By Katherine Hinds
Brown Alumni Magazine, December 1986.
Do you realize how much math has become part of society? Do you care? Phil Davis thinks you should pay attention
There's an old saying that one man's math is another man's personal nemesis. I will say right up front that I am one of those people who confronted mathematics in my early years — back when I was still calling it by its less formal name, arithmetic — and was overcome. It was not an equal match, math and me: The odds were against me and the evens didn't much like my style either. Consider, then, the fear with which I approached the idea of doing a feature story on Philip J. Davis, professor of applied mathematics and co-author, with Reuben Hersh, of a newly-published book on mathematics, Descartes' Dream: The World According to Mathematics (Harcourt, Brace, Jovanovich, 1986). I soon discovered that it was for people like me that Phil Davis is writing.
"Mathematics is generally a turnoff in our culture," he admits "Most people neither know much about it, nor care to. And if it comes their way the avoid it. That's a common reaction."
Davis's major point is that our society is becoming increasinglv "mathematized," and that we had better start paying attention to that fact, no matter how anxious we are. He makes the statement in his book that "the social and physical worlds are being mathematized at an increasing rate." And his prosaic moral: "We'd better watch it, because too much of it may not be good for us."
Phil Davis is the man to consult for math anxiety. His very presence is comforting. He is soft-spoken and gentle, with a quiet sense of humor. The nimbus of white hair framing his face nicely highlights the lines etched around his eyes. His pullover sweater and corduroy-clad appearance bespeaks that of someone who has spent decades in academia — the description "absent-minded professor" comes to mind and is not easily dismissed.
Although he says that "math came in for me when I was eleven years old," Davis confesses that writing was his major love in college. "I didn't think 1 could make a living at math, because before World War II there were rela- tively few jobs for professional mathematicians. Math has changed tremendously since the war. Ties were opened up between mathematics, science, engineering, and industry. And the computer business really started in mathematics. The computer business employs the greatest number of people in mathematics, although they may not think of themselves as mathematicians."
It's quickly apparent that Davis believes in digression as a conversational form "When I was in college, I didn't give much thought to where I would end up. We didn't think much about the future then. I did a bit of writing in college, took courses in it. I wrote fiction. I was in love with the short story. Wrote a lot of those." Did you ever have any published? "No. no. I had a book published, but never any fiction. No. Well. Wait a minute. I take that back. I had two short stories published — one in Harper's and one in William Buckley's magazine, which I forget the name of right now."
Although most mathematicians publish — either textbooks or other kinds of professional treatises — Davis has strayed from the fold. "I've published the textbooks, the professional material, but also I've done philosophical material. Descartes' Dream is both. I've done a lot of reading in other disciplines — more in the humanities than most mathematicians. I believe. I read literature, philosophy, history, art. My mind works more or less as a mathematical mind, encompassing other points of view. I draw on my acquaintance with these other disciplines when I write."
Davis says it's "good policy to have someone in mind when you are writing. When I wrote this most recent book. I had in mind a friend of mine who is a lawyer. He is very much interested in math, not afraid of it, and intrigued by historical and philosophical ideas. I kept him in mind, figuring that if I could explain what I was saying to him. I could probably explain it to anyone."
"Quite apart from our scientific, technological life, the world is becoming more mathematized every day," Davis begins. "That is to say, we're doing mathematics in a much more con- centrated way than we used to — mostly through computers. And the general public is unaware that most of it is there. The list of mathematizations is enormous. Just before you came in, I was making a list of these things. For ex- ample — lotteries, insurance, elections, matching people with jobs, matching people with other people, Grades. Grades is a billion-dollar business. Think of the SAT's, the GRE's, and IQ tests. Automatic teller machines — that's math. Voting. The whole financial world is based on math — stocks, bonds, btiving and selling, margins — all of that. Just look at the Wall Street Journal. I would say 75% percent of the symbols are numbers in that paper. Statistics in sports — read the sports section in the newspaper. It's somewhat less than the Wall Street Journal, but I would say probably up to 50 percent of the sports section is numbers. The world of medicine is mathematical. There is a tendency there to reduce people to a series of numbers — their blood pressure or whatnot. We know people are more than that. And this causes a problem. Numbers are not objective. We tend to think of mathematics as objective, but it's not. And that's where the philosophy comes in."
The mathematizations Davis rattled off occurred to him, he says, in less than five minutes. "Imagine how many more there are. I am trying to make people aware of what some of these things are, and make them aware of the increasing tempo of math. My book poses the question of how much of it is good for us, and invites readers to develop an enhanced awareness of this. I don't want people to think: math is just there. Addition is just there. How can it be bad? It's just like gravity. Well, mathematics is not just there. We put it there as we find it useful. No one has a gun at our backs forcing us to mathematize the world. We do it as we find uses for it, and then sometimes problems are created."
Problems, indeed. Mathematics is the science of abstraction, the authors of Descartes' Dream say, and, "The spirit of abstraction and the spirit of compassion are often antithetical. Whereas World War 1 was a chemists' war and World War II was a physicists' war, World War III will be a mathematicians' war."
This is the second book on which Davis and Hersh, a professor at the University of New Mexico, have collaborated. The first, The Mathematical Experience, was concerned with the nature of mathematics, its methodology, what it's like to be a mathematician. The book, according to Davis, is a "smorgasbord" of facts and fancy, history and philosophy, theorems and formulae — "a total immersion in mathematics." The authors looked at some of the less scientific aspects of mathematics: number mysticism, astrology, the use of geometric figures as talismans. Basically, The Mathematical Experience was an effort on the parts of Davis and Hersh to humanize a profession that we outsiders consider fairly monstrous. Davis told the Providence Journal that the book evolved out of a dissatisfaction with books that are "needless and futile attempts to establish mathematical knowledge as infallible."
One thing about mathematics that always frightened me was its sheer size. I viewed it as a huge, monolithic structure that I would have to conquer step-by-step. I couldn't just dive into it at any point and expect comprehension. As Davis and Hersh point out in The Mathematical Experience, the amount of mathematical knowledge has grown exponentially since its beginning, making math a far less manageable subject for human comprehension. A little more than a century ago. the authors explain, math could be divided into twelve distinct categories, including algebra, two kinds of geometry, and calculus. Today there are more than sixty such categories, including subjects such as nonassociative rings and algebras, abstract harmonic analysis, and quantum mechanics. Early in this century, they write, it was believed that any dedicated student was capable of knowing the whole of mathematics. Today it is estimated that mathematicians publish approximately 200,000 theorems a year in the more than 1,600 technical journals to which they subscribe.
Two hundred thousand theorems a year. That's a figure a seasoned mathematician would balk at, much less a math-phobe. Phil Davis says that the newly-published Descartes' Dream is the more approachable book of the two. In the preface, the authors offer this comforting advice: "Readers are encouraged to browse at random [through the essays in the book] and read whatever catches their fancy."
Imagine. Browsing through mathematics. I'm a born browser, so I gingerly tiptoed through the book. The writing stvle immediately soothed me — it's lucid, calm, engaging, and often humorous. This is a book about mathematics I could read — and it kept asking us to think about how much math is really good for us — a question I had asked for years, without considering the deep philosophical implications. The essays range across such topics as computer graphics as high art and the interplay of mathematics with our sense of time. Ethical relativism and non-Euclidean geometry are encountered, as is a chapter on "The Whorfian Hypothesis: Ends and Means in Computer Lan- guages." I leaped through those. I spent some time in the chapter on "The Computerization of Love," in which Davis and Hersh muse on the mathematization of dating.
"[T]here are hordes of mateless persons ('singles'), under a strong social or psychological imperative to mate, whether for the night or for a longer period, and with no traditional social vehicle by which they might be mated," Davis and Hersh write. "In such a situation, in a profit-driven, rapidly innovating society like the U.S., it could be foreseen that businesses would spring up attempting to fill this social need. The three most conspicuous types which have appeared are the singles bar, the personals column, and computer dating. To investigate the singles bar would require late evening fieldwork in a somewhat unscholarly environment, so we will limit our considerations to computer dating and the personals column." And they do, showing how people with human needs are reduced to the lowest common denominator — numbers. They reach the conclusion that in the personal columns and computerized dating, "mating becomes an impersonal, mechanistic procedure ... That is not to say that love or sex in our society at large is in immediate danger of being mathematized or computerized. We only say that in a certain, non-negligible segment of our society this has already happened to some degree. By no means do we imply that computer dating or the personals column are objectionable. On the contrary, thev may help and are unlikely to do harm. Nevertheless, they do stand as sign posts of the changes that are taking place in our society, and as indications of the way even our most intimate needs may be digitized, quantified, mathematized."
People take mathematics for granted. It has become something of Phil Davis's new anthem that we stop doing that and examine the deeper issues.
"Maybe a parallel example will help explain what I mean. Generally speaking, we don't teach biology in elementary school, we don't teach chemistry, anatomy, physiology. But at the same time, a certain amount of hygiene is taught — how to live right, how to brush your teeth, the effects of drugs, some sex education. But elementary-age students are not being taught the biology at the level of the molecules. You have to wait until college to learn about that. I'd like to see us get to a place where we could ignore some of the mathematical nuts and bolts so that we can start talking about a kind of mathematical hygiene. What things are good for us? What are not so good? How do they affect us?"
Mathematics is something that is part of our physical selves. We have ten fingers, we have ten toes — that's real, that's finite and comprehensible. How is it that something we rely on and "count on" can be something that we fear at a basic level?
"People don't examine math," Davis says, "simply because they do fear it. And it's going to be tough to overcome that fear. What has to happen" — for us to get to the level of mathematical hygiene — "is that over the next twenty-five years, as it becomes less necessary to teach the manipulative math stuff, schools could concentrate on what we're doing with it, and how it affects us."
Davis believes we should be talking about math and the way it affects us on many levels. "An interesting example is Daylight Savings Time. All of a sudden, we do something with the clock, with the numbers, and it affects us a certain way. Now let's discuss this. Let's not say that this is God-given, or something Washington has foisted on us. Daylight Savings Time imposes a reality — you add an hour, subtract an hour, and it changes traffic patterns with respect to the light and dark and the way people feel."
Most of the discussions about math education focus on what math should be taught at what age-level. "The focus is on the materials, the manner in which the teaching should take place. Do we use computers, overhead projectors, this kind of gimmick, that kind. 1 think, to a considerable extent, the purely manipulative aspects of math can be downplayed or phased out. Because a great fraction of the manipulation can be done automatically."
The world is changing at an incredibly quick pace, and the increasing mathematization has othet than educational and psychological implications. "The difficulty people have with math is, on average, a serious one. It affects women particularly. My own feeling about the math-phobia that women have is thai it is largely socially-conditioned. Certain subjects are still known as male subjects — for a girl to be doing them is to attribute something to them that is less than feminine, and I think girls find this hard at an early age. I see it even at the college level — most of the math concentrators at Brown are men; most of the physics concentrators are men. There is one woman professor in the applied mathematics department, one in the mathematics department. Personally, I think fear of math is handed down like a virus by elementary school teachers who are mortally afraid of it. They themselves have a fear of it, and it gets passed from teachers to stu- dents. I would guess that a fear of math sets in before a person is ten, and if that's the case, then there is hardly anything that will allay it in later years." If women are being socially-conditioned against mathematics at an early age, with little hope of overcoming their fears in later years, it's natural to wonder how the increasingly mathematized job market will be affected.
Phil Davis's work as an applied mathematician has been in numerical analysis — the mathematical bases of scientific computation, where real-life problems are reduced to numbers so a computer can tackle them. His career in mathematics began after graduating from Harvard (B.A., M.A., Ph.D.), when he went to work for the Bureau of Standards. "I was working on first- and second-generation computers," he says. "I was part of a team of eminent mathematicians assembled to investigate the mathematical bases of scientific computation. We were doing two things simultaneously: performing scientific computation that was necessary for other scientists at the bureau, while pursuing the craft of numerical analysis. I lasted twelve years."
Ultimately, Davis says it was his love for writing that led him to an academic setting. "Writing in a university context is an allowed activity. Teaching gave me the opportunity to write, whereas an industrial/governmental job wouldn't have given that to me.
"And I like teaching. The future of the world is in the hands of the teacher. And a good teacher is a bit of a ham. The more you ham up a presentation, the better it is, up to a point. I have a ham component. One of the problems in teaching math, though, is keeping it fresh. I like to try to teach as many courses as I can in my department, and invent courses from time to lime. Right now I'm trying to develop a course on the nature of prediction. This would be a course for students in the humanities: examining prediction, what it's about, how do you do it, how was it done in the past? With what success? What are the limitations? I'd like to look at all kinds of prediction, from scientific to dreams and intuition.
"And you keep your teaching fresh by reading, writing, doing standard courses in a slightly different way. and not worrying too much whether what you're teaching is going to connect up with the next course. And you keep it fresh by keeping your teaching close to your research, when you're teaching graduate students."
How do mathematicians decide what line of research to pursue? "Sometimes it happens out of necessity — you take a job, and that job requires that you do a certain kind of material. That happened to me during the war. I was in the Air Force reserve, working on aerodynamic theory and scientific computation, and that's what lasted for me professionally. Teachers are influential, especially in graduate school. They make an impression on you, and you work in that field. But, eventually, you have to fly for yourself, and I can't really explain how that happens."
Davis's interest in philosophical issues sprang from a teacher he had in college. "I took several courses with him and talked to him from time to time, but for years I didn't see how I could pursue my interest in the philosophy of math. It was always in the back of my mind. A lot of these seeds are planted early, and blossom later. Sort of like the century plants that you find in the Southwest desert. They blossom every hundred years."
In Davis's case, the blossoming was full-blown. His books have enjoyed extremely flattering reviews, and The Mathematical Experience has so far been translated into eight languages. "Writing a book about mathematics is a gamble," he admits, but one that has paid off handsomely.
Meaning can sometimes get lost in mathematics. This is Davis's main point. He and Hersh caution against blind faith in reason, logic, and mathematics.
After all he's said about the necessity for people to think about the increasing mathematization in society, Davis says, "On the other hand, not everyone has to know numbers. There's no lack of professionals to think about these things. Similarly, not everyone has to know surgery. Be a ballet dancer. It's nice that there are some ballet dancers. There may be as many mathematicians as society can stand."
In Descartes' Dream, he writes: "We should never forget that a stroll in the woods or a deep conversation with a new or old friend are beyond mathematics. And then, when we go back to our jobs, as administrators, teachers, or whatever, will we still remember that numbers are only the shadow, that life is the reality."
Words a math phobe can live by.
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Thanks to Katherine Hinds for her gracious permission to post this.