I was surprised to find myself in a regular school classroom— no easels, no tubes of paint. “Oh we don’t actually apply paint until high school,” I was told by the students. “In seventh grade we mostly study colors and applicators.” They showed me a worksheet. On one side were swatches of color with blank spaces next to them. They were told to write in the names. “I like painting,” one of them remarked, “they tell me what to do and I do it. It’s easy!”
After class I spoke with the teacher. “So your students don’t actually do any painting?” I asked. “Well, next year they take Pre-Paint-by-Numbers. That prepares them for the main Paint-by-Numbers sequence in high school. So they’ll get to use what they’ve learned here and apply it to real-life painting situations— dipping the brush into paint, wiping it off, stuff like that. Of course we track our students by ability. The really excellent painters— the ones who know their colors and brushes backwards and forwards— they get to the actual painting a little sooner, and some of them even take the Advanced Placement classes for college credit. But mostly we’re just trying to give these kids a good foundation in what painting is all about, so when they get out there in the real world and paint their kitchen they don’t make a total mess of it.”
“Um, these high school classes you mentioned…”
“You mean Paint-by-Numbers? We’re seeing much higher enrollments lately. I think it’s mostly coming from parents wanting to make sure their kid gets into a good college. Nothing looks better than Advanced Paint-by-Numbers on a high school transcript.”
“Why do colleges care if you can fill in numbered regions with the corresponding color?”
“Oh, well, you know, it shows clear-headed logical thinking. And of course if a student is planning to major in one of the visual sciences, like fashion or interior decorating, then it’s really a good idea to get your painting requirements out of the way in high school.”
“I see. And when do students get to paint freely, on a blank canvas?”
“You sound like one of my professors! They were always going on about expressing yourself and your feelings and things like that—really way-out-there abstract stuff. I’ve got a degree in Painting myself, but I’ve never really worked much with blank canvasses. I just use the Paint-by-Numbers kits supplied by the school board.”
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.
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.
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.
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?
Are you really trying to claim that mathematics offers no useful or practical applications to society?
Of course not. I’m merely suggesting that just because something happens to have practical consequences, doesn’t mean that’s what it is about. Music can lead armies into battle, but that’s not why people write symphonies. Michelangelo decorated a ceiling, but I’m sure he had loftier things on his mind.
it’s like rearranging the deck chairs on the Titanic!
But a problem, a genuine honest-to-goodness natural human question— that’s another thing. How long is the diagonal of a cube? Do prime numbers keep going on forever? Is infinity a number? How many ways can I symmetrically tile a surface? The history of mathematics is the
history of mankind’s engagement with questions like these, not the mindless regurgitation of formulas and algorithms (together with contrived exercises designed to make use of them).
This is intimately connected to what I call the “ladder myth”— the idea that mathematics can be arranged as a sequence of “subjects” each being in some way more advanced, or “higher” than the previous. The effect is to make school mathematics into a race— some students are “ahead” of others, and parents worry that their child is “falling behind.” And where exactly does this race lead? What is waiting at the finish line? It’s a sad race to nowhere. In the end you’ve been cheated out of a mathematical education, and you don’t even know it.
Art is not a race.
How sad that fifth-graders are taught to say “quadrilateral” instead of “four-sided shape,” but are never given a reason to use words like “conjecture,” and “counterexample.”