Delaware Today judged nearly 200 papers submitted by students at A.I. duPont High School and the Charter School of Wilmington. Here’s the winning essay by Zack Li, who will be a senior at the Charter School of Wilmington in the fall.
Math and Beauty
“When will I ever use this?” It is a phrase uttered almost religiously in the halls of lower learning, as students struggle with reading, writing, and more commonly, mathematics. At the first stages, this complaint is quickly silenced. Anyone can see that there is value in having a student learn fractions, or basic geometry. Good teachers would be quick with a rebuttal, describing some hypothetical situation that the student might have to face. As the student progresses through school, however, this question becomes more common, more pressing, more emotional, but sadly, less adequately answered. Why does a student need to know how to factor an equation? With calculators and computers so readily available, does math even need to be learned? Will they ever use this knowledge that they are being forced to absorb? The answer is that there is much more to mathematics than mere calculation (Wigner). Mathematics bestows upon humanity something that it has searched for since the mind first became conscious, something that wise men spend lifetimes searching for—understanding. Math is beautiful, both from the understanding that it bestows, and its inherent qualities.
From an objective standpoint, mathematics fulfills all of the qualities that mankind defines as “beautiful”. For instance, many consider nature to be a prime source of beauty—gorgeous, colorful flowers, intricate and complex creatures, and a wonderful harmony with itself. Yet at the heart of nature is mathematics. Every event, every nook and cranny of the universe can be explained with mankind’s system of logic and symbols. The gentle curves of a nautilus shell, explained by the Golden Ratio (Freitag); the fractal and recursive growth of trees; the turbulence of the tides, modeled by the Navier-Stokes equations (Clay Mathematics Institute); mathematics gives us an understanding of all of these things, and rather than detracting from our appreciation of their beauty, in fact adds to it. As physicist Richard Feynman said, “If you want to learn about nature, to appreciate nature, it is necessary to understand the language that she speaks in.” (Feynman) Bertrand Russell, Nobel winning writer, mathematician, philosopher, logician, and activist, claimed that “Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show.” (Russell, “Mysticism”) The subject has every quality that mankind associates with beauty. It possesses an amazing simplicity at times, yet can become extraordinarily complex at others. For example, in a mathematical proof, where one takes basic assumptions called axioms, and then logically builds from these to show that something else is true, something can be extraordinarily difficult to prove, but incredibly simple to state (Benjamin). An example of this is the Four Color Theorem, which asks if one can color in a map full of countries with only four colors, if neighboring countries cannot have the same color. It took almost a century, and 1,936 cases tested by computer, to show that four colors is indeed the maximum one needs (Rehmeyer). An interesting view on this is Paul Erdos’; when he saw a particularly short and direct, or “elegant”, proof, he would claim it was in “The Book”, referring to a book that God keeps in heaven of all the best mathematical proofs (Babai). If one compares mathematics to art, something which certainly has beauty, one can see math’s aesthetic qualities as well. If “art matters because looking at a beautiful painting or sculpture gives us an experience that nothing else can (Boone)”, then doing mathematics can result in the same level of experience, if not greater. The creators of both areas—artists and mathematicians, are often viewed by society as isolated and eccentric, and both are deeply intellectual subjects. Like art, there is a timeless quality to mathematics, just as there is to art, and the work done by Euclid is just as true now as it was thousands of years ago, just as paintings and sculptures hold the same emotion that the masons and painters put into them long ago. Both mathematics and art are intrinsic parts of human existence, and there have been examples of both from ancient times, and both are born from intangible concepts born in the mind (The Neolithic Era). As one Colorado State Professor argues, “Both disciplines are creative endeavors with analytical components”, and “mathematics is their deeper purpose to reveal truth about our world and ourselves” (Farsi).
However, there are many differences too, and mathematics, rather than being an art, actually transcends artistry. Art enables a society to look inwards and reflect, while mathematics allows society to look outwards to the universe around us. If art is about teaching how to feel, mathematics is about teaching how to see. And what we often see is beauty.
Mathematics also has a wonderful harmony, and unrivaled depth and order. Different sub-fields of mathematics often find that there is much in common, and the discovery of powerful links between seemingly completely disparate ideas happens often. These characteristics made 18th century scientist and mathematician Carl Friedrich Gauss call mathematics the “queen of sciences” (“Queen of Mathematics”). A thousand experiments will never best a single proof, for the thousand and first experiment may show different results, but a proof, if done correctly, is true for all time. Science can never equal mathematics regarding logical solidity—the entire field seems to be built on the firm bedrock of logic and reasoning.
Or is it? In the early 20th century, the British mathematician David Hilbert proposed a series of 23 problems that he believed were the most pressing questions in mathematics at the time (Joyce). Problem two asked for a formal proof that arithmetic was itself consistent, and without internal contradictions. Hilbert believed firmly in formalism, the mathematical philosophy that treated all of math like a game, basically scribbles on paper. Different marks on paper meant different things, and mathematics was just a product of the human mind, with different operations on the marks on paper as simply rules of the game. This view depends on the rules of the game to be consistent; it makes no sense for humanity to invent a game that doesn’t follow its own rules, so Hilbert expected mathematics to be proved consistent. However, soon after, Kurt Godel proved his two incompleteness theorems—essentially that mathematics can never be proven to be consistent using mathematics (Kennedy). The rules of the game were changed—this meant that there are statements in mathematics that are simply undecidable. It adds even more to the allure of math—there is a wonderful sense of mystery that there are certain things about the universe that humanity will never be able to know.
However, what we do know is incredibly useful. Like a powerful sports car, mathematics has both form and function. A fancy car, painted flashy colors and decorated with fins and spoilers, would feel cheap or deceptive if it did not perform well. Mathematics does not have this problem. Follow a commuter—let us call him Dave—through a typical morning, this becomes immediately apparent. The time is 6 AM, and the alarm clock rings. Its ringing and timing are a product of electrical engineering, which makes heavy use of mathematics and matrix operations. The intersections and traffic lights that Dave passes through have all been tuned by a civil engineer using mathematical models to predict traffic flow. The road that Dave is driving on has been calculated to endure thermal expansion and contraction. The GPS unit that Dave uses to get to work (he’s very bad at navigating) relies on special relativity and distance calculations to establish his time and location. When Dave finally arrives at work, he realizes he forgot to eat breakfast, and heats up a hot-pocket in the microwave. The hot-pocket’s assembly line process has been streamlined using mathematics to reduce cost, and the delivery of it to the grocery store where Dave bought it was planned using graph theory to reduce fuel use. The microwave he heats his food with was developed with ideas from quantum electrodynamics, which relies on group theory. According to Ian Stewart, if one were to put a red sticker on every man-made item that used mathematics, it would cover the entire globe (“Letters”). Modern lifestyles are entirely supported by applied mathematics, and it is awe-inspiring how human civilization would be nowhere where it is without it. It most certainly has both style and substance.
And what style it has! Like a classical symphony, one area of study flows smoothly into the next, and ideas from different fields can be used to great effect in others. There is a powerful unity to mathematics (Dorrie), and one of the common tools used to solve difficult problems is to transplant the problem to a different field. Mathematics is a system based on logic, all stemming from basic axioms, and there is a wonderful sense of near-perfection to the whole endeavor. The visceral reaction when one inserts a puzzle piece, and it fits snugly with its neighbors—that is mathematics. It is a world of amounts and figures, of magnitudes and directions, of the simple, and the incomprehensible—and yet all of it is built on reason.
However, beauty is still subjective at its core. A common thing to say is that “beauty is in the eye of the beholder”, and in the case of mathematics, the eye is the mind. Just as a sharper eye can see more details, a sharper mind can discern more intricacies of mathematics. However, looking closer at many things normally considered beautiful will reveal ugliness, such as in Gulliver’s Travels when he visits an island full of giants (Swift). In contrast, looking closer at mathematics reveals even greater beauty, as more subtleties are shown, and even more questions are presented. As the mind squints to get an even closer view, the blurry outlines of other parts of mathematics hover in the distance, a reminder of math’s inherent self-interconnectedness.
Mathematics has great elegance, both from an objective and a subjective standpoint. Humanity can describe nature, build monuments, explain physics, and develop an appreciation of the beauty of the world around us with math, and in doing so, one can see the beauty that mathematics itself has. It is a system with beauty, rigidity and mystery, and as much as mankind currently knows, there exists so much more to discover.