The world of mathematics is a never-never land to most humanoids, inhabited by forms that, despite their gentleness, are often mysterious and terrifying to the casual invader. It is a world that influences mightily the life of our species, yet its leading practitioners are as unrecognized and as little understood, even by otherwise well-educated Americans living among the blessings of mathematically-oriented science, as are obscure Chinese poets, Indian mystics, and practitioners of other arts that seem to have no impact on day-to-day American life.

I might add that it is not just the general public that misunderstands the role of the mathematician. University faculty members, who are (in the phrase of Professor Paul Halmos) generally on speaking terms with at least one mathematician, are just as badly informed -- sometimes worse, for some think that they know what mathematicians do, and they are often wrong. As Halmos has written, there are physicists who think their mathematical colleagues spend their time computing tables of useful coefficients, social scientists who think mathematicians spend their time dreaming up new ways of analyzing statistical data, and English professors who think mathematicians are computer programmers. That there exist people who call properly call themselves mathematicians while engaging in the activities just mentioned is true -- but many who practice these arts are anti-mathematical, and even would be insulted to be called mathematicians, while others who properly call themselves mathematicians do nothing in the areas mentioned.

You are familiar with the names Pythagoras, Newton, Bernoulli, DesCartes. I suspect, however, that if pressed to name the individual you thought to be the greatest mathematician of this century, many of you would offer the name of Albert Einstein, who was certainly a giant among physicists but not a mathematician. The names of Picasso, Stravinsky, Freud, Olivier, Pauling, Teller, Salk, etc., are widely reknowned as giants of modern culture, but how many nonmathematicians are familiar with the names of Gauss, Hilbert, Noether, von Neumann, Borsuk, Erdos, Thom, Fefferman, Knuth? -- all individuals of Promethean achievements in the "Queen and Servant of Sciences" during the last two centuries.

The subject of who is a mathematician and what a mathematician does is probably too complex for me to air thoroughly in this forum. (My mother used to badger me to summarize my research for her. The last time she studied mathematics was in high school, before World War II. I finally told her: "Ma, I went to college 10 years before I understood these ideas. I can't explain them to you in 25 words or less.") I hope, however, to shed light on the following: "pure" as opposed to "applied" mathematics; mathematical esthetics; and mathematical thinking.

The classification is not always clear-cut, but most mathematicians are one of "pure" or "applied." Roughly speaking, a pure mathematician is an abstract mathematician, one who studies mathematics for its own sake, for the fun of it. (Yes, for the fun of it. We certainly didn't choose abstract mathematics for the money!) Such disciplines as topology, number theory, analysis, modern geometry, and modern algebra are generally considered to be pure mathematics. Although many applications flow from these arts, their practitioners generally do not concern themselves with the building of better bombs, tractors, or video games.

"Applied" mathematicians, on the other hand, frequently are nearly indistinguishable from physicists, economists, computer scientists, or other students of phenomena that make their way into newspapers, government offices, and corporate boardrooms. Disciplines such as differential equations, mathematical statistics, numerical analysis, operations research, and certain aspects of computer science are generally thought of as applied mathematics.

The line between pure and applied mathematics is rather blurred. Most mathematics, pure or applied, has roots in the "real" world of applications. On the other hand, applications have a way of arising in the most unexpected places. Professor Halmos has written of the following: A few decades ago some people decided that it was necessary to place the study of mathematics on a firm logical foundation. They wanted to develop an axiomatic system for the study of mathematics. Today, such mathematicians are called logicians -- their subject, mathematical logic. Among their efforts were discussions of what a theorem is, what a proof of a theorem should consist of, and how someone should be able to recognize one of these objects in a crowded tavern (which was a common locale for these discussions). One of the notions they developed was the idea of an abstract "machine" to accomplish this task of recognition. After all, they were developing, in axiomatic fashion, sets of criteria that theorems and proofs of theorems should meet, so it was deemed theoretically possible to have a machine check a group of statements against these lists of criteria to determine suitability. Then an inspiration: if they could build such an abstract machine, why not a real one? From these beginnings, from the purest, most abstract of mathematical deliberations, one of the roots of the modern computer.

Nevertheless, most mathematicians regard themselves as belonging to one camp, and often are downright hostile to the other. There are pure mathematicians for whom a synonym for "applied" is "dirty" -- the great number theorist Leonard Dickson of the University of Chicago has been quoted as praising his Creator for the fact that "number theory is unsullied by applications." One wonders how Dickson would react to the knowledge that factoring integers, a fundamental topic of number theory, is at the heart of many data encryption schemes that secure electronic commerce. Conversely, there are applied mathematicians to whom "pure" means "useless." (These deprecations are also used by professors seeking to prevent graduate students from wandering into fields in which the professors are not active.)

These attitudes are regrettable. I am a topologist, concerned not with "the study of tops" as my sister-in-law would put it, but with properties of geometric form in infinite-dimensional space. I am a pure mathematician, and I freely admit the preferences of my youth for abstraction. I have always enjoyed teaching courses in certain applied areas and working with computers, but when I tease my students by telling them that "a day without mathematics is like a day without sunshine," what I have in mind is math for math's sake, "cocaine for the soul" as one colleague put it, the sculpted images and the rhythmic music of logical structure. For me, this used to mean abstraction, but as the practical matter of making a living has required over the years a greater knowledge of practical mathematics, I have found that necessity can be virtue, that there is joy in applications.

Why do some men and women paint, write poetry, compose music? Philistines who do not appreciate those arts often ask of what use they are. Just as the pleasure experienced by lovers of painting, sculpture, poetry, and music justifies activity in those areas, so also does the pleasure of mathematicians in their studies justify their theorems. The potential audience is smaller, but the principle remains. I might add that just as I feel the world is somehow better for having a professor of philosophy studying the essays of obscure thinkers of the distant past, so too is the world better for the existence of mathematics that offers little promise of improving the economic life of the world. (And as I have mentioned, important applications often arise in unexpected places.)

Professor Halmos notes that if you ask a physician why he chose a career in medicine, you are skeptical if he replies: "I wanted to relieve suffering humanity." We want our physicians to have some idealism, but we know many are motivated primarily by money or prestige. We may extend acceptance of a reply that expresses seriously a desire to aid the sick. Yet many with the talent to become medical doctors also have the talent to combine a rewarding career with the opportunity to serve the public in other fields. Why does such an individual specifically choose medicine? Other than a mother pushing a medical career, the explanation may well be enjoyment, for its own sake, of a high school biology course, which in turn convinced the future doctor that he or she might be good in, and enjoy, a medical career. And I suspect that most mathematicians choose their life's work similarly.

Spiritually, a mathematician is at least as close to an artist or a linguist as to a scientist. Does it surprise you that we speak emotionally of our studies? We talk of the power, beauty, depth, elegance, clarity, and difficulty of our theorems with the same sense of awe and wonder that lovers of sculpture and music have for acts of creative genius in those areas. We strive to understand and communicate, to examine and build structures of thought. The world treats us socially as it treats artists, also: For every painter who has heard "I could never draw a straight line!" or singer who has heard "I can't even carry a tune!" there is a mathematician met with "I can never balance my check-book!"

Perhaps an extreme example of a mathematician's sense of esthetics is to be found in the following story. Early in this century, there was a member of Harvard's Math Department who was known for his aristocratic demeanor. Every day he would come to class dressed in a finely-tailored suit, expensive shoes, sometimes wearing a stylish cravat. His particular pride and joy was a handsome gold watch attached to a long gold chain. It was this mathematician's habit to pull the chain and twirl the watch when he came to a pause in his lectures. One day, the chain snapped in mid-twirl during a freshman calculus class. The watch soared gracefully across the room and came to rest on a window ledge. Aghast, the scholar dashed to the window and picked up his watch. Miraculously, it was undamaged. Regaining his composure, the professor turned to his shocked audience and, referring to the path the watch had traced through the air, announced: "Gentlemen, you have just witnessed a perfect parabola."

So, what is mathematical activity? Is it what we teach our undergraduates to prepare them for careers in the social and natural sciences? Yes, to a certain extent, but in another sense, most of our courses are not concerned with what a modern mathematician considers to be at the heart of his discipline. After all, most college students progress no further than a year of calculus, a discipline that was on the frontier of human knowledge during the late 1600's, but is almost as fundamental for today's scientists as the addition table.

First of all, most practicing mathematicians are at least as interested in questions and how these questions are answered, as they are in the answers themselves. This shows particularly well in our encounters with students. Most teachers of mathematics have stories about students who want to know why the teacher penalized them on an exam for "the right answer." The teacher worth his salt replies that the "right answer" consists not only of the correct numerical result, but also a logically correct derivation of that result, and that a chain of errors that just by coincidence yields the correct numerical result hardly merits full credit.

We want our students to learn how to answer questions -- if, after a series of blunders, they just happen upon the desired number, we are rightly less impressed with them than with their peers who demonstrate proper understanding of the test question but, through minor carelessness, do not come up with correct answers.

To demonstrate the latter point, and to give an elementary view of how a problem may be solved with varying degrees of mathematical sophistication, I present the following question, whose answer is 511. Remember, the answer is 511. I tell you that now in order to emphasize my view that answers are byproducts of mathematics; that real mathematics is in solving, and not the solution to, a problem. I will present three distinct methods of solution, so if the method I happen to be discussing is not the method of your choice, please be patient -- I applaud your impulse to blurt "Why don't you just ..." but perhaps I will just ... with the next method. (This example is also due to Paul Halmos.)

The problem is as follows: Imagine yourself as the director of a tennis tournament with 512 players. You must be concerned with estimating the time required to hold the tournament. If players are eliminated each time they lose a match until only one remains to be designated the tournament champion, how many matches must be scheduled?

(Perhaps some of you recognize 512 as a power of 2, that is, 512 = 2 x 2 x 2 x ... x 2, and are alert to the possibility that a trick based on repeated doubling, or halving, may be employed. The really alert will also admit that I may have made the previous comment to throw you off-track.)

The first method of solution is quite simple. A bright grade-schooler could answer the question using this method. We reason that in round one there are 512 players requiring 512/2 = 256 matches, as a result of which the 256 winners will survive to round 2, which will require 256/2 = 128 matches to be survived by 128 winners, who go into round 3, etc. Thus we may form the following table:

The second method starts with the same observations made above. Now, we consider the problem-solver to be a bit more sophisticated mathematically, recognizing the sum required is of a rather special form, namely the sum of a geometric series (that is, a sum of terms such that the ratio between each successor term and its predecessor is constant, as is the case here -- for example, 256/512 = 0.5, 128/256 = 0.5, 64/128 = 0.5, etc.). Since the sophisticate knows that the sum of a geometric series is given by

a(1 - rⁿ)/(1 - r)

where a is the first term of the series (here, a = 256), r is the ratio of successors to predecessors (here, r = 0.5), and n is the number of terms (here, n = 9), the solver evaluates by calculator

256 x (1 - .5 PWR 9) / (1 - .5) =

and obtains the result, 511.

Before proceeding to the promised third method, let's compare the first two. The second solver used a method that resulted in a shorter calculation made possible by rather special circumstances: had the number of players entered in the tournament not been a power of 2, then at various stages in the tournament players would receive byes, so that the number of matches required could not be resolved by this method. Thus while the second method has the advantage of greater efficiency of calculation under a very limited set of circumstances that just happened to be satisfied in my statement of the problem, the first method has what computer scientists call greater "portability" --it is more easily adapted to cases in which the number of entrants is not a power of 2.

The third method of resolving the problem does not rely on notes concerning rounds, players per round, matches per round, etc. Instead (and here is the mathematical mind at work) we reason that since each match eliminates one player from the tournament, and that each tournament entrant except the eventual winner loses a match, there must be as many matches as there are non-winners of the tournament, 512 - 1 = 511 (and more generally, that such an elimination tournament in which N players are entered must similarly schedule N - 1 matches).

As compared with the first two methods, the latter is a model of mathematical elegance. It is clear, brief, and, as we have seen, readily generalizable. For those of you who would have known the answer even had I not given it to you, its brief explanation demonstrates why, as well as that, 511 is the desired number of matches. For all of you, it should be clear that the third method is even simpler than the first, quicker than the second, and is most easily extended to other cases.

The problem concerning the tennis tournament is far from the frontiers of modern mathematical research. Nevertheless, I hope that this contrast in styles of problem-solving will give you some sense of how mathematicians think. I hope the simplicity of the third method combined with the powerful generalization we obtained from its use will give you the flavor of mathematical discovery. If it is possible to reconstruct an ocean from a drop of water, or to describe the joys of skiing to a resident of an isolated equatorial area who has just felt for the first time shavings of ice from a modern freezer, then perhaps it is not too presumptuous of me to offer the above as an insight into mathematics.