Mathematics and Culture:
"A Mathematician's Lament"

The following is an excerpt from “A Mathematician’s Lament” by Paul Lockhart, provided courtesy of Paul Lockhart and the Mathematical Association of America.

The first thing to understand is that mathematics is an art. The difference between math and the other arts, such as music and painting, is that our culture does not recognize it as such.  Everyone understands that poets, painters, and musicians create works of art, and are expressing themselves in word, image, and sound. In fact, our society is rather generous when it comes to creative expression; architects, chefs, and even television directors are considered to be working artists. So why not mathematicians?

Part of the problem is that nobody has the faintest idea what it is that mathematicians do.  The common perception seems to be that mathematicians are somehow connected with science— perhaps they help the scientists with their formulas, or feed big numbers into computers for some reason or other. There is no question that if the world had to be divided into the “poetic dreamers” and the “rational thinkers” most people would place mathematicians in the latter category.  Nevertheless, the fact is that there is nothing as dreamy and poetic, nothing as radical, subversive, and psychedelic, as mathematics. It is every bit as mind blowing as cosmology or physics (mathematicians conceived of black holes long before astronomers actually found any), and allows more freedom of expression than poetry, art, or music (which depend heavily on properties of the physical universe). Mathematics is the purest of the arts, as well as the most misunderstood.

So let me try to explain what mathematics is, and what mathematicians do. I can hardly do better than to begin with G.H. Hardy’s excellent description:

A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.

So mathematicians sit around making patterns of ideas. What sort of patterns? What sort of ideas? Ideas about the rhinoceros? No, those we leave to the biologists. Ideas about language and culture? No, not usually. These things are all far too complicated for most mathematicians’ taste. If there is anything like a unifying aesthetic principle in mathematics, it is this: simple is beautiful.   Mathematicians enjoy thinking about the simplest possible things, and the simplest possible things are imaginary.

For example, if I’m in the mood to think about shapes— and I often am— I might imagine a triangle inside a rectangular box:

I wonder how much of the box the triangle takes up? Two-thirds maybe? The important thing to understand is that I’m not talking about this drawing of a triangle in a box. Nor am I talking about some metal triangle forming part of a girder system for a bridge. There’s no ulterior practical purpose here. I’m just playing. That’s what math is— wondering, playing, amusing yourself with your imagination. For one thing, the question of how much of the box the triangle takes up doesn’t even make any sense for real, physical objects. Even the most carefully made physical triangle is still a hopelessly complicated collection of jiggling atoms; it changes its size from one minute to the next. That is, unless you want to talk about some sort of approximate measurements. Well, that’s where the aesthetic comes in. That’s just not simple, and consequently it is an ugly question which depends on all sorts of real-world details. Let’s leave that to the scientists. The mathematical question is about an imaginary triangle inside an imaginary box. The edges are perfect because I want them to be— that is the sort of object I prefer to think about. This is a major theme in mathematics: things are what you want them to be. You have endless choices; there is no reality to get in your way. On the other hand, once you have made your choices (for example I might choose to make my triangle symmetrical, or not) then your new creations do what they do, whether you like it or not. This is the amazing thing about making imaginary patterns: they talk back! The triangle takes up a certain amount of its box, and I don’t have any control over what that amount is.

There is a number out there, maybe it’s two-thirds, maybe it isn’t, but I don’t get to say what it is. I have to find out what it is. So we get to play and imagine whatever we want and make patterns and ask questions about them. But how do we answer these questions? It’s not at all like science. There’s no experiment I can do with test tubes and equipment and whatnot that will tell me the truth about a figment of my imagination. The only way to get at the truth about our imaginations is to use our imaginations, and that is hard work.

In the case of the triangle in its box, I do see something simple and pretty:
If I chop the rectangle into two pieces like this,

 

I can see that each piece is cut diagonally in half by the sides of the triangle.

So there is just as much space inside the triangle as outside. That means that the triangle must take up exactly half the box! This is what a piece of mathematics looks and feels like. That little narrative is an example of the mathematician’s art: asking simple and elegant questions about our imaginary creations, and crafting satisfying and beautiful explanations. There is really nothing else quite like this realm of pure idea; it’s fascinating, it’s fun, and it’s free!

Now where did this idea of mine come from? How did I know to draw that line? How does a painter know where to put his brush? Inspiration, experience, trial and error, dumb luck. That’s the art of it, creating these beautiful little poems of thought, these sonnets of pure reason. There is something so wonderfully transformational about this art form. The relationship between the triangle and the rectangle was a mystery, and then that one little line made it obvious. I couldn’t see, and then all of a sudden I could. Somehow, I was able to create a profound simple beauty out of nothing, and change myself in the process. Isn’t that what art is all about?

This is why it is so heartbreaking to see what is being done to mathematics in school. This rich and fascinating adventure of the imagination has been reduced to a sterile set of “facts” to be memorized and procedures to be followed. In place of a simple and natural question about shapes, and a creative and rewarding process of invention and discovery, students are treated to this:

Triangle Area Formula:  A = 1/2 bh 
“The area of a triangle is equal to one-half its base times its height.”

Students are asked to memorize this formula and then “apply” it over and over in the “exercises.” Gone is the thrill, the joy, even the pain and frustration of the creative act. There is not even a problem anymore. The question has been asked and answered at the same time— there is nothing left for the student to do.

Now let me be clear about what I’m objecting to. It’s not about formulas, or memorizing interesting facts. That’s fine in context, and has its place just as learning a vocabulary does— it helps you to create richer, more nuanced works of art. But it’s not the fact that triangles take up half their box that matters. What matters is the beautiful idea of chopping it with the line, and how that might inspire other beautiful ideas and lead to creative breakthroughs in other problems— something a mere statement of fact can never give you.  By removing the creative process and leaving only the results of that process, you virtually guarantee that no one will have any real engagement with the subject. It is like saying that Michelangelo created a beautiful sculpture, without letting me see it. How am I supposed to be inspired by that? (And of course it’s actually much worse than this— at least it’s understood that there is an art of sculpture that I am being prevented from appreciating).