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Why Study Mathematics?
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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.
Jonathan's Corner
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Why Study Mathematics?
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