I Love Mathematics
A personal detail I don’t mind sharing is that my desired career path is that of a mathematics teacher. A physics teacher, too, but my love is, above all, for mathematics. Many may scoff at the idea of loving this subject, considering how drab and soul-depleting it is at school. For those people, I highly recommend reading an essay called A Mathematician’s Lament by Paul Lockhart (you can find a PDF of it for free, on the relevant Wikipedia page). In it, you’ll get to hear that mathematics is an art.
This might illicit an even greater scoff – solving equations for x, fulfilling menial and arbitrary tasks, an art? Believe it or not, yes. And as to why it’s easy to overlook its artistic beauty, you’ll find a decent description of the problem in the above mentioned essay, as well. The problem is that mathematics is portrayed as the language of physics, and by extension, of science in general. That’s certainly true, but to focus on only that aspect of mathematics is like throwing away many literary works, simply because they belong to the fiction category. There is a great pressure to teach and learn mathematics because it is useful, which ignores the fact that it is only useful in certain professions, and that there are large portions of mathematics which are “useless”. Consider the video which quite possibly landed in your YouTube recommendations at some point, “how to theoretically turn a sphere inside out”. It is an example of a part of mathematics called topology. Now, topology was born out of an honest, practical question, and is used to solve honest, practical problems. But this sphere-turning stuff? Absolutely, unquestioningly useless… just like a painting. And yet, a painting can be beautiful, and so can topology. It can be fascinating, exhilarating.
Math is the language the universe uses to communicate to us, but it is also the language we can use to explore the farthest reaches of our imagination, purely for the fun of it. Consider something as simple as i, the square root of negative one. Simple concept, really, quite useful in some physical calculations, but let’s just stop for a moment and really look at it. When we call it an imaginary number, it’s because it defies the basic axioms (assumptions) of mathematics, which themselves are based on principles of nature. And yet, all it took was saying “It’s not real, but let’s do it anyway and see what happens” to unlock something great. We turned the number line into a number plane, turning an infinity of numbers into an infinitely bigger infinity. We got the most beautiful formula in mathematics, e^pi*i = -1. We created something impossible, unreal, and found uses for it to describe the real. Every single one of these is incredible in its own right, and yet the idea of i is something a third grader can stumble upon, just by wondering what happens if the number under the little roof symbol had a minus.
Of course, if a third grader were to try it, or ask the teacher, there’s a decent chance they’d just get told that the number cannot be negative. This is a common trope in mathematics learning, where certain parts of the subject are omitted for the sake of making things simpler, i.e. more boring. In high school, when finding solutions for graphs, you’ll get told that there are no solutions if the graph doesn’t cross the X-axis. Put in a more algebraic way, a curve with an equation like x^2 + 1 = 0 has no real numbers that can replace X and have the equation still be true. Real numbers… but why restrict ourselves to those? There’s i to the rescue, fitting neatly and producing a valid equation. This isn’t simply a cheat, but an actual solution, one that encourages us to step out from the realm of simple XY coordinates on paper, and imagine three dimensions of movement.
Instead of treating math as a fascinating subject on its own, we’re obsessed with the idea that it’s there just to get us working with physics and chemistry and such. Everything in math therefore needs to have a purpose, a goal. When a new topic is introduced, the first question (after “Will this be on the test?”) is “What’s it good for?”, although it often isn’t said out loud. I used to not care about that, but at some point got swept up in this sort of thinking. However, the love of math as math remained, because even when a good reason couldn’t be conjured up, it didn’t make it any less interesting for me to learn. It’s like the answer wasn’t to justify learning mathematics to myself, but to justify to others why I want to learn it.
Mathematics is art, and the simplest proof of that is that, to enjoy math, one has to be willing to get dirty, to experiment, fail and create. From as far back as I remember, I found it enjoyable to just… play with numbers. When taking car rides, I would look out the window, at advertisements and posters, look at any numbers therein, and see if I can find some pattern. 25 and 49, both square roots of primes. Or how about this, take any multiple of 9, and add the digits together. The resulting number will always be 9 (if it’s a two digit number, just repeat the addition step). Sometimes it’d come almost immediately, and sometimes it would take a long time to figure out; sometimes I wouldn’t be able to find a pattern, but that’s okay, part of the process. One time, I wondered the following: take any regular shape with an odd number of sides, like a pentagon. Connect every single point to every other point, and what you’ll see is that, on the inside, another pentagon appears. This also means that the process can be repeated infinitely. Now, what is the size ratio between the small one and the big one? There’s no need for a practical application, no need for anything other than curiosity. I made the problem myself, and wanted to find out the solution myself; the solution itself was the goal, not a step towards another goal. At the time I couldn’t do it, because I had not yet learned trigonometry, which is crucial, but once I did, I revisited it and wrote up a proof. When the topic of radioactive decay came up during physics lessons later on, I noticed a pattern in the numbers, and wrote my own formula for calculating the decay, which was a fair bit faster than the textbook alternative. I didn’t have any innate understanding of radiation or of logarithmic functions; I just looked at some numbers, thought about it, and scribbled something down. Seemed to work, so I kept using it.
The above may sound like boasting. Don’t get me wrong, when I did the things mentioned, I certainly felt proud of myself. But it’s not meant to show how I am somehow superior, especially since such a comparison could easily get turned on me; Weiner Heisenberg, at age 23, barely older than I am now, revolutionised quantum mechanics, meanwhile I’m getting all worked up about some formula that’s just an edit of another formula that was just handed to me. Besides, for every lucky try, like the ones above, there have been many times when I tried something and gave up without finding a solution. I also wasn’t a stellar student, by any stretch of the imagination. Don’t ask me to recite the quadratic formula, because I don’t know it; whenever possible, I solved quadratics by completing the square, instead.
Algebra is not some evil contrivance, it is just a formal way to write the way we (or at least I) think when doing math in my head, and it gets naturally extended to accommodate ever larger and more complex equations. Paper is not where algebra happens; it happens in the brain, and paper is just an extension of the brain’s memory. Differential and integral calculus are absolutely incredible and mesmerising. Sure, the integration tables can be tough to memorise, I have problems with that too. But take some time to really think about it, about what really links a function with its integral, why you get area out of it. Start with some simple examples, like a straight line, and try to unravel the magic at work. When I first learned of the unit circle, I was so disappointed by just how underwhelming the classes were, even though the circle itself is a very elegant piece of geometry, with many creations just waiting to spring out of it for anyone willing to take a look.
I could go on for much longer, but I’d like to make a slightly shorter post, for once. My only hope for the future is that I will be able to transfer at least some of my passion over to my would-be students, to get them interested and see math for what it really is – a creative outlet for the mind, an adventure to be had and beauty to be experienced.