“When are we ever going to use this?” Those words have come from the mouths of many students through the years. And while more often the student is seeking an excuse not to work rather than an actual answer, it is an important question. We shouldn’t spend years studying a subject without point. And no subject gets as bad a rap for being useless nowadays as math. Constantly you’ll hear the same things: “When will I ever calculate the tax on 700 bananas in the state of Utah?” or, “Don’t we have calculators for that?” And it’s not even just students. Many adults will often relate to one another on the ground of not understanding math. I don’t blame them, as I said, we should not spend years studying a subject which has no use; however, there is an extremely great value to learning math. In fact, there are two: the beauty of math, and the utility of math.
“Math is beautiful” you've probably been told that before. But, no matter how many grueling equations, fancy graphs, and wacky diagrams you were shown, it didn't change your perspective on the subject.
“What does it matter that e=mc2? All I see is numbers and symbols.”
“So what a2+b2=c2? It's just another equation to get me through this homework set.”
The “evident” beauty of math that keeps some engaged with the subject even in their leisure hours is just unapparent to you; so, when you hear, “Math is beautiful,” you might wonder (and very fairly), “If math is so beautiful, why can't I see the beauty?” It is this I hope to answer foremost, in order to assure our understanding and appreciation of the math I hope to expound after.
五月雨に
鶴の足
短くなれり
This is a Japanese haiku written by the legendary master Matsuo Basho, considered one of, if not the most beautiful, poems of its genre, and yet, when you or I look at it, it is completely meaningless. Why? Is Japanese poetry unartistic? Are Japanese people smarter than us? No. We are unable to appreciate the beauty of this poem because of the symbols it is written with. Being completely unfamiliar with Japanese, the beauty of the poem is obscured to us.
ei+1=0
This is a mathematical equation written by the God-fearing mathematician Leonhard Euler, and is considered to be one of, if not the, most beautiful equations in mathematics. Like the haiku, we can't appreciate the beauty of this equation, not because it isn't there, or because it is too lofty for us, but because it is obscured by notation that we do not know; Behind the foreign symbols of both the Japanese haiku and Euler’s equation is a beauty understandable regardless of what language we speak. In fact, it may surprise you, but mathematical notation was almost entirely invented in the 17 century, which, compared to how long people have been doing math, is extremely recent. Before then, math wasn’t done in symbols and equations, rather in whatever people were speaking. The common representation of the Pythagorean Theorem as a2+b2=c2is rather modern; before even the use of letters as variables or a smaller two above a number to represent squaring, Pythagoras wrote the Pythagorean Theorem in plain Greek, and many after him in plain English.
“If”, you might wonder, “mathematical notation obscures the beauty of math, why was it invented in the first place?” While you may at first think that mathematical notation was the product of some 17th century conspiracy aimed at making all of our lives more difficult, I can assure you that notation does serve a purpose, several, in fact. The first has to do with art.
Shakespeare in his play Hamlet says, “brevity is the soul of wit.” That is, saying a lot with a few words can often be as important as what is being said; this is why comebacks are only good if they are short, haiku are 17 characters long, and why Euler's identity, the most beautiful equation in math, is so beautiful. Euler’s identity is not beautiful just because of what lies behind the symbols; the beauty of it is how much it manages to say with so few symbols. You could not have Euler's identity without modern mathematical notation. If you try to write it in plain English while maybe easier to understand, it has lost all brevity and by extension all wit. But the brevity of notation does not lend only beauty to math, but also, an extremely great power, one I wish to demonstrate briefly by way of example. Consider the following problem.
A boat begins on a small island and travels north 6.9 miles, and then turns and sails 4 miles due west. How far is it from its origin island? Given a grammar stage knowledge in math this problem is relatively simple. We know the two stretches of the boat’s journey are the sides A and B of a right triangle. And the boat's distance from the island is the hypotenuse (the third and longest side) of the same triangle. We all know the Pythagorean theorem and so we set up our equation.
6.92+42=c2
square the numbers: 47.61 + 16 = c2
add them: 63.61 = c2
and take the square root: 7.97... = c
Thus we arrive at the conclusion that the boat is now about 8 miles from its originating island. You may be unimpressed by the so-called “immense” power of mathematical notation. If that is the case, I would like you to consider how what we just did works; math, after all, is a deterministic science, built on logic and observation of the world around us, it should be a simple matter to understand each step of such a simply done solution, as well as why it works. Let's take a look at the second step in which we simplify the left side of the equation; we took 6.9 squared and simplified it as 47.61. Seems simple but ask yourself, “how do we get that number? What are we actually doing when we square numbers?” “Squaring” you might say “is just repeated multiplication; so when we say 6.9 to the power of two, we are saying 6.9 times itself two times; that is 6.9×6.9” Okay, so then, if squaring is to multiply, a number times itself, what is multiplication? To which you might respond in like manner as before, “Multiplication is repeated addition, so 6.9×6.9 is 6.9 added to itself 6.9 times” I hope you see what is wrong here, you can add a number to itself 6 times, or you can add it 7 times, but there is no in between, you can not add a number to itself 6.9 times. This is the power of notation, it is so concise and rigorously built that it allows us to leap over inconveniences, such as adding 6.9 to itself 6.9 times and solve problems while not even needing to understand what we are really doing. However, as soon as all we see is the notation which is the vessel of beauty and not giving any consideration to what is actually behind it, to what we are really doing, we're not going to be able to appreciate the beauty within the symbols.
It is the beauty behind the symbols which keeps mathematicians in love with the subject. I hope that anyone in this room who doesn’t know what squaring is is asking himself right now what the meaning behind these symbols is! If anyone is doing that, he is well on his way to discovering the beauty of math.
So, if not multiplying a number by itself, what is squaring really? The answer lies within the word itself. When we square a number, we are saying, give me the area of a square with each of its sides being the length of the given number. So 6.92 is the area of a square with each of its sides being of length 6.9. Looking at the Pythagorean Theorem in light of this new understanding of squaring, one can glean a new appreciation for the same, and even understand how it might be written in English rather than math, like so:
“In a right triangle (that is, a triangle with one angle measuring 90 degrees), if we construct three squares, each with side lengths corresponding to a different side of the right triangle, the combined area of the two smaller squares will be the same as the area of the largest square, the one opposite the 90 degree angle. I can not force you to see the beauty of this any more than one can force another to like broccoli, but I can present it to you for you to taste, to attempt earnestly to enjoy. So try it, roll it around in your mind and chew it up. Wonder why it is. Why do these two squares add up PERFECTLY to this one, not just in this triangle, but any triangle with a 90 degree angle? The Architect of the world founded this as an invariable truth in his creation. A work as beautiful as the trees, grass, and animals he made if we but have the eyes to see it.
One of the best books to open eyes to the beauty of math in God’s creation is The Phantom Tollbooth by Nortin Justin. In it, the main character, Milo, who is initially of the opinion that, “There’s nothing for him to do, nowhere he’d care to go, and almost nothing worth seeing.” ends up on an adventure through a land with wackiness to rival wonderland. At one point, in the kingdom of Digitopolis, Milo and his companion the Humbug meet the Mathemagician, the mathy monarch of the numerical land. Shocked at the seemingly supernatural fetes he performs, Milo gasps:
“‘How did you do that?’ ‘There’s nothing to it,’ [The Mathemagician said] ‘if you have a magic staff’ … ‘But it’s only a big pencil,’ objected the Humbug, tapping at it with his cane. ‘There’s nothing to it,’ [The Mathemagician said] ‘if you have a magic staff’ … ‘But it’s only a big pencil,’ objected the Humbug, tapping at it with his cane. ‘True enough,’ agreed the Mathemagician; ‘But once you learn how to use it, there’s no end to what you can do.’”
The Humbug is put off by the mundane nature of the Mathmagician’s magic staff, a simple pencil. He does not recognize that the power stems not from the pencil, but the user’s understanding of how to use it. In modern times, we have the opposite problem; rather than discrediting our tools, we attribute too much to them.
Rather than pencils, the modern tools for math are sleek and potent. They inhabit our pockets and live on our wrists. Computers are handy, ubiquitous, quick. Constantly, even diligent students may wonder if my phone does everything I’m learning in this class, faster and better than I ever could master, why am I wasting years learning it? I’ve heard a lot of answers to this question over the years: “You’ll look dumb when you pull out your phone for 5 + 7” or “You’ll impress your friends if you can do double digit multiplication in your head” While these answers may be true statements, they are terrible answers. If math was about looking suave, schools would be teaching magic tricks and acrobatics. No, these answers miss a fundamental flaw with the question asked, as well as missing the greatest utility gained by learning mathematics. The question states that everything learned in math class our phones can do better and faster than we can. This is a lie. The truth is, our phones do not do math, they do calculations. Those two things are NOT THE SAME. Calculation is to mathematics what hammering or sawing is to architecture. In the hands of a master, it can be used to accomplish truly great things, but someone who thinks a hammer is all he needs to make a castle will have only rubble, ruined walls, and stones robbed of their former beauty. No, mathematics is not a course in calculation, it is the opposite, it is the study of critical thinking, of which calculations to make. Your computer can divide six trillion five hundred and thirty nine by pi like THAT, but only if you tell it to, and only because it has been told how to do so. If it is not told how to do a task, a computer can not accomplish such. Even large language models such as ChatGPT can do no more than ape humanity through imitation of the massive quantity of data they have been trained on. Computers can not solve problems they don’t have the steps for.
This is the job of a mathematician: to discover the steps to solve a problem. That is to say: while any problem for which there is a process to discover the solution is a problem a computer can solve, only if it is given the aforementioned process (no matter how complex, long, or tedious) can it solve the same. So when we look back at the work of previous mathematicians (Pythagoras, Einstein, Alan Turing) and see them hailed for solving problems computers can compute instantly with ease, we must remember, the computers are only able to solve problems because these humans solved the problems first.
It is also necessary to affirm that the skills of a mathematician do not only begin their utility at the lofty height of the theoretical sciences; simply, it is most easy to see the gleam of mathematics in such an area due to its novelty and its requisition of mathematics. But for those aspiring to the less green offices of God’s kingdom, these skills are yet an indispensable asset. Though it is by the use of mathematics that we are taught, it is not just in the field of numbers and operators we find uses for what we learn. Consider the following: a man who wishes to hone his muscles will break down the skills he wishes to have into their simplest forms, lifting weights, doing pushups, and using machines designed specifically for practicing these skills. This equipment, these motions, while perhaps a measurement, are not the goal, rather they are the means by which a man maintains his muscles, which he uses otherwise. Just so is math but for our minds. With numbers it is we train our mind, not so we can solve yet more math, rather, so we may apply our problem solving muscles, which we tuned with the math, to whatever we face in life. It is for this reason that mathematical problems come in “sets” and are called “exercises”. As we are aided by trained strength in all we lift, push, and else, so by mastered mathematics in every problem for which a process to the solution is to be discovered; organization, time management, moving furniture, nothing we see but requires thought.
And while we can afford to miss a method to move out faster, there are matters of greater magnitude we as Christians necessarily need to understand. If we do not want to be duped with false theories of economics, biology, and politics, we must needs breed a critical thinking and an understanding of the hidden mathematical world God has set to govern his creation. How can one intend to spot thievery and lies among bills and papers written to confuse and obscure if he can not solve simpler problems with facts laid out clearly? You have to walk before you run.
From man’s creation he was intended to work; God set Adam in the garden to tend it, and after the fall told him he would, “Eat by the sweat of [his] brow.” Likewise has God made man to think. In the Garden Adam was charged with naming the animals, a task I doubt he did haphazardly or without thought. And we are charged likewise, Proverbs 25: 2 says:
It is the glory of God to conceal a matter; to search out a matter is the glory of kings.to search out a matter is the glory of kings.
The very first command God gives man in the Bible is the dominion mandate, to subdue the earth, to be kings. It is to us, therefore, to search out those matters which God has concealed, a task of critical thinking, beauty and, and great use, accomplishable only by true architects of reason.
Caleb Thoburn, pictured here with his older sister, Abigail, graduated from Oak Hill on May 23, 2025. This was his senior thesis.
Works Cited
Matsuo, Bashō, et al. Haiku Illustrated. 2020.
Wells, David. “Are These the Most Beautiful?” The Mathematical Intelligencer, vol. 12, no. 3, June 1990, pp. 37–41, https://doi.org/10.1007/bf03024015.
Florian Cajori. A History of Mathematical Notations. New York, Ny, Dover, 1993.
Juster, Norton. The Annotated Phantom Tollbooth. Knopf Books for Young Readers, 2011._
