trying to figure out how to use multimedia to help students learn calculus seemed like an interesting sideline to that research project. During my career, I have worked on projects to explain mathematics in the newspapers, on radio, and on television, and have always enjoyed the challenge of trying to use a new medium to get the message across. Getting a newspaper article past a hard-nosed news or features editor is very different from getting a piece into, say, the American Mathematical Monthly. And radio and television, whether scripted or “live and spontaneous,” are very different environments from the classroom or the mathematics department common room. What new challenges would arise in trying to use multimedia to explain calculus?
On the other hand, I knew of a number of projects nationwide to develop interactive texts on a variety of topics, including calculus. Why didn’t the publisher just approach one of them and offer to publish their product commercially, I asked. After all, for those projects, most of the initial development work will have been paid for by grants or institutional support. In contrast, I would have to start from scratch. This time, the publisher’s answer was, for me, almost the clincher. Her intention was, she said, to design and develop the product as a commercial product, with all the pressures that would entail. It would be sold directly to students (not sold by class adoptions), and hence would have to appeal to students on its own merits. It would have to sell for under $30. It would have to run on any reasonably contemporary PC or Mac. It was not intended as a “first text;” it would be called an Electronic Companion to Calculus, designed to complement any college-level first-year course, regardless of the textbook used or the method of instruction. It would have to be designed and produced on a tight budget built into the company’s business plan. These were just the kinds of issues I had faced writing for the press, radio, and television. Creating a good, honest product under those kinds of constraints presents a real challenge I had found I enjoyed meeting.
As I said, I was almost hooked. I had just one final question. The project would clearly be a joint venture. I know about calculus, and like every mathematician I have an opinion as to how it should be taught. But I knew little about developing multimedia and about using interactivity as a pedagogic device. Who would be the other members of the team? They would be crucial to any possibly success. So the publisher arranged for me to meet the others. Suffice it to say that after I had met them, I was highly flattered to have been approached to be the author. This was an impressive and highly talented team, with a wide variety of experience. We were about to enter uncharted waters, and the final product might “bomb.” (Few CD ROM products, if any, have been commercial successes to date, and some major publishers are already getting out of the CD ROM business.) But working with that team would be a marvellous experience, regardless of the outcome.
So I said yes. And promptly found myself living the kind of life generally associated with twenty-five year old computer whizzes working for Silicon Valley startup companies. With a full-time job as dean during the day, burning the midnight oil became my way of life for several months, as we deconstructed calculus and reassembled it from the ground up so that everything—both mathematical content and all the navigational apparatus of multimedia—could fit onto one screen. Whatever the topic, I had to make do with one small text area and one slightly larger image area. Paging forward was, my interface expert kept telling me, to be avoided wherever possible. Don’t use words, use diagrams, animations, and interactivity, she kept saying. Don’t present the material in a linear, cumulative way, she insisted. The user will want to jump around and explore, using the hyperlinks, very likely not starting at “screen 1” at all.
Using interactivity was the thing I found hardest to get the hang of, a sure sign of my half century age. Like most of my generation of instructors, I tend to think in terms of content, of explaining the method, and of conveying information-as-a-commodity. Interactivity is about process. Toward the end of the project, I began to appreciate the potential of interactivity as a pedagogic device. It really is very different from anything else in the instructor’s arsenal. Given a fresh start and a very large development budget, I think I could now take real advantage of what interactivity can offer. But as often in life, we had to live with what we had. We had all but spent our development budget and the business folk were pressing us to get the product out to market. For me this was an academic exercise. For everyone else, it was a business, and their livelihood depended on it!
Last week, I finished the project. For good or bad, the thing is ready to go on sale. Am I happy with it? Well, it’s not the product I would now like to produce, given what I know now. With lots more time and a much bigger development budget, I could really go to town! On the other hand, I have to say I do like it. I think we really have managed to take advantage of the medium to produce something new and useful. It does not—and was never intended to—replace anything else on the market. It was conceived from the beginning as a companion, designed to complement everything else that is available.
I believe it definitely has a “personality,” a very important feature for a stand-alone product of this nature, and a feature we struggled hard to develop. Though my name appears as the author, the final product represents the joint creative work of Belen, Bruce, Carol, David, Marian, Travis, Trish, Volker, and many others.
The point of this story? Not to advertise the product. The company has a marketing plan, and they don’t need my help in that arena. (Besides, the product is designed to sell directly to students, not to the faculty who I assume are the main readers of Devlin’s Angle.) Rather, my point is to say that if anyone is eager to develop multimedia educational products, you should get involved with professionals experienced in the area of multimedia before you put pen to paper (or finger to keyboard). One thing I learned early on, and kept being reminded of again and again, was how complex are the technical and cognitive issues that arise in multimedia. A multimedia product is not just a book and a few animations put onto a CD ROM. It’s a form of solitary experiental theater, and it has to be developed in that way.
I know we did not get everything right. I have ideas how we could improve lots of things; and I am certain there are flaws none of us are yet aware of. But on my own, with just my knowledge and experience as a mathematician and a writer, I could not have produced anything remotely like our final product. Writers of textbooks often complain about being “pressured” by editors to present (or delete) material against their will, or to change the presentation. With multimedia, those kinds of pressures are much more intense.
The mathematician who enters the world of commercial multimedia should be prepared to be just one cog in a very large and complex machine. To a great extent, the mathematical integrity of the final product rests largely on the author’s shoulders, and once the thing is out there, there will be no shortage of critics. So if you enter this new world, you will find yourself struggling to find new ways to get an old and familiar message across. In the case of an educational product, the medium is definitely not the message, of course. If there is something that you, as author, think has to be covered, it will be up to you to find a way to get it in that passes the muster of the others on the team, whose concerns will be with implementation, attractiveness, look-and-feel, ease-of-use, overall “theatrical” structure, cost, and marketability. When push comes to shove, its amazing what can be presented on a single interactive screen!
– Keith Devlin
Devlin’s Angle is updated at the beginning of each month.
Keith Devlin ( devlin@stmarys-ca.edu) is the editor of FOCUS, the news magazine of the MAA. The CD ROM package Electronic Companion to Calculus will shortly be published by Cogito Learning Media, Inc.
MAY 1997
Clash of the chess titans
On 3 May, in a hall in Manhattan’s Equitable Center in New York, the two most powerful chess players in the world will begin the first in a series of six games to see who is the best of the best.
One player will walk into the room looking the picture of health, having been training for the event for months, with a mixture of running, swimming, weight training, and other physical exercises—as well as studying past chess games and playing chess against a computer opponent.
The other player, a 2,800 pound giant, arrived a week ago in four 7-foot high moving crates delivered on a 16-wheel truck.
The first player is the Russian Grandmaster Gary Kasparov. The other player is Deep Blue, a special-purpose chess-playing computer designed and built by IBM. (The name Deep Blue is a derivation of the familiar term “Big Blue”, used to refer to IBM.)
Kasparov was born in Baku, the capital of the Russian republic Azerbaidzhan, in 1963. Since 1985, he has been the World Chess Champion.
Deep Blue was developed at IBM’s research facility in Yorktown Heights in upper New York State, under the guiding eye of an IBM scientist called Joe Hoane.
The last time these two titans clashed, in February last year, Kasparov lost the first game, won the second, drew games three and four, and won the final two. Man triumphed over machine. (See the March, 1996 Devlin’s Angle.)
Chastened, the team from Big Blue took their prodigy back to the labs and spent several months souping it up for this year’s rematch. The new model has 256 dedicated chess chips operating in parallel, as against 192 last year, and is claimed to be twice as fast as its predecessor. It can analyze 200 million chess positions in a single second. (A human Grandmaster averages about two a minute.)
Who will win this time round? Well, we’ll know the answer to that question soon enough.
A more interesting question, to my mind, is this: If Kasparov loses, what does it tell us about the human race, about computers, and about the future of intelligent life on the planet? Is Kasparov right when he says, of the coming match, “It’s about the supremacy of human beings over machines in purely intellectual fields. It’s about defending human superiority in an area that defines human beings.”
As he psyches himself up for the match, it is hardly surprising that Kasparov views the issue in that way. But wait a minute. Were it not for Kasparov, we would almost certainly already be at the stage where computers rule the world in chess—it seems clear that Kasparov is the only person alive who stands any chance at all against Deep Blue. And even if Kasparov wins this series, it will most likely be only a year or two before some “Deeper Blue” will be able to beat any human opponent, Kasparov included.
I view the outcome in exactly the opposite light from Kasparov. If there is one thing we have learned from forty years of research into artificial intelligence, it’s this: The things we usually regard as intellectually hard—chess playing, solving algebra problems, evaluating integrals, et cetera—are relatively easy to program a computer to perform at least as well as, and often much better than, a human being. But the things that practically every small child can do—use language fluently to communicate, understand a story, recognize a face, et cetera—defy anything but a very crude approximation by a computer.
With the arrival of electronic calculators, human beings lost their centuries old sole planetary rights to being able to do arithmetic.
The development of computer algebra systems such as Mathematica and Maple meant we likewise had to accept that machines could outperform us in algebra and much of calculus.
Well, now the same thing is about to happen in chess. “So what?” is my response. At least, “So what?” when it comes to the doomsayers who see the development of Deep Blue as the beginning of the end for humankind.
Let’s just step back and think a minute. No digital computer plays chess. Nor do they perform algebra. They don’t even perform arithmetic. These are all intensely human activities. What computers do is simply sit there and obey the laws of physics.
What makes them useful to us, of course, is that we (note that: we) design and build them so that when electric current flows through a computer, simply obeying the laws of physics, we can interpret its behavior as arithmetic, algebra, chess, or whatever.
In other words, like beauty, the arithmetic, the algebra, the chess game, or whatever is all in the mind of the beholder. It’s not in the machine.
So what we are about to witness in Manhattan is the latest testament to humankind’s marvellous intellectual abilities. We can now build machines that (i) we can regard as performing various tasks that humans alone among animals can perform, and (ii) when we regard their behavior in that way, then by golly those machines outperform us.
Well, that’s not entirely new. Throughout history, people have been building machines that can perform tasks better than humans. When Kasparov, referring to his upcoming match, talks about “an area that defines human beings,” I think first not of machines proving to be superior to humans but of our ability as people to build better and better machines, machines that can outperform us at certain tasks.
When I see a jet aircraft take off, it does not reduce for me the marvel I experience when I gaze on a humming bird.
Likewise, when I think about Deep Blue searching blindly through million upon million possible “plays”, looking for the one that maximizes a certain arithmetic function, I marvel at the way a good (human) chess player such as Kasparov can immediately home in, instinctively, on the two or three key moves worthy of further analysis. The fact is, Kasparov plays chess in a way that truly elicits our marvel—marvel at what the human brain can achieve.
For me, the match we are about to witness in New York is not at all emblematic of a new era where we are no longer unique in our intellectual abilities. Whether you look at the intellect of Kasparov or the science that has led to Deep Blue, it’s yet another testament to the power of the human mind, one of the feats of which has been the development of mathematics, leading to the science and the technology that has now given us Deep Blue.
– Keith Devlin
Devlin’s Angle is updated at the beginning of each month.
Keith Devlin ( devlin@stmarys-ca.edu) is the editor of FOCUS, the news magazine of the MAA. Computer chess is one of the examples he discusses in his latest book Goodbye, Descartes: The End of Logic and the Search for a New Cosmology of the Mind, published earlier this year by John Wiley and Sons.
JUNE 1997
Weighing the evidence
The recently concluded Oklahoma City bombing trial in Denver brought into the public spotlight the thorny issue of eyewitness testimony: how reliable is it, and how well are we able to attach value to the evidence supplied by an eyewitness? Who was it that took delivery of the Chinese food in the motel just prior to the bombing? Who was it that a survivor in the federal building saw hurrying away from the rental truck carrying the explosives moments before the blast?
People make errors, so any evidence has to be evaluated as to the likelihood of it being reliable. How well are we able to make such an evaluation?
The answer is that when it comes to making sense of probabilistic data, we often perform very poorly. Hundreds of thousands of years of evolution has equipped us with many useful mental abilities—our instinct to avoid many dangerous situations and our use of language are two obvious examples. However, evolution has not equipped us to handle statistical or probabilistic data—a very recent component of our lives. Where quantitative data is concerned, if we want to reach wise decisions, it is often safer to rely on mathematics. When we do so, we sometimes find that our intuitions are wildly misleading.
This was illustrated dramatically by the following example proposed by the psychologists Amos Tversky and Daniel Kahneman in the early 1970s. (I considered this example briefly in my July, 1996 column.)
A certain town has two taxi companies, Blue Cabs and Black Cabs. Blue Cabs has 15 taxis, Black Cabs has 85. Late one night, there is a hit-and-run accident involving a taxi. All of the town’s 100 taxis were on the streets at the time of the accident. A witness sees the accident and claims that a blue taxi was involved. At the request of the police, the witness undergoes a vision test under conditions similar to the those on the night in question. Presented repeatedly with a blue taxi and a black taxi, in random order, he shows he can successfully identify the color of the taxi 4 times out of 5. (The remaining 1/5 of the time, he misidentifies a blue taxi as black or a black taxi as blue.) If you were investigating the case, which company would you think is most likely to have been involved in the accident?
Faced with eye-witness evidence from a witness who has demonstrated that he is right 4 times out of 5, you might be inclined to think it was a blue taxi that the witness saw. You might even think that the odds in favor of it being a blue taxi were exactly 4 out of 5 (i.e., a probability of 0.8), those being the odds in favor of the witness being correct on any one occasion.
However, the facts are quite different. Based on the data supplied, the probability that the accident was caused by a blue taxi is only 0.41. That’s right, the probability is less than half. It was more likely to have been a black taxi.
How do you arrive at such a figure? The mathematics you need was developed by an eighteenth century English minister called Thomas Bayes.
What human intuition often ignores, but what Bayes’ rule takes proper account of, is the 0.85 probability (85 out of a total of 100) that any taxi in the town is likely to be black.
Without the testimony of the witness, the probability that it had been a black taxi would have been 0.85, the proportion of taxis in the town that are black. So, before the witness testifies to the color, the probability that the taxi in question was blue is low, namely 0.15. This is what is called the prior probability or the base rate, the probability based purely on the way things are, not the particular evidence pertaining to the case in question.
Specifically, Bayes’ method shows you to calculate the probability of a certain event E (in the above example, a blue taxi being involved), based on evidence (in our case, the testimony of the eyewitness), when you know:
(1) the probability of E in the absence of any evidence;
(2) the evidence for E;
(3) the reliability of the evidence (i.e., the probability that the evidence is correct).
All three pieces of information are highly relevant, and to evaluate the true probability you have to combine them in the right manner. Bayes’ method tells you how to do this. It tells us that the correct probability is given by the following calculation (where P(E) denotes the probability of event E occurring):
Compute the product
P(blue taxi) x P(witness is right),
and divide the answer by the sum
[P(blue taxi) x P(witness is right) + P(black taxi) x P(witness is wrong)].
Putting in the various figures, this becomes the product 0.15 x 0.8 divided by the sum [0.15 x 0.8 + 0.85 x 0.2], which works out to be 0.12/[0.12 + 0.17] = 0.12/0.29 = 0.41.
How exactly is the above formula derived? I’ll try to explain it for the given example, but you should be warned that it takes a very clear head to follow the argument. The principal lesson to be learned from Bayes’ rule is that computing probabilities based on less-than-perfect evidence can be done, but is not at all easy.
The witness claims the taxi he saw was blue. He is right 8/10 of the time. Hypothetically, if he were to try to identify each taxi in turn, under the same circumstances, how many would he identify as being blue?
For the 15 blue taxis, he would (correctly) identify 80% of them as being blue, namely 12. (In this hypothetical argument, we are assuming that the actual numbers of taxis accurately reflect the probabilities.)
For the 85 black taxis, he would (incorrectly) identify 20% of them as being blue, namely 17.
So, in all, he would identify 29 of the taxis as being blue.
Thus, on the basis of the witness’s evidence, we find ourselves looking at a group of 29 taxis.
Of the 29 taxis we are looking at, 12 are in point of fact blue.
Consequently, the probability of the taxi in question being blue, given the witness’s testimony, is 12/29, i.e. 0.41.
So much for the reliability of eyewitness evidence. If our intuitions can be so wildly misleading in the case of highly simplified examples, where all the figures we need are presented to us in a clean, neat fashion, what hope do we have in the far more messy real world that juries frequently have to deal with?
Fortunately, you can almost certainly regard this worrying question as purely theoretical, secure in the knowledge that you are unlikely to find yourself on a jury having to make such a difficult call. It has long been recognized that attorneys almost always object to the inclusion of any mathematician on a jury. After all, the last thing they want is a jury that tries to “complicate” the evidence of their star witness with questions about prior probabilities.
– Keith Devlin
Devlin’s Angle is updated at the beginning of each month.
Keith Devlin ( devlin@stmarys-ca.edu) is the editor of FOCUS, the news magazine of the MAA.
JULY-AUGUST 1997
Performance evaluation
Dear Professor G,
It is my duty as your dean to inform you of the results of the recent review of your performance as a member of the faculty.
I shall begin with your performance as a teacher. Here the student responses are mixed. While the majority of your students acknowledge that you seem to know the material, and some go as far as to say they find you inspiring, many complain that your expectations are far too high. The Review Committee notes that, although your courses have only ten basic requirements, most students fail your tests.
[As an aside, let me ask if you have considered lowering your standards so that more can pass? This has been shown in the past to lead to far higher student evaluations. Parents love it, and the institution as a whole goes up in the rankings for improved teaching effectiveness.]
There was considerable concern that you seem to have a belief in absolute truth, and have on occasion marked a student’s answer as “wrong”. This is a very outdated view of learning. As you are aware, it is our philosophy that all views expressed in the classroom are equally valid. Consensus should be our aim. In future, you should avoid giving the impression that you are in any way more knowledgeable than the students. Remember, truth is relative.
In addition, you rarely come to class, and your performance in responding to questions is not good. The few students whose approaches to you have met with a direct response, say that your replies are often very obscure and difficult to interpret.
It seems that, for the most part, you expect the students to learn for themselves, using your book as a source. Students find this particularly difficult due to the book’s thickness (which they find daunting), the dull cover of the present edition (Have you thought of finding a different publisher?), the lack of highlighted paragraphs to make learning for tests easier, the absence of flow charts to help solve problems, the complete absence of color illustrations, and your failure to provide exercises with answers in the back.
We note that all the other members of your department moved over to the reform method of instruction last year. The comment you make in your self-assessment, that your methods have worked well for several thousands of years, strikes the Committee as weak. Just because something has worked for thousands of years does not mean it will continue to work. Moreover, by adopting the reform approach, we become eligible to apply for large amounts of NSF funding, which will enable us to buy the expensive computers we need to teach using the reform method—had you ever thought of that?
As a first step toward changing your teaching methods, the Committee suggests that you consider making use of the computer and various visual aids in at least some of your classes. Evidence indicates that our students respond particularly well to video presentations. Moreover, by setting the volume on high, you will ensure that your colleagues teaching in neighboring classrooms will realize you are making use of modern teaching methods, and will start to regard you in a whole new light.
Many students complain that your office hours are infrequent. They are particularly upset by your habit of arranging them on Sundays and other days off.
Turning to your record in research, I regret to inform you that here the Committee has significant reservations. First, you appear to have just one major publication. Though many speak highly of it, praising the original thought it shows, others voice specific complaints: It is written in Hebrew, whereas English is now the accepted language for serious academic discourse; it contains no detailed references to your sources; and it was not published in a refereed journal. Furthermore, there is some suggestion that many sections of the book were written by others, though perhaps under your guidance.
Some members of the Committee observed that, while it may be true that you created the world, that was some years ago, and they wonder what you have done since then.
Others remark that the scientific community has been unable to replicate many of your results.
It is particularly worrying that you did not seek NSF support for your work. As you are aware, the Finance Officer depends on the overheads from research grants in order to balance the budget, and the President likes to include a long list of grants in her annual report to the regents.
On a technical matter, it appears that you did not apply to the ethics board regarding the use of human subjects.
Finally, in terms of service, our expectations in this area are, as you know, fairly minimal for someone such as yourself, at an early stage in your career. The only comment the Committee has asked me to pass on to you is that you seem to prefer to work alone. This goes against our desire to see far more collaborative work.
After much deliberation, the Committee has recommended, by a narrow margin, that your contract be renewed for a further year. However, we strongly urge you to address the specific points raised in this letter.
Wishing you the very best for the coming year, I remain,
YOUR DEAN
Note: The above letter was culled from several sources, and many parts have been circulating around the Internet—and before that on scraps of paper posted on bulletin boards—for years. Given that this is the season when such letters are sent out at colleges and universities all across the nation, I should add the familiar movie disclaimer that this is a work of fiction, much of it plagerized from unknown authors, and it should in no way be taken to refer to any real faculty member, either living or dead.
For those readers who seek to infer my own views from the above passage, I should point out that the linguistic coding scheme I am using is not the familiar one of the English language. Rather, the initial symbol-string “Dear Professor G,” denotes Andrew Wiles’ proof of Fermat’s Last Theorem, and the remaining (rather long) string is simply an end-of-file marker. In other words, what you have just read is Wiles’ proof.
– Keith Devlin
Devlin’s Angle is updated at the beginning of each month.
Keith Devlin ( devlin@stmarys-ca.edu) is the editor of FOCUS, the news magazine of the MAA.
SEPTEMBER 1997
Seeing red
All that recent media coverage of NASA’s Mars rover demonstrated the progress that has been made in building what are known as “autonomous robots”. Because it takes 11 minutes to send a signal from Mars to Earth, it was not possible for some earth-based engineer to control the exact path of the rover. It had to be programmed to make decisions for itself. The earthbound controllers used a computer mouse to tell the rover where to head for: they examined the picture supplied by the camera on the mother craft and displayed on the computer screen, picked a likely target, and simply mouse-clicked on the appropriate spot. The rover then had to figure out for itself the exact route to follow. It used a laser vision system to scan ahead for obstacles, and on board computers tried to find a way to negotiate around anything it detected in its path. It was slow going, with a top rate of about half a meter a minute.
Future, more ambitious explorations of distant planets will surely require robots with far greater autonomy than the present generation. The greatest limitation is almost certainly our inability to equip a computer control system with anything like proper vision, as enjoyed by practically all the animal kingdom. After decades of effort to develop computer vision, the most obvious lesson we have learned is that it is a very hard challenge and we still have a long way to go. In fact, if by “computer” you mean a traditional digital computer, it might be totally impossible to endow it with anything like the capacity of sight possessed by humans and many other living creatures.
The problem is, seeing is not simply taking in visual information the way we take in, say, food. The scenes we see are in fact created by our minds.
Though it is fashionable to talk about the mind as a computer, a better description is that it is a pattern former/recognizer. Our brains do this so reliably and so systematically, that it makes perfect sense to talk about the patterns created by our minds as if they are really “out there in the world”. We notice our ability to create images on those occasions when people report seeing something that we know is really just an accident, such as the face of Jesus in the pattern of rocks on a snowy mountainside. But seeing patterns in the environment is a basic ability we use all the time.
The problem facing the computer scientist trying to equip a computer with vision is how to simulate this pattern creation/recognition ability using digital technology. Giving a robot “eyes” is no problem: simply fit the device with one or more cameras. Using two cameras and a bit of mathematics, you can ensure that the device has stereoscopic vision, giving it a sense of depth, as with the Mars rover. But what then?
The camera provides the computer with a massive array of numbers (pixels), representing the light intensity (and the wavelength, if the system is supposed to have color vision) at each point in the field of view. Using some fairly sophisticated mathematics, you can program the computer to pick out in that array things like straight lines and nicely shaped curves. But how do you decide what they represent?
The current approach to the problem of pattern recognition is to provide the computer with a large stock of known patterns and program it to try to find the best match. The problem is what do you mean by “best match”? In what sense is one huge array of numbers “close” to another? The problem is squarely in the laps not of the computer scientists but the mathematicians.
Some of the most promising work in this area of late has been carried out by mathematicians at Brown University in Rhode Island. One of their main projects has been to find ways to help radiologists interpret brain scans to diagnose illness. The computer is provided with a database of reference scans, and then tries to find the one closest to that of the patient. Brown mathematicians have developed a technique called brain warping that examines the changes that have to be made to the patient’s scan to make it identical to one of the reference scans. The fewer the number of changes, the better the match. Since the computerized scans consist of massive arrays of numbers, this is a difficult challenge. The most successful approach found so far uses mathematical objects called Lie groups, which were initially developed for use by physicists.
One of the leaders in developing the mathematics of pattern recognition is the Fields Medal winning mathematician David Mumford, who recently left his position at Harvard University to join the group at Brown. For Mumford, the challenge is to find mathematical ways to describe what it is exactly that we recognize when we identify a picture or a scene. For example, we can recognize the face of a friend we have not seen for many years, even though many pixel-scale details of the face have changed dramatically. (At the scale of a pixel-array, a wrinkle in the skin can be like the Grand Canyon.) What is more, we can recognize our friend from different angles, under different light conditions, and at different distances.
Mumford adopts a probabilistic approach, using techniques of statistical mechanics. The important thing about an image, he says, is not what you actually see but how it compares with what you expect to see. This leads to the probabilistic methods of Bayesian statistics—the same kinds of reasoning that are used to determine guilt based on DNA evidence: What is the likelihood that the given pixel array is such-and-such an array in the database?
A fundamental question that Mumford has to address is: What exactly do we mean by an image (as opposed to a pixel array)? He approaches this problem in much the same way that Euclid approached geometry: Try to formulate simple axioms from which everything else follows. Euclid’s axioms allow us to say precisely what is a triangle, what is a circle, et cetera. Mumford’s axioms are intended to give us a way to say what we mean by an image. (Not a particular image; rather the general concept of an image.)
Scale invariance and decomposability are two of Mumford’s axioms for images. Scale invariance says that the probability of a particular match being the “right” one should not change when you alter the scale. (Your friend’s face is the same however far away it is.) Decomposability says that any image can be split up into a collection of smaller images, each of which is somehow simpler or less cluttered than the original one, but still an image. (Your friend’s face consists of eyes, nose, mouth, et cetera.)
A third of Mumford’s axioms is his “blue sky hypothesis”: Any image will likely contain regions with no objects in them. That is not to say that parts of the field of view are devoid of objects. Rather, when we view a scene, we concentrate on certain parts and aspects of it—our image is selective. (We see our friend’s face, but not what surrounds it.) According to Mumford, the blue sky regions are as important a part of an image—part of what makes it an image (i.e., something created by our minds when our eyes gaze out at the world)—as are the parts we identify and name.
Only time will tell how far work such as Mumford’s will get us in the quest to explore other planets. Though the mathematics is impressive, it may well be that building robots that can make their way safely around completely alien territory is simply not achievable using mathematics and digital technology.
– Keith Devlin
Devlin’s Angle is updated at the beginning of each month.
Keith Devlin ( devlin@stmarys-ca.edu) is the editor of FOCUS, the news magazine of the MAA. He discusses limitations to what can be achieved using mathematics computer technology in his most recent book, Goodbye, Descartes: The End of Logic and the Search for a New Cosmology of the Mind, published earlier this year by John Wiley and Sons.
OCTOBER 1997
The K-12 view from Oregon
What do you get if you take two Englishmen, a Canadian, and an American, and deposit them in Corvallis, Oregon, for a day in front of nearly 400 K-12 mathematics teachers?
The answer is, considerable agreement and an enthusiastic round of applause. At least, that was what happened on October 2, when Oxford University mathematician Sir Roger Penrose, Canadian author and Science News writer (and former science teacher) Ivars Peterson, emeritus UC Santa Cruz mathematician and chaos pioneer Ralph Abraham, and yours truly, all converged on the campus of Oregon State University for a one-day “Math Summit”.
Organized by Terry Bristol, President and CEO of Oregon’s Institute for Science, Engineering, and Public Policy, at the request of Oregon Superintendent of Public Instruction Norma Paulus, the Math Summit brought together four hundred K-16 mathematics teachers and mathematics education professors.
The day began with twenty-minute presentations by each of we four guests. “Outline what you think is important in a mathematics education to prepare students for life in the next millennium,” was our single instruction.
Given the remarkable degree of agreement between our four responses, an observer might have been forgiven for thinking we had conferred beforehand. But that was not the case. Conversation over dinner when we all met up the night before had been largely the usual mathematicians’ gossip, ranging from who was doing what, where, and with whom (the mathematical variety, I hasten to add) to black holes (an old Penrose co-discovery) and Penrose’s recent discovery of an aperiodic tiling of the plane with regular hexagons. (The hexagons have oriented edges and there are restrictions on how they fit together.)
Emphasize understanding, not algorithmic skill, all four of us cried in agreement.
Show students how mathematics all fits together and relates to other aspects of life, we insisted.
Present the big picture, not just the details, we argued.
Make sure it is fun, said Roger and Ivars.
Move away from the current emphasis on a uniform curriculum, on “standards”, and on frequent testing.
Train teachers well, and then trust them to do their job.
Education is about people, we kept saying, not about curriculum or facts. Put the emphasis on math teachers, not the curriculum.
Remember that mathematics takes place in a historical and a social context. Our teaching may be more effective if we do not try to isolate mathematics from those two contexts.
People remember their teachers (or some of them—the very good or the very bad), and when they remember what they were taught it is usually by association with a particular teacher.
Education won’t happen if you try to impose a single approach. People are individuals, and what might work brilliantly for one person might not fly at all with another.
In the end, Ivars Peterson summed it up like this: The question we had been asked to address was “What will tomorrow’s mathematics education look like?” A far better question would be “What will tomorrow’s mathematics teacher look like?”
And so it went. Four individuals, all singing the same tune unrehearsed. But would it work? Can it happen?
Having set the agenda with our opening presentation, the job facing Sir Roger, Ivars, Ralph, and myself was to spend the remainder of the day interacting with the 400 attendees, in different sized groups as the day progressed, as everybody discussed what we had said (and sometimes what we had not said), and asked that crucial “How?” question.
The goals that most of the teachers seemed to share with the four of us was clearly tempered by a strong sense of reality all round. No one I spoke to was not acutely aware of the power of the parent lobby where education is concerned. In a democratic society, if the majority of parents want hours of drill, standardized tests, and school league tables, then by golly they’ll get them. Demonstrable accountability is the buzz word, trust in the individual teacher so often seems to be non-existent.
So, enough of what three ivory-towered academics and a science writer had to say. What did the K-12 teachers themselves think? Sitting in on the various discussions, I jotted down verbatim some of their comments, and I’ll reproduce some of them here. I could provide introductions and links. But on balance, I think the naked quotations speak louder than any rhetoric I could add. After all, we are all mathematicians, and mathematics is the science of patterns. You look for the pattern in the following thoughts.
“Students feel disconnected and incompetent in the math class.”
“Divergent thinking is better than convergent thinking.”
“Too often we teach the way we were taught.”
“The really big interest killer is timed tests.”
“Math doesn’t make sense to 90% of kids.”
“Kids need to understand what they are doing, not just doing it.”
“We break math down too finely, and we lose the big picture.”
“Kids need a system to make sense of their world. When we move too quickly, they revert to their previous system, and this leads to problems in understanding.”
“Facilitator is a better word than teacher or professor.”
“Parents don’t like a non textbook approach.”
“Problems with calculus come more from a lack of algebra understanding than a lack of algebra skills.”
“We teach width but not depth.”
I’ll stop there. You surely get the picture. With so much agreement, you’d think we should be able to make sweeping changes. But, as I observed earlier, none of us present, neither we four presenters nor the 400 attendees, were under any illusions that in the world of education, changing the system is anything other than, at best, a painfully slow process. On the other hand, as one elementary school teacher said:
“It’s easy to change the world of a five or six year old.”
Maybe there are answers, and maybe we can find one of them by starting with that teacher’s observation.
Or, take the insight of one middle school teacher who said that when you have a student who has failed (say) Algebra 2 twice, and you suspect that he or she is going to continue to fail if you continue with the drill and testing approach, then you have the enormous freedom to try a different approach. For those children—and there are plenty of them—there is nothing to lose and a great deal to gain by trying something different. Maybe it is with the present “failures” that we can prove the effectiveness of a different path.
I don’t have answers. But I met a lot of highly committed, resourceful, good people at the Oregon Math Summit. If they are given a little freedom, I would think there is a fair chance they will generate plenty of ideas, and among them, maybe some answers will start to emerge. The question is, in the land of the free, will they be allowed that little bit of freedom?
– Keith Devlin
Devlin’s Angle is updated at the beginning of each month.
Keith Devlin ( devlin@stmarys-ca.edu) is the editor of FOCUS, the news magazine of the MAA, and the author of Mathematics: The Science of Patterns, a Scientific American Library paperback published by W. H. Freeman in 1996.
NOVEMBER 1997
A Nobel formula
The discovery of a single mathematical formula in 1970 led to this year’s Nobel Prize for economics, shared between Stanford University professor of finance (emeritus) Myron Scholes and economist Robert C. Merton of Harvard University. The prize would undoubtedly have been shared with a third person, Fischer Black, but for the latter’s untimely death in 1995.
Discovered by Scholes and Black, and developed by Merton, the Black-Scholes formula tells investors what value to put on a financial derivative, such as a stock option. By turning what would otherwise be a guessing game into a mathematical science, the Black-Scholes formula made the derivatives market into the hugely lucrative industry it is today.
So revolutionary was the very idea that you could use mathematics to price derivatives that initially Black and Scholes had difficulty publishing their work. When they first tried in 1970, Chicago University’s Journal of Political Economy and Harvard’s Review of Economics and Statistics both rejected the paper without even bothering to have it refereed. It was only in 1973, after some influential members of the Chicago faculty put pressure on the journal editors, that the Journal of Political Economy published the paper.
Industry was far less shortsighted than the ivory-towered editors at the University of Chicago and Harvard. Within six months of the publication of the Black-Scholes article, Texas Instruments had incorporated the new formula into their latest calculator, announcing the new feature with a half-page ad in The Wall Street Journal.
Modern risk management, including insurance, stock trading, and investment, rests upon the fact that you can use mathematics to predict the future. Not with 100% accuracy, of course. But well enough so that you can make a wise decision as to where to put your money. In essence, when you take out insurance or purchase stock, the real commodity you are dealing with is risk. The underlying ethos in the financial markets is that the more risk you are prepared to take, the greater the potential rewards. Using mathematics can never remove the risk. But it can tell you just how much of a risk you are taking, and help you decide on a fair price.
The idea of using mathematics to predict the future goes back to two seventeenth century French mathematicians, Blaise Pascal and Pierre De Fermat (he of Fermat’s Last Theorem fame). In a series of letters exchanged in 1654, the two mathematicians worked out the probabilities of the various outcomes in a game where two dice are thrown a fixed number of times. For example, suppose Mary and Bill are playing a best-of- five series, and after three throws, Mary is ahead two to one. What would be a fair stake for you to wager on Bill winning the series, if I offer to pay out $100 if he wins? Pascal and Fermat showed how to find the correct answer. According to their mathematics, the probability that Bill will go on to win the series is 25%. So, if I allow you to take the bet for exactly $25, then I am making you a totally fair offer. A stake of less than $25 would be more attractive to you than to me; a stake of more than $25 would be unfair to you. Of course, we would both still be gambling. The mathematics does not eliminate the risk. It simply tells you what the fair price is.
What Black and Scholes did was find a way to determine the fair price to charge for a derivative such as a stock option. The idea with stock options is that you purchase an option to buy stock at an agreed price prior to some fixed later date. If the value of the stock rises above the agreed price before the option runs out, you buy the stock at the agreed lower price and thereby make a profit. If you want, you can simply sell the stock immediately and realize your profit. If the stock does not rise above the agreed price, then you don’t have to buy it, but you lose the money you paid out to purchase the option in the first place.
What makes stock options attractive is that the purchaser knows in advance what the maximum loss is: the cost of the option. The potential profit is theoretically limitless: if the stock value rises dramatically before the option runs out, you stand to make a killing. Stock options are particularly attractive when they are for stock in a market which sees large, rapid fluctuations, such as the computer and software industries. Most of the many thousands of Silicon Valley millionaires became rich because they elected to take a portion of their initial salary in the form of stock options in their new company.
The question is, how do you decide a fair price to charge for an option on a particular stock? This is precisely the question that Scholes, Black, and Merton investigated back in the late 1960s. Black was a mathematical physicist with a recent doctorate from Harvard, who had left physics and was working for Arthur D. Little, the Boston-based management consulting firm. Scholes had just obtained a Ph.D. in finance from the University of Chicago. Merton had obtained a bachelor of science degree in mathematical engineering at New York’s Columbia University, and had found a job as a teaching assistant in economics at MIT.
The three young researchers—all were still in their twenties—set about trying to find an answer using mathematics, exactly the way a physicist or an engineer approaches a problem. After all, Pascal and Fermat had shown that you can use mathematics to determine the fair price on a bet on some future event, and gamblers ever since had used mathematics to figure the best odds in card games and roulette. Similarly, actuaries used mathematics to determine the right premium to charge on an insurance policy, which is also a bet on what will or will not happen in the future.
But would a mathematical approach work in the highly volatile, new world of options trading which was just being developed at the time. (The Chicago Board Options Exchange opened in April 1973, just one month before the Black-Scholes paper appeared in print.) Many senior market traders thought such an approach could not possibly work, and that options trading was beyond mathematics. If that were the case, then options trading was an entirely wild gamble, strictly for the foolhardy.
The old guard were wrong. Mathematics could be applied. It was heavy duty mathematics at that, involving an obscure technique known as stochastic differential equations. The formula takes four input variables—duration of the option, prices, interest rates, and market volatility—and produces a price that should be charged for the option.
Not only did the new formula work, it transformed the market. When the Chicago Options Exchange first opened in 1973, less than 1,000 options were traded on the first day. By 1995, over a million options were changing hands each day.
So great was the role played by the Black-Scholes formula (and extensions due to Merton) in the growth of the new options market that, when the American stock market crashed in 1978, the influential business magazine Forbes put the blame squarely onto that one formula. Scholes himself has said that it was not so much the formula that was to blame, rather that market traders had not grown sufficiently sophisticated in how to use it.
The award of a Nobel Prize to Scholes and Merton shows that the entire world now recognizes the significant effect on our lives that has been wrought by the discovery of that one mathematical formula.
– Keith Devlin
Devlin’s Angle is updated at the beginning of each month.
Keith Devlin ( devlin@stmarys-ca.edu) is the editor of FOCUS, the news magazine of the MAA, and the author of Mathematics: The Science of Patterns, a Scientific American Library paperback published by W. H. Freeman in 1996.
DECEMBER 1997
Move over Fermat, Now It’s Time for Beal’s Problem
Texas millionaire banker Andrew Beal has just offered a major cash prize to the first person to solve “Beal’s Problem”, a mathematical teaser that has a tantalizing appeal similar to that of Fermat’s Last Theorem, which attracted hundreds of professional and amateur mathematicians until its solution three years ago.
Dedicated followers of mathematical gossip will know that Fermat’s Last Theorem said that for exponents n greater than 2, the equation
xn + yn = zn
has no whole number solutions for x, y, and z (apart from trivial answers where one of the unknowns has the value 0).
First proposed in the seventeenth century by the great French mathematician Pierre De Fermat, the “theorem” resisted numerous attempts at solution until British mathematician Andrew Wiles of Princeton University found a proof in 1994. Wiles’s achievement was portrayed in a BBC television Horizon documentary The Proof last year and described in a small rash of popular books.
Much of the general interest in Fermat’s challenge can be attributed to the offer of a cash prize, the Wolfskehl Prize, to the person who first proved the theorem. Established in 1908 by Paul Wolfskehl, a German physician and amateur mathematician, the prize lost much of its value in the German inflation of the 1930s, but was still worth about $50,000 when Wiles collected the award earlier this year.
Beal’s Problem is like Fermat’s, but instead of focusing on equations with one exponent, n, there are three: m, n, and r. Thus, Beal’s equation looks like this:
xm + yn = zr
The idea is to look for whole number solutions to this equation where the solution values for x, y, and z have no common factor (i.e., there is no whole number greater than 1 that divides each of x, y, and z). Beal has conjectured that if the exponents m, n, and r are all greater than 2, then his equation has no such solution for x, y, and z.
Just as Fermat’s equation can be solved if the exponent n is equal to 1 or 2 (the best known solution for n=2 is the “Pythagorean triple” x=3, y=4, z=5), so too you can find solutions to Beal’s equation if you allow one of the exponents, m, n, r to be equal to 1 or 2. for example:
11 + 23 = 32
25 + 72 = 34
Also, just as it was known in the early 1980s that, for any exponent n, the corresponding Fermat equation could only have a finite number of solutions, so too it is known that for any three exponents m, n, and r greater than 2, the corresponding Beal equation has only a finite number of solutions. In the Fermat case, Wiles showed that the finite number for any exponent is in fact zero. Beal’s conjecture is that it is zero for each of his equations as well.
Fermat’s last theorem is the special case of Beal’s conjecture you get when the exponents m, n, and r are equal. So the new problem is a generalization of the problem Wiles solved three years ago.
So, is Beal’s conjecture likely to achieve the fame of Fermat’s last theorem? Like Fermat’s problem, Beal’s question is easy to state, involving nothing more complicated than simple equations to be solved for unknown whole numbers. And like Fermat’s last theorem, Beal’s conjecture postulates that there are no solutions of the specified kind.
Now throw a cash prize into the mix, and you have all the ingredients for a new mathematical saga.
This year, Beal’s prize stands at $5,000. Thereafter, it will increase by $5,000 each year for every year the problem remains unsolved, up to a limit of $50,000, the same amount as the Wolfskehl Prize when it was finally awarded.
Much is known of Fermat. But who is Beal? He is forty-four years of age, the father of five children, and the founder, owner, and chairman of Beal Bank, the largest locally owned bank in Dallas. He recently founded Beal Aerospace, which is designing and building a next-generation rocket for launching satellites into earth orbit. And, of course, he is a keen amateur mathematician, who has spent many hours pondering Fermat’s last theorem.
Beal has appointed a committee of three distinguished professional mathematicians to screen potential prize-winning proofs of his conjecture. Anyone who thinks they have a solution should send it to Professor Daniel Mauldin at the University of North Texas in Denton, Texas, USA, who chairs the prize committee. But beware. There are indications that, as with Fermat’s last theorem, solving Beal’s problem might require a large dose of heavy mathematical machinery. But then again . . .
– Keith Devlin
Reference R. Daniel Mauldin, A Generalization of Fermat’s Last Theorem: The Beal Conjecture and Prize Problem, Notices of the American Mathematical Society Volume 44, Number 11, December 1997, pp.1436-1437.
Devlin’s Angle is updated at the beginning of each month.
Keith Devlin ( devlin@stmarys-ca.edu) is the editor of FOCUS, the news magazine of the MAA, and the author of Mathematics: The Science of Patterns, a Scientific American Library paperback published by W. H. Freeman in 1996.
profkeithdevlin.org 