*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 ideas.

It’s about the *othe*r 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?”

It’s about digging deeper.

**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 19^{th} 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 20^{th} 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.

**Q: Did they ever build it?**

A: Yeah. The guy’s name was Alan Turing. Today we call those machines “computers.”

**Q: *stares blankly, jaw slowly unhinging***

A: Exactly.

This enormous project to prove everything—one of the purest mathematical enterprises ever undertaken—didn’t just end with a feeble flicker and a puff of smoke. Far from it.

Sure, it didn’t accomplish its stated goals. But by clarifying (and, at times, revolutionizing) ideas like “proof,” “truth,” and “information,” it did something even better.

It gave us the computer, which in turn gave us… well… the world we know.

**Q: So the pure mathematics being done today might, someday, give us a new application as transformative as the computer?**

Maybe.

But you shouldn’t hold any specific piece of work to that standard. It won’t meet it. Paper by paper, much of the pure math written this century will never see daylight. It’ll never get “applied” in any meaningful sense. It’ll be read by a few experts in the relevant subfield, then fade into the background.

That’s life.

But take any random paper written by an early 20^{th}-century logician, and you could call it similarly pointless. If you eliminated that paper from the timeline, the Jenga tower of our intellectual history would remain perfectly upright. That doesn’t make those papers worthless, because research isn’t a collection of separable monologues.

It’s a dialogue.

Every piece of research builds on what came before, and nudges its readers to imagine what might come next. Those nudges could prove hugely valuable. Or a little valuable. Or not valuable at all. It’s impossible to say in advance.

In this decades-long conversation, no particular phrase or sentence is necessarily urgent. Much will be forgotten, or drift into obscurity. And that’s all right. What’s vital is that the conversation keeps on flowing. People need to continue sharing ideas that excite them, even—or perhaps *especially*—if they can’t quite explain why.

**Q: So, pure maths… come for the pretty patterns, stay for the revolutionary insights?**

A: That about covers it.

Reblogged this on myspacesourav and commented:

This piece covers it up all…..the need to think, the need to keep on researching when we failed. Thank you for this beautiful post.

why failed mtc?

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lovely piece

It quite clear

Hello. I have been engaged in considering the practical applications of pure math philosophy, for lack of a better term. You blew my mind when you identified the invention of the computer as a byproduct of conceptualizing a system to organize true, false, and indecipherable. This is intrinsically interesting to me, because it seems like a practical benefit to the human race, which implies that if we continue to explore the manipulation of those 3 things, we will continue to advance. It seems to me that this is the edge of human understanding, so I wonder if there were any pursuit more worthwhile. However, my only critique on the position of mathematical philosophy is the idea that mathematical representation is an imposed boundary, which is to say, the idea of indecipherability becomes invalid. Curious what your position on this idea might be.

the lover math must the read a article (y)

Pure Maths, to me is like Drawing or sketching. It is not the amount lines in a maths drawing that matters.It is not even whether most people actually understand the drawing. It is quite simply that someone eventually sees it and gets it… they understand the intention of the mathematics just like the artist and they run with it – they too doodle the next piece of the jigsaw as it were and on we go …..communicating what we think we know to someone. Perhaps someone we’ve never met ….perhaps years later after we have died even – like a cave drawing reaching out to the present understandings and speculations across space and time?

Music, I used to be a professional cellist, might accent and underscore the action portrayed video devices (movie scores, newscast fanfares et al), might mask the human anxieties in a waiting room or conversations in a restaurant, might pander to a customers’ yearning hearts or restless feet… But deep in the recesses of the soul lies a language without words. Some perceive it clearly in terms of sound: woven tapestries of pitch, timbre, rhythm, melody. Pure purposeless music…..

I broke a hand, stopped playing music (applied or pure) and knew immediately that I would study mathematics because that language of the soul can be spoken through the elements of mathematics, too. Physics is just the tip of a very large iceberg.

Nice as usual, thank you! A small correction: Kurt Goedel was a (mathematical) logician, not a philosopher.

Small correction to AEC’s comment: Gödel was a logician, mathematician and philosopher. https://en.wikipedia.org/wiki/Kurt_G%C3%B6del.

Thank you very much for this

It was so helpful

Everything else I read was confusing my bearings!!!

Reblogged this on James Revels Composer.

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I really enjoyed your article but the following assertion is incorrect.

“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.”

Gödel’s theorem only proved this for formal systems capable of arithmetic on the natural numbers.

https://en.wikipedia.org/wiki/G%C3%B6del's_incompleteness_theorems#Systems_which_contain_arithmetic

The Wikipedia entry you provided states,

‘ In 1931 and while still in Vienna, Gödel published his incompleteness theorems in Über formal unentscheidbare Sätze der “Principia Mathematica” und verwandter Systeme (called in English “On Formally Undecidable Propositions of “Principia Mathematica” and Related Systems”). In that article, he proved for any computable axiomatic system that is powerful enough to describe the arithmetic of the natural numbers (e.g., the Peano axioms or Zermelo–Fraenkel set theory with the axiom of choice), that:

1. If the system is consistent, it cannot be complete.

2. The consistency of the axioms cannot be proved within the system. ‘

I mention this because Gödel’s work has been wrongly portrayed by many. For example, in New Age and ‘alt med’ circles it has been used to avoid and undermine legitimate skepticism of beliefs, assertions, and practices in those circles. It won’t be long before the ‘alt-fact’ crowd starts doing the same.

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Great fun reading this at 4:36 AM. Thanks a lot, a really enjoyable piece of writing and I look forward to reading more in the future.

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Pretty stupid to say that pure math is useless. Wow. Good job OP.

This is a good enough explanation of pure math, but a terrible outlook that I wish you had not written.

well explained