# Applied Mathematics vs. Pure Mathematics

June 28, 2016

Or, the Many Uses of Uselessness

One of the joys of being married to a pure mathematician—other than finding coffee-stained notebooks full of integrals lying around the flat—is hearing her try to explain her job to other people.

“Are there…uh… a lot of computers involved?”

“Do you write equations? I mean, you know, long ones?”

“Do you work with really big numbers?”

No, sometimes, and no. She rarely uses a computer, traffics more with inequalities than equations, and—like most researchers in her subfield—considers any number larger than 5 to be monstrously big.

Still, she doesn’t begrudge the questions. Pure math research is a weird job, and hard to explain. (The irreplaceable Jordy Greenblatt wrote a great piece poking fun at the many misconceptions.)

So, here’s this teacher’s feeble attempt to explain the profession, on behalf of all the pure mathematicians out there.

Q: So, what is pure math?

A: Picture mathematics as a big yin-yang symbol. But instead of light vs. dark, or fire vs. water, it’s “pure” vs. “applied.”

Applied mathematicians focus on the real-world uses of mathematics. Engineering, economics, physics, finance, biology, astronomy—all these fields need quantitative techniques to answer questions and solve problems.

Pure mathematics, by contrast, is mathematics for its own sake.

Q: So if “applied” means “useful, ” doesn’t it follow that “pure” must mean…

A: Useless?

Q: You said it, not me.

A: Well, I prefer the phrase “for its own sake, ” but “useless” isn’t far off.

Pure mathematics is not about applications. It’s not about the “real world.” It’s not about creating faster web browsers, or stronger bridges, or investment banks that are less likely to shatter the world economy.

Pure math is about patterns, puzzles, and abstraction.

It’s about the other ideas that come before, behind, next to, or on top of those initial ones.

It’s about asking, “Well, if that’s true, then what else is true?”

Q: You’re telling me there are people out there, right this instant, doing mathematics that may never, ever be useful to anyone?

A: *glances over at wife working, verifies that she’s not currently watching Grey’s Anatomy*

Yup.

Q: Um… why?

A: Because it’s beautiful! They’re charting the frontiers of human knowledge. They’re no different than philosophers, artists, and researchers in other pure sciences.

Q: Sure, that’s why they’re doing pure math. But why are we paying them?

A: Ah! That’s a trickier question. Let me distract you from it with a rambling story.

In the 19th century, mathematicians became obsessed with proof. For centuries, they’d worked with ideas (like the underpinnings of calculus) that they knew were true, but they couldn’t fully explain why.

So at the dawn of the 20th century a few academics, living on the borderlands between math and philosophy, began an ambitious project: to prove everything. They wanted to put all mathematical knowledge on a firm foundation, to create a system that could—with perfect accuracy, and utter permanence—separate truth from falsehood.

This was an old idea (Euclid put all of planar geometry on a similar footing 2000 years earlier), but the scope of the project was new and monumental. Some of the world’s intellectual titans spent decades trying to explore the rigorous, hidden meanings behind statements like “1 + 1 = 2.”

Can you imagine anything more abstract? Anything more “pure”? Curiosity was their compass. Applications could not have been further from their minds.

Q: So? What happened?

A: The project failed.

Eventually, the philosopher Kurt Gödel proved that no matter what axioms you choose to start with, any system will eventually run into statements that can’t be proven either way. You can’t prove them true. You can’t prove them false. They just… are.

We call these statements “undecidable.” The fact is, many things can be proven, but some things never can.

Q: Ugh! So it was just a massive waste of time! Pure maths is the worst.

A: Oh, I suppose you’re right.

Of course, the researchers tried to salvage something from the wreckage. Building on all this work, one British mathematician envisioned a machine that could help us decide which mathematical statements are true, false, or undecidable. It would be an automatic truth-determiner.