I readily acknowledge that there are many math classes which are drudgery and a general waste of time, and that many people have had experiences with mathematics which give them good reason to hold a distaste for the discipline. However, it is my hope that I may provide readers with an insight that there is something more to mathematics, and that this something more may be worthwhile.
Let’s begin by looking at the reasons that the reader may already have come across for why he should study mathematics:
- There are certain basic computational skills that are needed in life. People should be able to figure out whether a 24-pack of their favorite soda for $3.89 is a better or worse deal than a 12-pack for $1.99.
- It builds character. I suffered through mathematics for such-and-such many years. So should you.
Of course, nobody explicitly says the second reason, but it may very well seem that way—like one of the hush-hushed truths that the Adult Conspiracy hides from students the same way it hides the fact that there is no Santa Claus from little children. And the first reason is something that many non-mathematical administrators believe.
But those are not the real reasons that a mathematician will give for why a nonmathematician should study mathematics, and what kind of mathematics a nonmathematician should study.
The first question which should be addressed is, “What is mathematics really about?”
The answer which many nonmathematicians may have is something along the lines of, “Mathematics, at its heart, is about learning and using formulas and things like that. In gradeschool, you learn the formulas and methods to add, subtract, multiply, and divide; then in middle school and high school it is on to bigger and better formulas, like the formula for the slope of a line passing through two points. Then in college, if your discipline unfortunately requires a little mathematics (such as the social sciences requiring statistics), you learn formulas that are even more complicated and harder to remember. The deeper you go into mathematics, the more formulas and rote methods you have to learn, and the worse it gets.”
The best response I can think of to that question is to respond by analogy, and my response is along the following lines:
A child is in school will be taught various grammatical rules, sentence diagramming, and so on. These will be drilled and studied for quite a long while, and it must be said that this is not the most interesting of areas to study.
An English teacher who is asked, “Is this what your discipline is really about?”, will almost certainly answer, “No!”. Perhaps the English student is proficient in grammar, but that’s not what English is about. English is about literature—about stories, about ideas, about characters, about plots, about poetic description, about philosophy, about theology, about thinking, about life. Grammar is not studied so that people can suffer through learning more pointless grammar; grammar is studied to provide students with a basic foundation from which they will be able to use the English language. It is a little drudgery which is worked through so that students may behold an object of great beauty.
This is the function of the formulas and rules of mathematics. Not rules and formulas so that the student is prepared for more rules and formulas, but rules and formulas which are studied so that the student can go past them to see what mathematics is really about.
And what is mathematics really about? Before I give a full answer, let me say that it is something like what English is about.
The one real glimpse that someone who has been through high school may have had of mathematics is in the study of geometry. There are a few things about high school geometry that I would like to point out:
- In geometry, one is given certain axioms and postulates (for example, the parallel postulate—given a line and a point not on the line, there is exactly one line through the given point which does not intersect the given line), definitions (a circle is the set of points equidistant (at an equal distance) from a given point), and undefined terms (point, line). From those axioms and postulates, definitions, and undefined terms, one begins to explore what they imply—theorems and lemmas.
- In geometry, rote memorization is not enough—and, in fact, is in and of itself one of the least effective approaches to take. It is necessary to understand—to get an intuitive grasp of the material. Learning comes from the “Aha!” when something clicks and fits together—then it is the idea that remains in the student’s memory.
- Geometry builds upon itself. One starts with fundamentals (axioms, postulates, definitions, and undefined terms), and uses them to prove basic theorems, which are in turn used along with axioms and postulates to prove more elaborate theorems, and so on. It is like a building—once the foundation has been laid, beams and walls may be secured to the foundation, and then one may continue to build up from the foundation and from what has been secured to the foundation. Geometry is an edifice built on its fundamentals with logic, and the structure that is ultimately built is quite impressive.
- Geometry is an abstract and rigorous way of thinking. (More will be made of this later.)
- Geometry is about creative problem solving. The aforespoken edifice—or, more specifically, what is in that edifice—is used by the geometer as tools with which to solve problems. Problem solving—figuring out how to prove a theorem or do a construction (which is a special kind of theorem)—is a creative endeavor, as much as painting, musical improvisation, or writing (and I am writing as one who does mathematics, paints, improvises, and writes).
Imagine a dream where there are many pillars—some low, some high—all of which are too high to step up to, and all of which are wide enough to stand upon.
Now imagine someone dreaming this dream. That person looks at one of the pillars and asks, “Has anyone been on top of that pillar?” Then one of the Inhabitants of his dream answers, “No, nobody has been on top of that pillar.” Then the person looks at another of the pillars, which has a set of stairs next to it, and asks, “Has anyone been on top of that pillar over there?”. The answer is, “Yes, someone has, and has left behind a set of steps. You may take those steps and climb up on top of the pillar yourself, if you wish.”
And this person continues, and sees more pillars. Some of them stand alone, too high to step up to, and nobody has been to those. Others have had someone on top, and there is always a set of steps which the person left behind, by which he may climb up personally. And the steps go every which way—some go straight up, some go one way and then another, some seem to almost go sideways. Some are very strange. Some pillars have more than one set of steps. But all of them lead up to the top of the pillar.
The person dreaming may well have the impression that one gets atop a pillar by laying down one step, then another, then another, until one has assembled steps that reach to the top of the pillar. And, indeed, it is possible to climb the steps up to the pillars that others have gone to first.
But that impression is wrong.
And the person sees what really happens when the guide becomes very excited and says, “Look over there! There is a great athlete who is going to attempt a pillar that nobody has ever been atop!”
And the athlete runs, and jumps, and sails through the air, and lands on top of the pillar.
And when the athlete lands, there appears a set of stairs around the pillar. The athlete climbs up and down the stairs a few times to tidy them up for other people, but the stairs were produced, not by laying down slabs of stone one atop another, but by jumping.
Then the guide explained to the dreamer that the athlete had learned to jump not only by looking at the steps that others had left, but by jumping to other pillars that already had steps, instead of using the steps.
Then the dreamer woke up.
What does the story mean?
The pillars are mathematical facts, some proven and some unproven.
The pillars that stand alone are mathematical facts that nobody has proven.
The pillars that stand with steps leading up to them are mathematical facts that have been proven.
The steps are the steps of proofs, the little assertions. As some of the steps are bizarre, so are some proofs. As some pillars have more than one path of steps, so some facts have more than one known proof.
The leap is a flash of intuition, by which the mathematician knows which of many steps will take him where he wants to go.
As the steps appeared when the leap was made, so the proof appears when the flash of intuition comes. The athlete then tidied up the steps, as the mathematician writes down and clarifies the proof, but the proof comes from jumping, not from building one step on another.
The athlete was the mathematician.
Finally, the athlete became an athlete not only by climbing up and down existing steps, but also by jumping up to pillars that already had steps—one becomes skilled at making intuitive leaps, not only by learning existing proofs, but also by solving already proven problems as if there were no proof to read.
As one philosophy major commented to me, “Mathematicians do proofs, but they don’t use them.”
That flash of insight is the flash of inspiration that artists work under, and in this sense a mathematician is very similar to an artist. (What do a mathematician and an artist have in common? Both are pursuing beauty, to start with…)
This character of mathematics that is captured in geometry is true to geometry, but the actual form that it takes is largely irrelevant. Other branches of mathematics, properly taught, could accomplish just the same purpose, and for that matter could just as well replace geometry. Two other disciplines which draw heavily on applied mathematics, namely computer science and physics, have essentially the same strong points. I would hold no objections, for that matter, if high school geometry classes were replaced by strategy games like chess and go.
Mathematics is about puzzle solving; I would refer the reader to works such as Raymond Smullyan’s The Lady or the Tiger? and Colin Adams’s The Knot Book: an Elementary Introduction to the Mathematical theory of Knots. There are many people to whom mathematics is a recreation, consisting of the pleasure of solving puzzles. If mathematics is approached as memorizing incomprehensible formulas and hoping to have the good luck to guess the right formula at the right time, it will be a chore and a torture. If it is instead approached as puzzle solving, the activity will yield unexpected pleasure.
My father has a doctorate in physics and teaches computer science. He has said, more than once, that he would like for all of his students to take physics before taking his classes. There is a very important and simple reason for this. It is not because he wants his students to program physics simulators, or because there is any direct application of the mathematics in physics to the computer science he teaches. There isn’t. It is because of the problem solving, the manner of thinking. It is because someone who has learned how to think in a way that is effective in physics, will be able to think in a way that is effective in computer science.
This applies to other disciplines as well. Ancient Greek philosophers, and medieval European theologians, made the study of geometry a prerequisite to the study of their respective disciplines. It was not because the constructions or theorems would be directly useful in making claims about the nature of God. Like physics and computer science, there was no direct application. But in order to study geometry, one had to be able to think rigorously, analytically, critically, logically, and abstractly.
Thinking logically and abstractly is an important discipline in life and in other academic disciplines that consist of thinking—it has been said that if you can do mathematics, you can do almost anything. The main reason mathematics is valuable to the non-mathematician is as a form of weight lifting for the mind. Even when the knowledge has no application, the finesse that’s learned can be useful.
To the non-mathematician, mathematics is a valuable discipline which offers practice in how to think well—both analytic thought and problem solving. Mathematics classes will most profitably be approached, not as “What is the formula I have to memorize,” but with ideas such as those enumerated here. The nonmathematician who approaches a mathematics class as an opportunity for disciplined thought and problem solving will do better, profit more, and maybe, just maybe, enjoy the course.
It is my the hope that this essay have provided the nonmathematician with an inkling of why it is profitable for people who aren’t going to be mathematicians to still study mathematics.