This is one of my least-favourite questions about maths. It’s a question on the level of “Is Mo Farah the world’s greatest athlete ever?” - to some people, the answer is clearly yes, and to others, just as clearly no (Bubka? Beamon? Lewis? Gebrselassie ((Nice joke on the radio the other day: in a recent survey of former Ethiopian athletes, 65% were found to be highly Gebrselassie…)) ?) and there’s no real way to prove it either way.

The case for discovery

The discovery argument rests on maths ‘just being there’, like America or buried treasure. You need a degree of ingenuity and persistence to discover a mathematical truth - you start from a place you know, follow paths others have forged before you, possibly leaving maps behind them, and then you machete through some mathematical undergrowth to find a sparkling New Thing that wasn’t there before. If you’re lucky, you might even get to put your name on it, although Stigler’s Law suggests some chump with a better agent will probably get the credit.

It involves wrong turns, it involves fighting off mythical beasts (the deadly Deadline, and the Distraction Bird of Doom), and you’re never quite sure if you’re going to make it out of the search alive. (In other news, I think universities should provide mental health support for all PhD students as a matter of course, but that’s a different story).

Asimov said, “the most exciting phrase to hear in science… is not ‘Eureka!’, but ‘That’s funny…’”. That’s where finding things out mathematically comes from: you notice something odd, you try a few examples, and you notice that something seems to work. Then you start to question why.

The case for invention

Inventing something is a different affair altogether: it involves putting things together to create something new. It involves finding the right piece to slot into the right hole to make the whole contraption work.

A mathematical proof works very much that way: you know where you want to get to, and - once you have a bit of experience - you start to make a sketch of what the proof ought to look like. You highlight possible problems. You make a note of what ought to be straightforward. Then you try stuff: you find that the technique you liked the look of doesn’t work, so you fill in the gaps a different way.

The question is invalid, starting as it does from licensed premises

My feeling is - in a typically wishy-washy way that you’d be more likely to associate with an economist than a mathematician - that mathematics involves both discovery and invention skills. At least in the kind of maths I understand (three dimensions and the truth!), discovery accounts for noticing that something is true and the initial testing of ‘does that work in this case?’ and ‘can I find a counter-example?’ After that point, proving carefully that the thing you’ve discovered is true is a process of invention.

The Mathematical Ninja thinks mathematics is largely a process of sculpture, though, and I don’t want to argue with them.

* Edited 2015-03-23 to fix footnote. * Edited 2021-05-17 to correct the Mathematical Ninja’s gender.