Why is math useful?

One of the most frequent questions many (if not all) mathematicians get at some point during their career is, "Why is this useful?" I kind of both love and hate this question; hate, because it misses the point about why I study math, but love, because it gives me license to be an unapologetic geek for at least the next five minutes. Now, "this" can refer to just about anything in math. For instance, it can be whatever the mathematician in question is researching, or it can be some beautiful mathematical result, or it can be something this mathematician is teaching their students. And if you're a pure mathematician, this can be a difficult question to answer.

I should probably explain what I mean by "pure mathematician." Loosely speaking, mathematics falls on a spectrum, with one end of the spectrum being "pure math" and the other end being "applied math." Applied mathematicians work on research that has an immediate application to the non-mathematical world. For example, applied mathematicians may try predicting economic trends, or creating predator-prey ecological models, or developing new encryption technologies. There's a foreseeable "end goal" that exists outside of mathematical intrigue. Pure mathematicians, on the other hand, do research in math for its own sake. For example, a theorem in pure mathematics could look like, "There are no integers a, b, c, and n with n > 2 for which a^n + b^n = c^n." If you set n = 2, you may recognize a similarity between this statement and the Pythagorean theorem. But with n > 2, this result is known as Fermat's last theorem, and was unproven for around 350 years and only proven in 1994. The mathematical community was left in shock when this was finally proven. However, many people outside of mathematics will look at that and say, "Well, I guess that's cool, but how is that used?" In this case, most mathematicians can only respond with, "How is this used?? It's freaking awesome!!"

My Ph.D. program is in pure mathematics, and my family asks me this question a lot about whatever I'm studying in a given semester. To be sure, fields like complex analysis, differentiable manifolds, measure theory, and algebraic number theory DO have significant applications in physics, economics, engineering, and computer science. But that's not what I'm studying, and it's definitely not why I'm studying it. So I'm going to do my best to answer the question: Why do we, as a human society, study pure mathematics? We can break it into two different questions:

1. Why is it useful to study pure math?

2. Why do pure mathematicians enjoy doing pure math?

First question first. There are at least two uses to studying pure mathematics: one for the individual, and one for society. The individual usefulness is the same usefulness we find in studying world literature, or history, or philosophy, or social science. Usually you don't go to the gym because you actually expect to be lifting weights in your day-to-day life; you go to the gym to become stronger and healthier. Am I going to use the lessons I learned from reading Shakespeare and Homer in high school ever in my life? No, probably not. Nor will I use the lessons I learned from my first-year undergrad religious studies course in Indian mortality rites. Or the lessons from the history class I took as a sophomore on the history of the science of gender and race. Practically speaking, I will very likely never work in an occupation where I will use the lessons I learned in any of those classes. But that doesn't mean these classes weren't useful or enriching. On the contrary, they inform the way I see the world. They have changed and refined the way I think. I can make an informed opinion when I watch the news or read an editorial, not using what I learned about Darwin or Freud, but using the "mental muscles" that those classes exercised. Studying liberal arts courses makes you a stronger member of society for having exercised your ability to reason and argue.

The same is true with mathematics. To other math instructors out there, if your students ever ask you why studying radical expressions and logarithms and trigonometry is useful, and they're not interested in going into a STEM field, you can quote me here. Mathematics is an exercise in logic. You may never use trig identities in your day-to-day life. You may never use the quadratic formula again after high school. But you can execute an algorithm. You can accomplish a specific computational task. You can use rules and results and information at your disposal to arrive at a new result. You can recognize a pattern and predict future events based on that pattern; you might even be able to prove that your pattern is true. You can make an argument in the form, "if P is true, then Q is true; if Q is true, then so is R; if R, then S. Since P is true, S must be true." The power of using logic in everyday life, in everything from politics to religion to current events, cannot possibly be overstated. You exercise these "mental muscles" by studying mathematics.

The usefulness to society is a little trickier, because it's somewhat unpredictable. Some truly incredible innovations and discoveries have come out of studying pure mathematics. Alan Turing, the father of modern computer science, was doing completely theoretical work in developing a machine that could tell true statements from false statements, but his machine was imaginary; he never invented a computer himself. His research was in logic, a very pure field. G. H. Hardy, a number theorist of the early 20th century, happily believed that none of his work would find application in any field of natural science, and in particular defense research, but we owe much of our modern encryption technology to his work in prime numbers -- including encryption and decryption techniques used by the NSA! But pure mathematics is an ongoing conversation spanning the centuries, and most results will likely fade into obscurity decades from now. No single result can be expected to have an application outside of mathematics when the result is first published; if an application exists, it will make itself apparent in the ongoing human enterprise of scientific exploration.

But if an application never makes itself apparent, the result is far from meaningless. This is the second question: why do we enjoy doing math when it doesn't have utility? I won't try to speak for all mathematicians, but for me, I think it's for the same reason i enjoy solving puzzles. Math is nothing if not a universe of puzzles, and if you've been working on one of these puzzles for several hours, or days, or even months or years if you're doing active research, when you finally begin to solve it, it feels like you're flying. I can't point to an earthly experience like it, but mathematicians live for it. And if a result like Fermat's last theorem, or the four-coloring problem, or bounded prime number gaps, gets solved, it is a glorious day for mathematicians, because humanity has just solved one of these cosmic universal puzzles.

Wait, hold on. "Cosmic universal puzzles?" I promised this blog would involve spirituality, so here it comes. These aren't like the puzzles you buy in a toy store. I can't point to where they come from. Some mathematicians think that mathematics is a purely human construction, and it doesn't "exist" outside of human thought. Personally, I disagree. Maybe we think of the questions ourselves, but the rules of logic governing how we answer them were discovered independently by ancient cultures around the world. These questions come from logic, and logic exists independently of human thought, just like different metals exist independently of human classification of iron v. lead v. silver. I believe that logic is a deeply embedded part of the universe, and mathematics is a natural consequence of logic. Of course, I also believe in God, and if I am to believe in God in any respect, I must believe that God is a creator of the universe. So it follows that these questions, these puzzles, are questions and puzzles about the universe, not about physical aspects of the universe, but how God created it.

Mathematics is, among other things, a cosmic puzzle game that we play with the universe, and to some people, a game that we play with God. You've gotta admit, there's something pretty beautiful about that.