Why Mathematics?

Everywhere around me, I see intelligent science-minded people going off to study engineering, medicine, and computer science. Many of them are very skilled at mathematics; yet, when I ask them if they had considered studying mathematics, the answer is almost universally no. I find this to be quite a peculiar phenomenon given the relevance of mathematics in these fields. It seems like “STEM” is missing the “M”, as it were.

Humans have been doing mathematics for thousands of years. In fact, mathematics is perhaps the only subject where results derived thousands of years ago are not only still relevant, but crucial to our modern understanding of the world. In this sense, there is a timeless quality to mathematics. We still use Euclid’s axioms, the Pythagorean theorem, and Archimedes’ formula for the volume of a sphere. Al-Khwarizmi’s method of completing the square is taught in every high school classroom. We use these concepts not for their historical interest, but because they are true!

We don’t know the rules of the universe; that’s what we’re trying to figure out when we do science. So as we experiment and inevitably make mistakes, we are constantly revising our model of the rules. The fundamental difference between mathematics and the other sciences is that, in mathematics, we choose the rules and then see what happens. This means that everything is internally consistent as long as we use the same axioms today that we did two thousand years ago. For this reason, Professor Robert Gunning of Princeton University has called mathematics “perhaps the only truly cumulative human activity” because “nothing once proved is really lost”—we simply build upon what already exists. This may not seem especially unusual—until one considers that we could never say the same thing about ancient notions of astronomy, medicine, or any other branch of science. (I certainly hope you’re not relying on any ancient medical advice.)

This also illuminates why mathematics is so useful as a tool in the other sciences. When we develop a tentative model for the rules of the universe—for example, a new physics theory—we can develop this set of rules into a coherent mathematical framework, determine its implications, and then compare these predictions against what actually happens. In other words, mathematics becomes the tool we use to rigorously apply the scientific method.

Then again, we can also come up with arbitrary rules just to see what happens. This is essentially the difference between “pure” and “applied” mathematics: are our axioms designed to model some kind of real problem? This is not a permanent distinction, because often pure mathematics becomes applied once we find a suitable situation where we can use it. For example, Euler’s theorem (in number theory) was “pure” math for almost two hundred years before we discovered that it could be used as the basis for digital encryption; it is now a tool used by many applied mathematicians (and every person with a web browser).

So, why then should I—or anyone else—choose to study mathematics? I believe that the most compelling argument lies in its beauty.

The idea of beauty in mathematics is quite tellingly characterised by the acclaimed photographer Mariana Cook:

I have photographed many people: artists, writers, and scientists, among others. In speaking about their work, mathematicians use the words “elegance,” “truth,” and “beauty” more than everyone else combined.

This may seem confusing to many because the typical perception of mathematics is that of numbers and equations—where is the beauty in that? But this misses the point of higher mathematics. Saying that math majors only learn to solve increasingly difficult equations is like saying that English majors only learn to write increasingly long sentences. Broadly, we might say that mathematics is about the manipulation of abstract structures, and numbers and equations are just some of those structures. All fields of mathematics share an emphasis on proof, a form of logical reasoning that guarantees the truth of a conclusion based on given premises.¹ The abstract structures of mathematics and the things we prove about them are beautiful because they hint at objective truths, at least insofar as they convey a relationship between whatever assumptions we choose and the consequences that necessarily follow. Proofs themselves can also be beautiful, if difficult results are obtained in relatively few steps using an elegant argument.² At the heart of all these notions of beauty is abstraction, a technique which mathematicians employ to discover the common grain of truth between disparate ideas. The result? A collection of theories which encapsulate a startling variety of phenomena in our universe.

I make the case for mathematics not to convince you specifically to study it, but to highlight the differences between the public perception of advanced mathematics and the reality of it. By doing so, perhaps I will pique the interest of someone who did not think mathematics was their cup of tea. You do not need to multiply ten-digit numbers in your head to study mathematics.³ What you really need is an unstoppable curiosity, a love of abstraction, and an appreciation of mathematical beauty. And that, I think I can handle.