For centuries, much of our understanding of the world has been conveniently cleaved into two poles – the Mathematicians as opposed to the Romantics; the empirical world view as opposed to the emotional world view; the logical, mathematical mind as opposed to the creative, artistic mind. These dichotomies have pervaded and subtly influenced our modern lifestyles – it seems that mathematics is for the studious; art and the humanities are for the spirit. These two dimensions of the human existence seem immiscible, irreconcilable even. This rift has left mathematics relegated to a position analogous to the position occupied by a kitchen knife in the kitchen – it is cleanly efficient in serving a clear purpose, but once its purpose has been served, it is quickly sheathed and stowed away. There is none of the appeal possessed by an ornately decorated piece of cutlery, for instance. However, I believe that mathematics not only serves a practical purpose, but possesses a supreme and profound beauty to the appreciative mind.
To the cynical eye, mathematics may appear to lack beauty, for it is unforgiving. The study of mathematics is characterised by the dogged desire to prove a fixed outcome. The world of mathematics is necessarily binary, where every response is either right or wrong. This departs significantly from common perceptions of beauty, which is inclusive and accepting of varied responses. Moreover, mathematics appears highly esoteric, drawing only the appreciation of the most cerebral. Hence, some might argue that mathematics lacks the inherent, intrinsic appeal many a work of beauty possesses. Finally, many critics have lamented that mathematics is stripped bare, devoid of ornamentation or decoration that breathes life and beauty into human creations. By dint of this austerity in aesthetic appeal, it cannot be said to be beautiful.
However, such accusations are little more than misrepresentations and misinterpretations of what mathematics entails. Within the realm of mathematics, there are many possible ways to interpret a question, and accordingly, many possible ways to arrive at an answer. This allows for diversity in thought to pervade mathematics. Moreover, the beauty of mathematics lies in its universality – it is democratic and catholic. Mathematics is fundamentally built upon a priori reasoning, which means that a thinking brain is the only apparatus needed to enter the enigmatic world of mathematics. The study of modern physics, in contrast, demands billions of dollars of investment before any real findings can be made, as evinced in CERN’s construction of the Large Hadron Collider, which cost a hefty 1.2 billion Euros. The notion that anyone, armed with just pen and paper, can make groundbreaking discoveries previously hidden from mankind is not only truly fascinating, but also a testament to the universal allure of mathematics. This concept of accessibility has long been fundamental to the nature of beauty – truly beautiful creations are easily appreciated and accessed by the common man. This is best illustrated in the case of music. With a few mere years of musical education, many of us are able to pry open the world of musical geniuses, reproducing the most imposing of Beethoven’s symphonies, the most soothing of Chopin’s nocturnes, or the most jovial of Mozart’s sonatas in their full glory. By virtue of their accessibility, these masterpieces have been etched into the halls of eternity, a shining star of beauty amidst our quotidian lives. Similarly, the accessibility of mathematics has allowed it to touch and inspire the lives of many, magnifying its beauty tremendously.
Beyond this, mathematics is a common language that transcends the labyrinth of culture and language, expressing ideas without a shred of ambiguity. The articulation of mathematics in a common language – a lingua franca, if you will – allows for people to convey ideas in their original, preserved form across time and space. While the modern reader may never know exactly what Shakespeare had in mind when he wrote Macbeth, people today have no doubts about Newton’s intentions when he proclaimed that force is equal to mass times acceleration, despite living in a very different age today. Amidst the tides of ever-changing social norms and linguistic nuances, the lingua franca of mathematics stands proudly as a firm, unmoving sculpture, a distillation of the undiluted brilliance of great geniuses before us. Indeed, there is great beauty to be had in the eternal.
Furthermore, where some people may see harshness in the laconic nature of mathematical expression, a deeper analysis will reveal that in conciseness lies great elegance. Supreme beauty is conveyed not with excessive ornamentation, but with profoundness in simplicity of expression. Beautiful and well-written poems do not contain any measure of superfluous words; musical masterpieces do not force in more notes for the sake of achieving complexity in performance; truly sublime paintings do not contain any more brush strokes than are needed to portray their subjects. Instead of condemning the concise nature of mathematics for being bare and unrefined, we should appreciate the beauty of being able to encapsulate great truths in a short series of unassuming numbers and letters. Every word of Maxwell’s four equations on electromagnetism is crucial to our understanding of this otherworldly force, allowing for a few mere letters to harness a great power. Adding more layers of decoration and opulence is a simple task anyone with sufficient resources can undertake; cutting back and preserving only what is necessary, on the other hand, is the true crucible of a beautiful mind.
However, a nagging question persists – what distinguishes mere beauty from supreme beauty? Beauty is defined by sensuous enjoyment; it is simply anything that pleases the senses. Supreme beauty, however, serves a deeper purpose – it seeks to empower mankind, elevating humanity from an uninspired existence that is solely driven by the pursuit of a cornucopian utopia. Mathematics achieves this, by placing man in a position of omnipotent authority. Man reigns supreme within the realm of mathematics – we are free to define the boundaries of this virtual world, and lay down axioms of truth upon which greater truths might be discovered. There are no limitations within the world of mathematics – mathematicians are empowered with the ability to dream up concepts and realities such as multiple dimensions which cannot be perceived in the physical world we live in. In many cases, the pursuit of supreme beauty is accompanied and harmonised by the pursuit of subverting physical limits, expressing mankind as the triumphant party in the feuding battle between man and his environment. Architecture presents many examples of such a struggle to overcome the constraints of natural forces. Brunelleschi’s attempt to build Santa Maria del Fiore – the largest dome built in the world – embodied this desire to rebel against gravity, casting physical boundaries aside in the pursuit of supreme beauty. Notably, Brunelleschi’s architectural tour de force was achieved only with a close application of mathematical concepts. Brunelleschi utilised a revolutionary brick layering technique that took advantage of geometrical concepts to distribute the weight of the dome evenly, preventing gravity from foiling his architectural marvel. Hence, mathematics has clearly entered the world of sublime, supreme beauty, for it uplifts man to be the master of his environment, a position from which mankind can create boundless and unrestrained beauty.
Perhaps the true mark of supreme beauty, ironically, is that we are never made aware of its existence – a supremely beautiful creation achieves its purpose efficiently and simply, never attracting unnecessary attention. As mathematics quietly serves its purpose, solving problem after problem, it may be easy for it to fade into the background, losing its beauty in the eyes of the masses. This is a great pity. Against the backdrop of the cacophony of dissenting and clashing voices that has become commonplace in the modern world order, sometimes what we should treasure most as beautiful is not the divergence in thought among the arts, but rather the austere simplicity of mathematics that reassuringly tells us that one plus one is always two.