Utility and Futility of Mathematics
by Daniel Lee January 2013
Received Honorable Mention for Best Essay Award at Korea-American Scientists and ENgineers Association YGNITE 2013 Conference.
Abstract Usefulness of math rarely comes into question. No educational curriculum is built without it; no scientific discipline is free from its principles and notations. But utility of math is not free from criticisms, and the assertions like “math is a game played with meaningless symbols on paper” maintains silent subscribers. Is math really “useful”? In this essay, I question the “usefulness” of mathematics and conclude that mathematics, at its core, is not “useful” - and that its futility is perfectly okay.
Our gathering of the honey of the imaginative world is not immediate; it takes work. But though it requires traveling some distance, merging with something not of our species, communicating by dance to our fellow creatures what we've done and where we've been, and, finally, bringing back that single glistening drop, it is an activity we do without contortion. It is who we bees are." Barry Mazur, from Imagining Numbers: Particularly the Square Root of Minus Fifteen.
I think my calculus teacher in highschool enjoyed being asked the ominous question, “When will I ever use Calculus in my life?” She would grin lightly, taking pleasure in the opportunity to imbibe a significant knowledge onto the dull, unsharpened mind who had mustered enough courage (and ignorance) to ask that infamous question. She would gently quip back at him with plethora of reasons why he should care to learn calculus; that calculus, in fact, was employed to predict the trajectory of a football, used to describe the planetary motions, that civil engineers use calculus to build beautiful and durable skyscrapers. All reasons culminate to a single potent argument: that calculus, and more broadly mathematics, is indispensable for scientists and engineers.
No scientifically literate person would deny the utility of mathematics in science. Open any scientific literature and you cannot get very far without encountering mathematical notations or two, and no respectable university curriculum of science and engineering discipline is without a calculus sequence. A sensible, intelligent response for utility of mathematics is a definitive YES.
But why do I, as a student studying mathematics, sense an equally strong albeit subtle hushed motion against the utility of mathematics? An instance; I am in constant pressure to justify my studying mathematics in college- the immediate reaction when I disclose my major is not of approval but of amusement or puzzlement. Another - generally, mathematics is appreciated by only when its applications are revealed. All those abstract theorems that require an ideal agent who can draw perfectly straight line on an infinitely long blackboard infinitely many times is much less discussed and appreciated. Very few people poke around those abstruse mathematical concepts. Even less finds it worthwhile.
Why the discrepancy?
I think it’s because the child who had dared ask THE question has a point. Chances are that he will not have to invoke Green’s Theorem, or any other theorem he had to memorize for his calculus course, for any of his professional endeavors. Perhaps the faculty of logical tools he unconsciously acquired by studying mathematics may come in handy, but Calculus 101 certainly does not deserve all the credit for a person’s ability to think logically if it is so, mathematicians should be the most reasonable, logically empowered persons in the world, which is an ambitious generalization that not all people will agree). The point is this. Even if we understands, and perhaps agrees, with what that inquisitive child is asking, we have learned to simply dismiss the child’s inquiries. The question cries heresy. When someone rebukes, “HOW DARE YOU, BENEFICIARY OF ALL THE MODERN SCIENTIFIC ADVANCES, ASK THAT QUESTION?” we don’t have a good comeback. Enough people had commented on the child’s ignorance, we had seen few examples of math being useful, and we have accepted the usefulness of math as a fact. I think that inquisitive child who had questioned the usefulness of mathematics is still within many of us. We have simply fed it with adequate reasons to keep his mouth shut.
I dare propose the alternative - that it is quite okay for us to question the utility of mathematics.
In fact, the question itself is a source of tremendous amount of humor in the mathematical community. In my analysis class one day, upon a nice little proof about fields that stated that the identity element of addition, 0, is less than the identity element of multiplication, 1, our professor remarked, “And now we have proved that 1 > 0. call your mother tonight and tell her that your tuition is being well spent.” He often made comments how we should not complain about writing long proofs, for it took Bertrand Russell, a famous mathematician, took painstaking 379 pages to prove that 1+1 = 2 in his magnum opus Principles of Mathematics. Stupendous effort for something a five year old knows to be true.
I don’t claim that mathematics is not useful. Beyond its excellent use in counting coins, mathematics functions as the language of the cosmos, an elegant translator of the subjective perception of reality to objective, tractable pieces that scientists and engineers digest and manipulate. A swing of a pendulum can fully be characterized by a parabola; the essence of waves is captured through the mathematical lens; the intricate web of social networks, a purely social phenomenon, are dissected and illuminated under the lights of mathematics. Mathematics strikes the chords of the rational universe that our rational mind can muse upon - it breaks down the subjective reality into a digestible pieces that our minds, with training, can grasp. It reveals patterns, describes patterns, and predicts the future of patterns. It’s difficult to think of a world without mathematics. In fact, we probably cannot pluck away mathematical intuitions and still make sense of the world.
But here is what I do propose: that mathematics is shockingly useful. As much as the society advanced from the milestones of mathematical discoveries, the most forcible drive for mathematical achievements has rarely been to meet the society’s expectation of its utility. Mathematicians labors not to reap the fruit his work through its application but for the fruition of mathematical truths.
The nature of work for mathematician, as I perceive it, is this. Given a system of axioms (truths that needs no explanation) and of logic, mathematicians project a cogent universe and, like an inquisitive child, roam freely and joyfully, investigating the properties and rules that underlie that universe. Observations are made, conjectures are proposed, theorems are proved. Exclamations sound from various landscapes, connection between different landscapes are made and pondered upon with great enthusiasms, and brave pioneers conducts courageous and intelligent study of unknown landscapes. Then, applied mathematicians, or physicists, or any scientists may come to compare mathematicians’ findings with our own universe. The greatest success of mathematics has been attributed to this phase: it turns out those mathematical truths beautifully dovetails with the workings of our own universe.
Inspirations for mathematics have been our own universe, but there are countless instances the mathematical discoveries superseded its reflection in the natural world. Quantum mechanics, which is seriously out of tune with our intuitions, has been made sense through mathematics. Évariste Galois studies in symmetry in the 19th century played a vital role in proposition the Standard Model of particle physics in the 20th century. Mathematics did not anticipate these applications; it existed, independently, from these applications. And almost eerily, they turned out to be the perfect language to describe our universe.
Mathematics is an art form and does not pride itself on its utility. By wonderful and baffling coincidence, it served as an incredibly useful tool for pragmatic endeavors, but mathematics would have remained in value even if we had found no use for it. If anything, it’s fun and interesting. It is a worthwhile way to spend one’s day. For some of us anyway.
That heretical child who dared to ask the question roams freely in my mind today. Will I ever use that the edge length of Koche curve, after n interactions, tends to infinity as n approaches infinity? Or that the sequence (n+1)/n is a Cauchy sequence? Maybe, but most likely not. But it was certainly fun trying to prove that they do. And worthwhile intellectual exercise at that.
