If all the maths you’d ever seen was at school, then you’d be forgiven for thinking numbers were boring things that only a cold calculating robot could truly love. But, there is a *mathematical proof* that you’d be wrong: Gödel’s incompleteness theorem. It comes from a weird part of maths history which ended with a guy called Kurt Gödel proving that to do maths, you *have* to take thrill-seeking risks in a way a mindless robot never could, no matter how smart it was.

The weirdness begins with philosophers deciding to have a go at maths. Philosophers love (and envy) maths because they love *certainty*. No coincidence that Descartes, the guy you have to thank for x-y graphs, was also the genius who proved to himself that he actually existed and wasn’t just a dream (after all, who *else* would be the one worrying about being a dream?). Maths is great for worriers like him, because there’s no question of who is right and who is wrong – show a mathematician a watertight proof of your claim and they’ll stop arguing with you and go away (disclaimer: this may not to work with maths teachers…).

However, being philosophers, they eventually found a reason to worry again. After all, maths proofs can’t just start from nothing, they need some *assumptions*. If these are wrong, then the proof is no good. Most of the time, the assumptions will have proofs of their own, but as anyone who has argued with a child will know, eventually the buck has to stop somewhere. (“Why can’t I play Mario?” “Because it’s your bedtime.” “Why is it bedtime?!” “BECAUSE I SAY SO!”) Otherwise, you go on giving explanations forever.

The way this usually works for maths, is mathematicians agree on some *excruciatingly* obvious facts to leave unproved, called *axioms*. Think “1+1=2”, but then even more obvious than that (in fact, Bertrand Russell spent hundreds of pages *proving* that 1+1=2 from these stupidly basic facts!). This means that mathematicians can go about happily proving stuff from their axioms, and stop worrying. Peak boring maths.

But the philosophers *still* weren’t happy. Mostly, it was because the mathematicians massively screwed up their first go at thinking of obvious ‘facts’. How massively? The ‘facts’ they chose turned out to be nonsense. We know this because they told us things which flat-out contradicted each other. You could use them to ‘prove’ anything you like – *and the opposite at the same time*. You could ‘prove’ that God exists, *and* that He doesn’t – and no matter which one of those you think is true, we can all agree that they can’t *both* be right! In other words, the axioms the mathematicians chose were *inconsistent*.

Philosophers’ trust in maths was shattered (after all, it was *their* job to prove ridiculous stuff). Before they could trust another axiom ever again, they wanted some cast-iron proof that they weren’t going to be taken for another ride by the new axioms. But where could *this* proof start off? If we had to come up with a whole other list of axioms for it, then we’d need a proof for *them* too… This was all a bit of a headache.

The only way out the mathematicians and philosophers could see was to look for a proof that the new axioms were consistent, using only those new axioms themselves. This turned out to be very, very hard. In fact (and this is where Gödel steps in) it turned out to be *impossible*.

Cue Gödel’s incompleteness theorem. It says that any axioms that you can think of are either inconsistent – nonsense – or *aren’t good enough* to answer all of your maths questions. And, sadly, one of those questions *has* to be whether the axioms are inconsistent. In short, all good axioms are *incomplete*.

This may sound bad, but it’s really an exciting thing. It means that if you want to do maths, you really *do* have to take big risks, and be prepared to see your whole house of cards fall down in one big inconsistent pile of nonsense at any time. That takes serious nerve. It also means mathematicians have the best job security on the planet. If you *could *just write down axioms and get proof after proof out of them, like a production line, then you could easily make a mindless robot or a glorified calculator sit down and do it. But thanks to Gödel’s incompleteness theorem, we know for sure that will never happen. Maths needs a creative touch – a willingness to stick your neck out and try new axioms just to see what will happen – that no robot we can build will ever have.

*Joe Double*

[…] Maths proves that maths isn’t boring […]

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