The Eternal and All-Encompassing Nature of Mathematics
To those who have grown to harbor disdain towards mathematics for any reason whatsoever, I am sorry. This writing is for you.
I set one goal for this essay - to convince my reader that mathematics is more than a mere annoyance of incomprehensible symbols and formulas. I shall not address the ways we are taught mathematics in school - that is not a matter I would like to discuss in this essay. Instead, I aim to tackle the common misconception that mathematics is a boring, uninteresting, and unattractive field of study. For this purpose, my approach will be rather whimsical, and I mean in no way to impose my views on my readers. All I hope is that I may inspire in them a new way of seeing mathematics. Maybe after reading this, they will consider more than one possible attitude towards the subject…
Let us begin.
Mathematics is the most misunderstood, misinterpreted and aridly explained subject in all of human history. For the purposes of this essay, I have one request of my reader: to abandon the notion that mathematics is solely a science of manipulating numbers and performing operations with them, and to set aside any resentment they might bear towards it on that basis. For math is a thousand things before it is this. Math is first and foremost a philosophy – a way of seeing the world around us. It is fundamentally about imagination, creativity, flexibility, and vision.
Suppose the following, a ballet dancer moves her body in an elegant motion, which immerses you into a captivating performance. You watch as that body twists and molds into different shapes and curves. They are continuous, they bend and cross as the dancer swings her arms front and back, and front, and back... I take each of these imaginary curves and I plot them on a Cartesian coordinate system in my mind. They are beautiful, unpredictable, changing and intangible. Now watch how I can give each one meaning - a symbolic representation that I can then replicate and study over and over.
And repeat this dance forever.
Y = f(x) = x, a perfect diagonal line at 45 degrees, a stretched body with its powerful muscles.
Y = f(x) = x^2, a gentle parabola, described by the slightly curved, lifted arms of the ballerina.
Y = f(x) = x^3, an off-set motion – one arm bends downwards, while the other bends upwards.
The mesmerizing dance of mathematics.
This is what math is – a way of seeing the world. I interpret my surroundings in mathematical language to invent new imaginary worlds. In that sense, math resembles more what we generally call art, rather than a rigid rational science. If math is art, we can compare it to what is often considered to be its counterpart: literature, and namely, poetry. A poet gathers inspiration from reality and redefines it to craft new imaginary worlds. A mathematician does the same. Imagination plays an analogous role in both. Surprisingly, the latter might demand an even higher magnitude of this quality. Influential German mathematician David Hilbert states the following about a student who lost his aspirations for mathematics and switched to pursuing poetry: “He didn't have the imagination […] to make a good mathematician – [so] he became a poet” (qtd. in Tomov). A striking statement. Perhaps the reason we fail to appreciate math is because we focus too much on the raw formulae, and we fail to see the bigger picture - the imagination and creativity that lie as if knitted into the well-known symbols. But what IS the bigger picture?
Math is nothing but a serene dance woven into all the cells and rocks, and matter in the universe.
Margaret Wertheim writes about this phenomenon – that math is present even within species exhibiting much lower thinking capacity compared to our own. Corals, slugs, even atoms move and mold in complex geometric patterns that baffle even the most adept of mathematicians. In her essay ‘How to play Mathematics’ Wertheim delves into meticulous mathematical pattern-chasing which suggest that math is almost too fitting, oddly fitting, of our world. It is as if there is a cosmic blueprint that seeps into and gives structure to our entire universe and all of life and non-life in it. Curious. Why does the falcon always swoop down towards its pray in a perfect logarithmic spiral? Why do atoms move in torus (doughnut-shaped) patterns and diffuse into clouds? Why do corals form hyperbolic crochets with their bodies? I am starting to see something here.
Math might be present all around us.
I like to imagine mathematics as an all-encompassing weave – an ether of intangible fibers of concepts that flows eternally around all parts of the universe. It is fascinating to think about. And what is even more fascinating is our human attempt to tap into this ceaseless weave and channel its power to our own goals and desires. The centuries-long struggle to make sense of this force that envelops us – this is what makes mathematics irresistibly interesting. Mathematicians in history have been like tailors who have tried to push this cosmic thread through their needles and string it into forms that we understand. Hence, the symbolic formulas we know from textbooks. However, the underlying thought process behind these formulas is really what engages the mind of the mathematician and it is a wonder that most of us never get to experience for ourselves. So let us try to change that and uncover, at least a little, what math is really all about.
What is the area of a rectangle? We know the formula but let us try to derive it intuitively. A rectangle has two pairs of identical sides. To find the area it covers all we need to do is measure the space inside these four sides. Let us fill this space with identical small circles in straight rows and columns. We can have them as small as we want, so the space left between them would be infinitely small and so negligible. To find the area then, all we need to do is find the sum of these circles’ areas. We could use simple addition. As we keep adding more circles, we will eventually reach our desired answer. That, however, would be quite tedious, so we stop and try to think about our approach. If our mass of circles fills the space, won’t the sum of all vertical or horizontal lines of circles give us the area of the whole rectangle? It will. It will, because the circles are so small, as are the gaps between them, that it would be just as if we were lining up lines without any space between them. But now instead of counting circles, we are counting lines. Hmm. What if we just multiplied the area covered by a line of small circles by the area covered by a line of small circles on the adjacent side? It would give us the whole space. Eureka! Well, isn’t finding the area the same as multiplying the sides? It turns out to be true. And there we have it:
The area of a rectangle: A = a * b.
People who dislike mathematics often ask questions like: “Who invented this? Who thought of this? Who would ever care about this?” I ask a counter-question: Wasn’t what we just did fun? We rediscovered a widely known formula, but what made the process engaging was that we employed our thinking and creativity in finding the answer. What many do not understand, as Lockhart reminds us, is that “mathematics, like any literature, is created by human beings for their own amusement” (6). It is precisely the journey of discovering the answer for myself that has made mathematics meaningful to me. It is not my knowledge of how to calculate the rectangle’s area from the very start. It is my struggle with the problem and the thought process that led me to its solution.
And here comes my next point.
Mathematics is not hard because it is incomprehensible; it is hard because it is a creative process which revolves around solving a problem (Lockhart 11). It is for the same reason, that math is also not boring. Consider this. We have already established that math is an ethereal weave that flows all around us. And it is only through our inventive and imaginative minds that we can actually bend some of the fibers of this waltzing ocean of concepts. But really what we haven’t talked about is how our connection with this weave actually reflects on us. Think about how we resolve conflicts in our daily lives. It is similar to the way we solve math problems. The short answer is: we don’t. We avoid the most troubling of matters like building strong relationships, tackling environmental issues that concern us all, pursuing our dreams even. Why? Because they are difficult conflicts to resolve. Ones that are synonymous with struggle. And humans do not like to struggle. It is hard to be honest with that person when you realize you do not want a relationship with them anymore. It is hard to dedicate your time and energy to preserving the environment. It is hard to put in the hours to reach your goals. Our response to these dilemmas often is to throw in the towel. To turn a blind eye on it and light the fire under that cauldron of denial. Put off the talk with that person for as long as possible, say “someone else will do it” and tell yourself: “I can do it later” and have “later” turn into “never”.
Thus brewing our own inner hell.
This is why we procrastinate. We have an innate tendency to avoid confrontation. Mathematics is misunderstood and disliked because it points exactly to that shortcoming of our species. The fear of self-confrontation. Math points to our heart. It points to our inner hell, and our perpetual denial of its presence. It is precisely that denial, that reluctance to look into ourselves that impedes most of us from seeing the beauty of mathematics. Solving a difficult math problem is akin to confronting our demons. It is often an arduous battle. However, the sublime feeling of accomplishment when we finally resolve it is worth all the struggle. Every single encounter we have with this wonderful subject is an adventure of its own. Like any good adventure, there is something between the place we are in and the place we want to reach – the start of the problem and the desired solution. And only a challenging journey can take us there. That process of discovery, of facing our fears, of imagining solutions, of bending and adapting the established to craft the undiscovered. The struggle with ourselves – that’s what mathematics is about. Math is perhaps the most honest of arts, the most humane of the humanities, and the most artful of the sciences. It is wonderful we have managed to tailor even a small fragment of this heavenly force into our world.
I hope that I have managed to convey my perception of the art of mathematics so far, but I have done little to ground my daydreams into concrete proof. So what if after all math isn’t so effective at describing our world? Professor of electrical and electronics engineering Derek Abbott states the following: “mathematics is the product of the human imagination that we tailor to describe reality” (qtd. in Zyga). You would think Abbott and I agree. Except I am an optimist and he is not. He points to various shortcomings of mathematics to suggest it is not always so effective in describing our world as we like to think. One major reason is that we cherry pick suitable problems which allow us to apply our known mathematical solutions. Another is that even counting has limitations – how do we define where a banana begins and where it ends? Do we understand integers? We have no mathematical definitions besides our visual comprehension. Also, our definitions are biased by our own nature – would we speak of the Sun as something that burns for so long if we lived for billions of years? All these peculiar oppositions threaten to burst the bubble of our established magical view of mathematics.
However, I think that, on the contrary, they further solidify what has been said. The fact is that mathematics, as we know it, is an invention of the human mind, derived from our innate curiosity, imagination and capacity to see patterns and reinterpret reality. The keyword here is imagination. I imagine mathematics as the ubiquitous force that spans thousands of galaxies and I find the patterns that make it applicable to my world. Of course, our understanding of math is biased heavily by the nature of our species. And it is only natural that it is limited to our perspective of the universe. How could we have another view of things? This is why I believe these symbols and encapsulations of concepts are unique to our civilization. It is OUR interpretation.
Also, it is natural that we praise math as being effective and applicable to the problems we know how to solve. As mentioned earlier, we avoid confrontation – some questions remain unanswered for centuries, and just because we wield the tools does not mean we are capable of arriving at the answer. Math is greater than us and we have tamed but a small part of it. Abbott argues that math is made by humans and it is limited and purposeful when it delivers compact and simple explanations. This may be true of our world. Nonetheless, it is marvelous that we have managed to see so much of math’s applications in the universe, be they biased or not.
To sum up, I have but four things to recompile. Number one: mathematics is not simply a study of manipulating numbers; it is fundamentally a way of perceiving and interpreting the world around us. Number two: we exist in a world enveloped by a mass convergence of intangible fibers, which weave and flow through our entire universe, and it is our eternal struggle to knit these fibers that makes mathematics meaningful to us. Number three: mathematics is not hard because it is complicated, it is hard because it is about solving problems and self-confrontation. And number four: our understanding of mathematics may be biased by our nature but it is only natural that they are, it is amazing we even have them.
My sole purpose in this essay has been to depict math as a gentle and honest study that comes from and points to the heart. In the end, learning and applying equations is not the reason why we love math. We love math because it helps us learn about ourselves. And that should be reason enough for us to care. And if you don’t find that interesting or at least engaging in a sense, my dear reader… Maybe that would be reason enough for you to pick up a pen and paper. You just might find more than you came looking for. I myself am off to think about some rectangles.
Lockhart, Paul. “A Mathematician’s Lament.” Mathematical Association of America, 2009.
Shapiro, Stewart. “Mathematics and Reality”, Philosophy of Science, vol.50, no. 4, pp. 523-548, Dec. 1983, JSTOR.
Tomov, Lachezar. “The Mathematician as Poet”, Peat Nekogash, vol.29, Jun. 2020.
Wertheim, Margaret. “How to Play Mathematics”, Aeon, 7 Feb. 2017. Edited by Marina Benjamin.
Zyga, Lisa. “Is Mathematics an Effective Way to Describe the World?”, Phys.org, 3 Sep. 2013.
This essay was originally submitted as an argumentative essay assignment for ENG 102 Writing Academic Research Papers (Spring 2022) with professor Olga Nikolova.