Beauty vs. Utility in Mathematics

Mathematics is the language of the universe, it happens to be embedded within all that surrounds us. It is the way in which the various quantities of our universe are harmoniously entangled with one another. Not only does mathematics have the ability to coherently convey what our tools of linguistics cannot, but it also has a beautiful quality of revealing the unknown, given that we humans are able to interpret it. This tells us about another rather elegant quality about mathematics: that its hidden treasures are only illuminated upon when we humans are able to decipher them. This makes mathematics not only a language but also an art form; its interpretation lies at the discretion of those subject to it and even its beauty to be perceived, lies in the eye of the beholder.

Mathematics happens to be a subject of dread for many primarily due to the way it is taught to us in our schools; we are expected to memorise and regurgitate formulae in our exams with little understanding of the phenomena and marvels which have been discovered by mathematicians throughout the ages. The exposure which we have had to what mathematics is, is quite constrained during our high school years. We believe that at its core, mathematics can be appreciated by each and everyone - this is irrespective of whether a deep passion exists, to pursue it for life.

Within the massive realm of mathematics itself, there exists a duality: pure and applied mathematics. Despite having their individuality as separate areas of mathematics, in today’s world, these domains of mathematics exist in solidarity to compile solutions for real-world problems. The ensemble of pure and applied mathematics enables the common man - with no background in mathematics whatsoever - to be truly able to appreciate the power of this art.

It can be argued that without pure mathematics, there is no possibility for its applied counterpart to exist - yet, on the other hand, it can also be argued that without applied mathematics, pure mathematics has little to no utilisation in our practical world. The integral property of applied maths is that it is founded by principles of pure maths: the axioms and laws constructed by pure mathematicians govern applied mathematics. Although we can say that to apply to our practical world, we tend to make simplifications to highly complex and intricate mathematical identities, yet, we have to also understand that these simplifications hold true and hence, are not redundant.

Putting aside the argument of the degree of utilisation of pure maths, since we established mathematics as an art form, we seek to also convey our message that an entity’s utilisation does not equate to its beauty. Although many may disregard pure mathematics on its own and prefer applied mathematics to it as the former on its own has little value in terms of utilisation in the real world, this cannot be said the same for its value when considering beauty. We seek to establish that mathematics is an art and art, indeed, has beauty. However, the integral quality of art is that its beauty is subjective and dependent on the perceiver - the same can be applied for these domains in mathematics.

Pure mathematics is a form of art and its beauty and conveyance of the same is subjective; it differs for those perceiving it. When talking about its utilisation solitarily - it may appear to be of little use but the point is not to appreciate its use rather to appreciate that its beauty does not rely on its use. There is a certain aesthetic beauty which is present within pure mathematics - which may take a certain eye to see it, but once it is acknowledged, it is surely such that enables you to fall deeply in love with the very truth of mathematics.

It is this aesthetic beauty of pure mathematics that attracts people to it. Mathematicians believe that mathematics allows them to create order out of the chaos that we live in. This order can be translated into symmetry, patterns and precision. These things, from an evolutionary standpoint, correspond to beauty. This is the reason why pure mathematicians seek out aesthetic beauty and are less concerned with its utility. In some cases, the simplifications that come with application actually takes away from the beauty of pure mathematical concepts because these simplifications often involve approximations which diminishes the precise nature of these concepts.

Another aspect that draws people into this discipline is that aha! moment where one suddenly realizes the solution to a problem or understands a previously incomprehensible concept. When one experiences a eureka moment, they are filled with a sense of wonder and profound joy. This feeling is very addicting and helps to drive one’s curiosity further.

Sometimes, the beauty of mathematics comes from its unexpected quality. Take for example the following function f(z)=z^2+c, where c is a complex number, and we start with z=0 and we plug f(0) back into the function and repeat this infinitely. Depending on the value of c, this function either converges or diverges. When one plots the values of c for which the function converges on the complex plane, we get the Mandelbrot set, which is a magnificent and beautiful fractal. In this case, we get an unexpectedly beautiful piece of fractal geometry from a rather mundane function.

Mathematics also allows humans to work with infinite and infinitesimal quantities that are otherwise unfathomable. In calculus, both the derivative and integral of a function both deal with infinitesimal differences in the independent variable of the function. And with induction, one can prove a statement for an infinite number of cases. In group theory, the symmetrical properties of objects are represented by sets and operations on those sets. Thus, by representing symmetry in this manner, it makes it possible to analyse and understand symmetry and its fundamental properties. It is to be noted that both of the above branches of mathematics have undergone certain levels of abstraction and it is this removal from the real world that allows them to reveal truths about elemental ideas like infinity and symmetry.

To many, only when the complexity which exists within the domain of pure mathematics, is watered down to suit practicality, then only can it be appreciated. This could be seen as an inherent quality of us humans to be ignorant to ideas which we find too difficult to understand. Yet, it’s true that one can still appreciate mathematics and simply feel apprehensive of what pure mathematics brings to the table. Moreover, the way we appreciate these domains of maths as separate ideas may be attributed to the way each of our minds are wired: some of us may discover beauty in applicability whereas, others in abstractionism.

But when solely considering applied maths, it cannot be denied that the way in which it is manifested by bringing pure mathematical ideas to life is truly wonderful! Take for instance, calculus: the study of instantaneous or rather, infinitesimal change: absolutely revolutionary in what it is able to provide for us as humanity in the realms of physics to economics to even how we perceive things on the daily. Although the fundamental theorem of calculus, essentially purely mathematical in nature, is the foundation of calculus as a whole - the way it has evolved and developed to solve problems in various other fields is how it can be truly appreciated.

By virtue of applied maths, these pure mathematical concepts are being used in physics: to explain and act as a tool to solve in mechanics and electromagnetism, for instance. Calculus seems to almost be made for application; the ways in which it can be used on the daily are also prevalent. Weber’s law is one such law which makes use of differential calculus at its core: it forms a relationship to describe how we humans experience things. It states that the extent to which we are able to recognise change is dependent on the ratio of change that has occurred to the total stimuli; calculus is able to quantify our experiences and define it as a law which can be used in business and psychology to explain the science of human reactions and perceptions.

The world has mathematics embedded in every nook and cranny: the possibilities of application of mathematics can be thought to be figuratively infinite. As we advance as humans, we will discover more realms in which we can apply our pure mathematics to understand phenomena. Essentially, applied mathematics provides humanity with a vital tool, a tool which can be seen as revolutionary, as it enables us to quantify our lives in order to better comprehend it. Of course, it’s immensely difficult to quantify things such as human emotions, yet a quantification nonetheless is able to take us one step closer to understanding the world and ourselves. Another sub-domain of applied mathematics which helps model and quantifies human behaviour is game theory, the marriage of human sciences with probability. Game theory is a sound manner to explain the interdependence of human behaviour with stimuli, again, an attempt to combine mathematics and applicability to real-life scenarios.

Linear algebra and computational mathematics enable algorithms and statistical models to be built to forecast the weather or even explain our ongoing COVID-19 pandemic. Graph theory has become fundamental to pictorially representing variables and how they’re related to convey to the common man. This is the power of applied maths: it makes itself comprehensible to someone with no mathematical background whatsoever, even to an extent to fall in love with it. Applied maths can be thought of as a medium through which anyone can interact with principles of pure maths indirectly; it is a mode through which we are exposed to the very intricate ideas of pure mathematics.

Applied maths performs the cohesion of pure maths with our very lives to give us the product of a sound understanding of the workings of various other realms: be it physical, socio-economical, psychological or even artistic.

The domains of mathematics are like the two sides of a coin. While it is known that pure mathematics leads to developments in its applied counterpart, it is also true that applied mathematics can lead to advancements in pure mathematics. For example, Issac Newton developed calculus to study physical phenomena like gravity and motion, and now it is one of the most important branches of pure mathematics. From this, one can see that even though pure and applied mathematics are separate fields, there is a cyclical relationship between them, where progress in one often results in progress in the other.

Pure mathematics and applied mathematics can be seen as being two different art forms which could explain why one would appeal more to another. Pure mathematics could be thought of as abstract art: art that requires perspective and an eye to interpret or conceive what ‘beauty’ is. Whereas in the case of applied mathematics, the art counterpart would be realistic art, where principles of abstractionism are utilised to portray real situations we are more enlightened to. We understand that beauty is a very subjective concept, quite especially in the realm of mathematics where more often than not, the only ones able to see this beauty are those who have a deep passion for it. However, we would like to convey that in order to see beauty, it isn’t imperative to love; it is just imperative to appreciate.