Have I scared you yet? Talking about math seems like one of the easiest ways to terrify people, make them feel stupid, and cause them actual pain. I think that’s a shame, because math is AWESOME. I’m going to try to convince you of that in the next few hundred words.
What is math?
Math is the study of patterns and logic. Any repeating pattern or system analyzed rigorously and logically can be math. The coastline of Britain? Sure! The spread of diseases? Absolutely! What the relationship between the number of sides of a shape where all sides are the same length and the area of the shape? Definitely.
Since any pattern or system is up for grabs, math is incredibly creative. You get to just pick whatever rules or approach or framework you think might yield useful or interesting results and see where they lead you. Let’s say you were interested in triangles. You could take the points of the triangles as coordinates (like (2,5) and (3,4) and stuff) and do all kinds of calculations to see what the area was. OR, you could not care at all where the points are and just take lengths of sides. OR, you could not bother with calculations or algebra at all and do the whole thing geometrically. You can even prove things with gifs!
Ok, why are you getting so excited right now?
Because that makes math AWESOME. Anything can be explored. Look, pick some rules you’ve decided to follow. See where they lead. BOOM, you’re in an entire world of your own making. Those rules are axioms. Seeing where they lead, logically, means you prove things with them, sometimes you call those conclusions theorems. And now you have a totally new mathematical world. It might as well be writing fiction or LARPing.
You think I’m kidding, but I’m not. There are 5 axioms called the Euclidean axioms. They are as follows.
- A straight line can be drawn between any two points
- A finite line can be extended infinitely in both directions
- A circle can be drawn with any center and any radius
- All right angles are equal to each other
- Given a line and a point not on the line, only one line can be drawn through the point parallel to the line.
You can have all kinds of fun with just these. Take a piece of paper and see if you can convince yourself, even informally, that these seem to be true. Use crayons, markers, pencil, whatever.
Now throw them all out. Fuck ‘em. We’re going to start our own mathematics with blackjack and hookers. What if instead of a straight line being the shortest distance between two points, a semicircle is. Seriously. You just invented geometry on a convex (curving inward, like the inside of a beachball) plane. It’s called hyperbolic geometry.
It looks like this.
And people freaking crochet hyperbolic curves.
You could have done basically the same thing by saying, ok, I learned in like 7th grade that all triangles have 180 degrees. Well, what if they don’t? A triangle is a shape with three sides, right? What if I want more than 180 degrees? Well, you can have whatever you want. In math, the only rule is that you have to follow your own rules. What those are, you get to decide. So draw a triangle with more than 180 degrees. Ok, it’s hard, I grant you. It seems like it would have to have more than three sides. What are we missing? What assumptions are we making? Oh! That the sides have to be straight! What if they curved out! Like a triangle you blew air into?
Congratulations, you just invented elliptical geometry. It’s the geometry that explains why planes fly like the curved line instead of the straight one:
Because the earth is curved, not flat, so the geometry changes. And triangles, just like you wanted, have more than 180 degrees.
Then Why Does Math Feel So Awful to Learn?
There are a few reasons why people hate math. For one, no one teaches math as something fun and creative. They teach it as something boring and rote, where the rules are set up beforehand and totally unchanging. To get fun math, you have to go to youtube to see people like Vi Hart make math the beautiful thing it is.
Second, because math is so abstract, it can be hard to visualize, and it makes it feel mysterious, even after you understand the problem. Like in biology, once you understand why evolution works, you get it. You know how it works. Sometimes, I’ll prove something for a class, and I’ll know it’s right, and that everything follows logically, and still not really know what I just did. For instance, visualize a line for me. That’s one dimension. What’s the two dimensional form of that? Right, a square. And three dimensions? A cube, great. And next? That question is the intellectual equivalent of moving both index fingers together in front of someone’s face, asking them to follow the fingers with their eyes, and then suddenly moving them in different directions. You just don’t know what happened to you. (For readers of Flatland, there’s a reason the sphere gets very upset when asked if there are more than three dimensions)
That shape, by the way, is called a tesseract, and it looks like the picture below in three dimensions, even thought it’s a four dimensional thing. But what does the next one look like? At some point, visualizations run out and logic and proof must take over.
Thirdly, math has a language, and it’s not an easy one to learn. There are all the symbols, for one: numbers, logical operators, less than, more than, exponent, subscript, and on and on until you think you’ll drown in them. And then there are the rules for how they fit together. This implies that. Why again? Oh yes, because this. And that makes sense because? Oh, right. But eventually, if you follow math far enough, you develop a deep respect for mathematical notation, its minimalism, its utility, and you begin to deeply distrust anyone who says, “Math would be fun, but why are there so many symbols?” (Though of course, there’s tons of math to be done without them. You get to make the rules, remember?). But you also get to criticize notation, decide that some is better than others, and take sides on Newtonian vs Leibnizian differential notation.
But because it is in some sense, a language, I wish it was taught like one. I wish that young children read proofs without fully understanding them, just as we are encouraged to read texts in Spanish without looking up every single word. I wish we contented ourselves with the gist of the proof, the point, so that we learned to prioritize the meaning over the form, just as we may not be able to word-for-word translations of our French teacher’s request, but we know it’s time to sit down.
What’s the Point of Math? And of this blog post?
Well, math is beautiful. And fun. And creative. And the point of this blog post was to convince you of that. But if pure practicality is important to you, know that if you set up the rules properly, in that they reflect the way the universe works, then you’re going to get empirically verifiable predictions from the logical conclusions of the rules. That’s how physicists knew there had to be a Higgs Boson long before we could even in principle find one. That’s how we started building bridges.
Math can describe with incredible accuracy how the world works. But it can do so much else besides. It is a powerful discipline, and it deserves our respect. For some of us, it has commanded our reverence.