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Math: Usefulness vs. Elegance

By Philip P | Thursday, July 14, 2022

Every once in a while, we ask ourselves, “why am I doing this?” We sometimes question why we are doing the things we do now, or why things happened as they did: why we go to school, why we exercise, why water is essential for life, or why diagrams and pictures on exam papers are always blurry and monochromatic?.

For students, probably the more commonly asked question is why we study math. Math is one of the most disliked subjects in school. Even before we delved into advanced math, I didn’t understand its functionality in the real world.

But, as we all are aware, math’s contribution in shaping today’s worlds are quite significant. Its practical applications in improving science & technology are quite evident. Also, literally almost every major requires math, so it’s pretty important to be good at it.

For that reason, I started to like studying math, and I think it is the same reason for many other students as well. When we study things that can be used to solve real-world problems, we are much more interested in them because they’re useful. We study, say, Calculus because everything in this world is changing and moving, and Calculus allows us to describe and model changing systems. We study Statistics because of its usefulness to analyze data and make predictions. Or, if you’re like me, studying math is also useful in understanding math memes. :D

There is the sense of purpose that acts as motivation. Undeniably, math is not the easiest subject in the world, so we still need to give some extra effort. But, we know that practice makes perfect. It’s like practicing music. I practice my scales even though I dislike them, because I know by doing I will become a better musician.

However, that motivation will not last long because of the kinds of math you’ve learned or will learn. Think of some math problems, preferably harder ones, you’ve solved in school, from any field (Trigonometry, Calculus, etc) you like. What do you think about them?

Aren’t they somewhat…random? I mean, they came out of nowhere without any background context – like someone made them up just to make our lives difficult. Not all of them, though it seems like the majority are. They don’t seem to answer any problems relevant to our real lives.

Think about all the weird and unrealistic functions you have to integrate, as if those functions will ever exist. Think about all the unusual shapes that their areas you have to find, as if that has any use. Or, perhaps the problem seems to be realistic but at the same time it is heavily idealized, like “The teacher will be giving out homework according to the function f(x), or “A runner is running on a triangular-track that has sides a,b,c such that (a+b+c)(a+b-c)=3ab…” and….well you see where this is heading.

The point is that lots of math problems are pointless (see what I did there), and seem very disconnected from practicality. So, what’s the point in doing the hard work of solving them? This is the point where students who do math for ‘work’ will have a hard time and get demotivated easily, eventually nurturing a dislike for math because of its abstractness, weirdness, and irrelevance.

So then, how can others love math? Surely it’s not just because of innate talent or some random fascination with numbers and calculation. There must be something attractive about solving those ‘artificial’ problems.

The answer is simply…those questions are beautiful; math is beautiful. To us, this may sound strange as how come a thing that only consists of numbers and formulas be considered beautiful. We use the word ‘beautiful’ to instead describe things like art, music, fictions, scenery, etc –not academic subjects, specifically math. But really if you think about it, math is a lot like art or fiction, and you’ll see it even more clearly when encountering difficult, strange problems.

For example, you are asked to integrate a weird function that is a combination of nesting natural logarithms and trigonometric functions. Right off the bat, you don’t know where to start. You stare at the function for a while to get some clues what to do with it. One hits you, then you decide to try it out. Sadly, that method doesn’t work. So, like a good detective, you search for another clue, and think outside of the box. Then, you came up with an ingenious idea to solve the integral. And finally, when you arrive at the answer, it turns out to be very beautiful – which makes solving this integral very satisfying and fun!

Words cannot describe how elegant math can be. If you’re still doubting the elegance of math, I highly encourage y’all to check out some math content on the internet, preferably YouTube. There are a bunch of Math YouTube channels out there that really display how elegant, fun, and satisfying math can be. Some of my personal favorites are 3Blue1Brown, BlackPenRedPen, Numberphile, MichaelPenn, Tibees, FlammableMaths, and of course some of our very own Schoolhouse lessons. They all have different math content, different expertises, and different ways of explaining math – like characters in a story having distinct personalities and perspectives.

Nevertheless, I think to truly experience the beauty of mathematics is to really interact with it. Do those problems yourself, work through the steps. If you’re struggling, that’s good. That means you’re learning. After all, where’s the fun if you can do all of the problems with no effort?

So, that’s the real reason why math problems can be so useless and unrelated to real life. While being able to use math to solve real world problems is satisfying, I truly believe that it is much more satisfying to experience the true beauty of math.

for some fun math!

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