Mathematics for Its Own Sake

Phillip Williams

Why should a person learn mathematics? I often ask my freshman students this question on the first day of class. Usually the question evokes a range of standard answers. In fact, I’ve found that most people (not just college students) will give the same sorts of answers when posed this question. Answers generally involve why mathematics is important, why we need it. People will say that we need math because it is useful in all sorts of ways: We need math to build computers, or to construct buildings, or to send rockets into space. We need math to estimate our grocery bill, or to balance our checkbooks, or to keep a personal budget. We need math for all that science accomplishes, and to understand our universe and how it works. We also need math to be informed citizens, to understand how numbers are used or misused in the public sphere.

These are all true, and good, answers. But there is one answer my students almost never offer, and it is this: We should learn mathematics for its own sake. Occasionally someone mentions that math is beautiful, and that perhaps this is a (mostly likely secondary) reason to learn about it. No one, however, directly states that we should study mathematics simply in order to have a better understanding of mathematics itself, that we should explore and appreciate it on its own terms.

People easily accept the idea that a person might study art, or even science, for its own sake. These pursuits are commonly accepted as intrinsically interesting and worthwhile. But math for its own sake? While it is generally accepted that there are some odd people who are inclined this way (the sort of people who become mathematicians), rarely is it suggested (by anyone other than a mathematician) that this is the main reason that most people should learn mathematics. My hope is to convince you that it is.

There are two things we are almost never taught in school to do, things that mathematicians routinely do:

  • Come up with a mathematical question (a question about which the subject is mathematics itself).

  • Try to answer a mathematical question for ourselves, through our own ideas and reasoning (even if we know the answer is already out there).

For most of the math done in school, the questions, as well as how to answer them, are provided to the student. Students are not expected to try to come up with either of these on their own; nor are they taught that doing so is really the main point of mathematical thinking. There is almost nothing for students to do! And yet, ironically, doing math in this way can seem extraordinarily difficult, and also tedious.

So let me give an example of a mathematical question. It is not an original question; in fact, I bet that readers may already know how to get the answer.

Question: If I walk 30 feet north, and then 40 feet west, how far am I from my original starting point?

I often pose this question to freshmen; many of them know, right away, how to answer it. They remember that a “distance formula” exists, or that the way to approach such a problem is to draw a triangle and employ the Pythagorean theorem: The sum of the squares of the two legs of a right triangle is equal to the square of the hypotenuse:

a2+b2=c2
>

Using this, if a = 30 and b = 40, we can see that c = 50.

After a student successfully solves this for me, I then ask the following: How do you know that the Pythagorean theorem is true? And further, how do you think that anyone ever came up with the Pythagorean theorem in the first place? Why is it the case that the sides of a right triangle share this special relationship, where if you attach squares to the two smaller sides you get the same area as when you attach a square to the larger side? Someone had to discover this.

I try to get students to see that discovering this is not impossible for them to do—that coming up with such a formula, and understanding why it is true, is not reserved for geniuses. The math formulas that exist came about, for the most part, because ordinary people were having ordinary thoughts about their experience. For example, in our world we can move in different directions. How does our movement in one direction relate to our movement in another (perpendicular) direction? Ancient people, with much less understanding of nature than we have, asked and successfully answered this question. Almost anyone with enough curiosity can do the same. Math is, in this sense, universal. To ask interesting questions, and think about the possible answers for yourself, only requires that you be a human being.

But what has to be appreciated is that even if a good question has already been asked and answered, there is still great value in undertaking to answer it for yourself. Even if you already know what the answer is supposed to be, doing this is valuable. Ultimate originality is not important—what matters is that some of the questions, and at least some of the avenues you follow and methods you use in answering them, must be original to you. In fact, if you don’t regularly do such things, then you will never really understand the “answers” that you think that you know; further, you will never really enjoy doing math if you don’t approach it on its own terms, as a reality to be observed and engaged. It will always be external to you, and detached from your mental life.

You might never hit upon the Pythagorean theorem as your own idea (to me, it is still somewhat mysterious how the original formula occurred to anyone, although I have some thoughts about that), but you can still take a crack at trying to prove it. I encourage readers to do so; dozens of different proofs exist—America’s twentieth president, James A. Garfield, is even responsible for one of them! Draw some pictures (the claim is essentially visual, and you can prove the Pythagorean theorem without ever writing down a number), and try to see for yourself why it might be true. Even if you get stuck, or never come to a complete answer without assistance, the attempt will be worthwhile.

Asking and answering mathematical questions routinely leads to new questions, so there is never a shortage of new and interesting things to ponder. For example, the right triangle mentioned in the example above has whole number edges (30, 40, and 50). Are there other such right triangles? How many? Can we describe what they look like? Asking such questions leads to consideration of whole number solutions to the equation

a2+b2=c2
>

This is a question of arithmetic rather than geometry (everything in math is connected!). We can discover, and completely describe, what whole number solutions to this equation look like—all of this was worked out by mathematicians in the past, and the answers are beautiful and interesting; in short, there are infinitely many solutions, and they can all be described by a particular formula. This leads to other questions: for instance, can we get any whole number solutions to this similar-looking equation?

a3+b3=c3
>

The answer actually turns out to be no. This is part of the content of the famous Fermat’s Last Theorem (1637), proved in the 1990s by Sir Andrew Wiles, which basically says that the equation

an+bn=cn
>

has no integer solutions (other than the “trivial” ones, where some of the numbers are zero).

Here is another path we could follow. We could ask: What about the triangles that don’t have whole number edges? How can we measure the length of their sides? This leads to the contemplation of irrational numbers, and various ideas about approximation.

In a different direction, we could ask: Is there a 3-D version of all of this? After all, we live in a 3-D world. So how do we find distances in three dimensions? If we go up by x, left by y, and forward by z, how far have we traveled from our starting point? On a whim, I once asked a version of this question to my students, and was quite pleased to discover a student who came up with the solution on the spot. Again, I encourage you to think about this for yourself. The answer can be obtained from the 2-D Pythagorean theorem with a bit of clever visualization. (Send your answers to me at [email protected], and the editors of AQ will post them, with my comments, on the National Association of Scholars website: https://www.nas.org/articles/the_2d_pythagorean_theorem.)

Every question we ask and answer in mathematics leads to new questions. Math is, perhaps, the only field of study in which we are guaranteed never to run out of good questions.

What is the point of all of this? Is it really worthwhile for everyone to think of mathematics in this way, or is this an activity that ought to be reserved for the few odd people who seem to want to engage in it? I am not always sure that everyone can discover the interest in mathematics that I have. But what I do know is that if most people are to find true value in learning mathematics, they have to learn it in the right way.

A mind that has been trained to think mathematically, as described above, to ask and answer the sorts of questions that mathematicians ask, and to learn to try to answer them, will hone itself to perceive mathematical structures in all of life. Seeing how mathematics applies to the world around you—and seeing mathematics in the world around you (that is, seeing things that raise new mathematical questions)—becomes natural to you only when you have spent some time contemplating mathematical structures for their own sake.

I am not arguing that an emphasis on application is not important when one is studying or learning mathematics. In fact, the connection to the world around us is the source of good mathematical questions; the two realms are intimately related. Einstein, I’ve heard, conceived of his theory of general relativity because he found himself imagining the experience of a person in a free-falling elevator. This led him to contemplate new ways of thinking about the relationship between gravity, matter, acceleration, and space. And this led him to think about non-Euclidean geometry, a subject that had already been explored by mathematicians before him.

Einstein obtained these insights into the physical world because he was very familiar with the mathematical world. His intimate familiarity with mathematics caused him to see how a mathematical idea (non-Euclidean geometry) would be useful in resolving certain physical questions. We are not all Einstein, but we can all contemplate the mathematical world, and doing so will, I’m convinced, change how we look at the world around us, and make clear to us the ways in which mathematics enriches our understanding of it.

I’ve stressed the importance of curiosity, imagination, and discovery in the process of doing mathematics. There is another side to the mathematical process that is equally important, and perhaps equally valuable to shaping a person’s mind: the strict, logical standards of mathematics. Regularly engaging in mathematics requires one to develop a firm grasp of logic and argumentation. Everything in mathematics must be proved, rigorously, from what is already known. In math, every argument must have airtight logic, or it is rejected.

I believe that these hard logical standards of mathematics develop a strong eye for all of the common logical fallacies, for incomplete arguments and instances of circular reasoning, and for the subtle yet essential places where a plausible-sounding argument fails. When you do mathematics, you have to focus your mental energy precisely where your argument is incomplete, and so you train your mind to seek out what has not yet been established. Doing mathematics therefore helps you to pinpoint quickly where an argument might break down, and to create counter-examples to statements asserted without justification. This develops a healthy skepticism about what can easily be deduced from what is already known. This is particularly valuable, for instance, in the humanities and social sciences.

A lot more could be said about the logical aspect of mathematics, but that is perhaps a subject for another article. It suffices to say for now that studying math for its own sake refines our logical, as well as imaginative and perceptual, abilities. For these reasons, and others, it brings us to a place where we see the world around us differently, with new and better eyes.

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