This is a short post on something that I get asked a lot: What is “math research”?

To my understanding, there are two kinds of math research, which I’ll refer to as “applied” math research and “pure” math research.* There’s lots of overlap between these two things, but for clarity’s sake, I’m going to talk about them as if they’re really different.

“Applied” Research

In applied math research, you use math to solve problems outside of math. This is the kind of research that’s familiar to most people: using math to answer questions in biology, finance, physics, computer science, and so on. To do applied math research, you often have to figure out how to connect existing math to your problem from one of those fields.

“Pure” Research

In pure math research, you’re actually developing the math itself. This is the one that’s hard to explain.

It’s hard to explain because it’s not compatible with the way most of my peers have learned math. I’ll quote what I’ve said about this before:

I think most people learn math backwards. For most of my life, I had the impression that math is a neat package of rules. The teacher would say something like “The square root of 2 is irrational,” “,” or “Dividing by zero is undefined.” Then I would memorize the rule and apply it to a set of problems.

In this framework, there’s no place to develop new math. Math is done. It’s an ancient set of instructions, and math class is about understanding, applying, and following those instructions.

This is a widespread but wrong interpretation of what math is. Even without delving into research – say, by taking an upper-level math class at Queens College – it’s clear that math is something else.

I’ve tried to define “what math is” in previous posts, but I’m not really content with any of those answers anyway. I’ll say this, though: when I do math, I’m exploring concepts. And when I do math research, I’m trying to push those concepts a bit farther, to prove something about math that no one’s proved yet.

In a way, it’s sort of like arguing a thesis in an essay. You claim that “Gatsby represents the American Dream,” and then try to draw from existing concepts (like a definition of the American Dream, a line that Gatsby said to Daisy, and a statement from a respected F. Scott Fitzgerald scholar), pull them together with logic, and make your case.

The “stuff” of math is completely different from the “stuff” of literature, and so the style of argument is different as well. Most notably, math concepts are clearly defined, and it’s possible to prove things – not just to make a convincing case, but to actually present an airtight logical proof. Mathematicians don’t just agree that there are probably infinitely many primes, or that it’s extremely likely that there are infinitely many primes. Euclid (and others) actually proved it.

Pure math research is usually about proving things.

It’s sometimes a frustrating process: it’s one thing to notice a pattern, but proving it is something else entirely. To prove things, you have to engage with conceptual, logical structures, and try to grasp how they naturally fit together. It’s a lot more abstract than dealing with coffee tables or test tubes.

That’s also what’s so rewarding about it. When I get to really “see” the structures behind the symbols on the page, it’s incredibly beautiful.

That’s why I do math, anyway.

By Kieff – Own work, Public Domain, Link

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*Note: I’m borrowing the terms “pure” and “applied” a bit incorrectly here. Often, in what people call “applied” math research, you still have to develop new math to help answer the question. And often, “pure” questions do have motivation from, or application to, another field. There’s no clear dichotomy between the two. To my understanding, “pure” and “applied” math are more like two ingredients of a smoothie than two completely different flavors of ice cream.