Unlearning Math Part 2 — Math is Art
This is the second in a series of articles exploring why math is often so adamantly avoided, and why I think it’s worth playing with anyway.
In our last chat, we discussed the need to interact with math differently. That requires a significant paradigm shift in how we perceive math, because clearly very few people are getting excited about memorizing formulas and plugging numbers into equations on repeat.
The thing is, there’s a major disparity between what most people see as math and what mathematician’s see as math. Quite a few mathematicians are working to bridge this gap and redefine mathematics for the general public. Here are a few that I particularly love:
- Eugenia Cheng — This incredible human is the reason I decided to abandon computer science and study pure math. I watched a video of her keynote at Lambda World 2017 over a weekend during my first term of college and made an appointment with my advisor the following Monday to change my major. She’s also written a stellar book that I highly recommend called How to Bake Pi.
- Francis Su — Wrote a book called Mathematics for Human Flourishing that fundamentally blew my mind and reignited my excitement for math during a period of major burn out at the beginning of COVID. He pushes the idea that doing mathematics is deeply tied to being human.
- Paul Lockhart — Wrote an incredible essay called The Mathematician’s Lament that I thoroughly believe every human deserves to read. It will shake your conception of math to it’s very core and has served as the motivation and inspiration for this series of articles. You’ll see me referencing this piece a lot.
In order to build up our own understanding of what mathematics is, let’s take a look at how some of these and other prominent mathematicians have defined mathematics:
“Mathematics is the study of anything that obeys the rules of logic, using the rules of logic” ~Eugenia Cheng
“A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it’s because they are made with ideas” ~G.H. Hardy
“Mathematics is a subject that allows for precise thinking, but when that precise thinking is combined with creativity, openness, visualization, and flexibility, the mathematics comes alive” ~Jo Boaler
“Mathematics is for human flourishing” ~Francis Su
“Mathematics is the art of explanation” ~Paul Lockhart
Art. That’s what mathematics truly is. It’s an intensely creative act. The facts and formulas you were forced to memorize in school are simply the byproducts of mathematics. As Lockhart says in his essay, “the art is not in the ‘truth’ but in the explanation, the argument. It is the argument itself that gives the truth its context, and determines what is really being said and meant”. Because let’s be entirely honest, the things you drilled and memorized in school are precisely what computers were made to handle. Storing facts and numbers, calculating formulas given specific values, arithmetic, solving equations… computers will always be able to do these things better, faster, more efficiently, and with fewer errors than we can. Yes, there is a place for it, just as there’s a place for memorizing and practicing vocabulary words when we’re learning to read and write. But notice that no one considers vocabulary to be the main goal, or even an overwhelming component, of studying literature. We see vocabulary as a tool. The real value comes from the creativity, the exploration, the aesthetic of the literary works, of which vocabulary is merely one piece.
So if the things most of us learned in school are simply the vocabulary of mathematics, what is mathematics itself? Paul Lockhart describes the mathematician’s art as the following: “asking simple and elegant questions about our imaginary creations, and crafting satisfying and beautiful explanations”. That’s hard to conceptualize through words alone, so let’s walk through an example.
In the video above, the first thing we come across is a simple question: If we draw a line from each vertex of a square to the midpoint of the opposite side, what is the area of the square created in the center? Great, it’s a wonderful question about an imaginary, but logical, scenario we’ve crafted in our mind.
Since we’re playing with these ideas in our head, we’re able to move the triangles around to see that when combined with the quadrilaterals, we get squares. It’s simple, it’s beautiful. And we can immediately see that our inner square is one fifth the area of the original square. At the end we get a formula, which is great if we ever come across this exact scenario again. But the real beauty lies in how we found that formula. This idea of moving around components of a shape to form new shapes is a beautiful technique that we can use in other ways to answer other questions. In fact one of the many proofs of the Pythagorean Theorem uses a very similar strategy.
Kind of stunning, right? Unfortunately, when most of us were taught the Pythagorean theorem in school, we were handed a formula relating the three sides of a right triangle to each other and told to find the hypotenuse of a bunch of triangles. Everything that makes the Pythagorean theorem a masterpiece was swept under the rug and you were left with some arithmetic to work out on your own. No wonder so many people find mathematics boring and pointless! If the entire purpose of learning the Pythagorean theorem is to be able to compute the side length of a right triangle, write a couple lines of code in Python and be done with it already!
And contrary to what you might have been taught, there’s rarely one way to do something mathematically. There are numerous proofs of Pythagoras’s theorem, for example, and when you start to see mathematics as art, you also start to develop your own aesthetic and taste. Some proofs will appeal to you more than others, just as surrealism and lofi might do it for you while your friend is more into pop art and acid jazz. Here’s an outline of a few of the more well-known proofs of the Pythagorean theorem. Do you notice yourself gravitating towards one over the others?
The point is, it’s these questions we pose, the mental puzzles, thought experiments, and beautifully-crafted, logical explanations that form mathematics. Unfortunately most people aren’t exposed to these things until their second or third year of studying math in college. Imagine how different your experience of math would be if it was treated as an art from the very beginning, rather than an exercise in memorization and regurgitation. Imagine if we saw it’s value in it’s beauty and creativity rather than simply it’s utility. Imagine being allowed to play with it, ask your own questions, be confused, make mistakes, and strengthen your creative problem solving skills in the process.
I know it feels like a stretch, but ask yourself this: is math education really serving us as it is? What are we actually gaining from it at this exact moment in time? And while you’re pondering this, let me leave you with one last snippet from The Mathematician’s Lament.
SIMPLICO: But don’t you think that if math class were made more like art class that a lot of kids just wouldn’t learn anything?
SALVIATI: They’re not learning anything now! Better to not have math classes at all than to do what is currently being done. At least some people might have a chance to discover something beautiful on their own.
SIMPLICO: So you would remove mathematics from the school curriculum?
SALVIATI: The mathematics has already been removed! The only question is what to do with the vapid, hollow shell that remains. Of course, I would prefer to replace it with an active and joyful engagement with mathematical ideas.
SIMPLICO: But how many math teachers know enough about their subject to teach it that way?
SALVIATI: Very few. And that’s just the tip of the iceberg…
