Maths, but not as you know it…
Another fun-filled hour of your favourite two things – maths and rock music – courtesy of Tom Rocks Maths on Oxide Radio. This week I’m joined by two of my students from Teddy Hall, Fran and Tom, who will be explaining their favourite mathematical topics, taking part in a bumper numbers quiz, and sharing some of their music tastes. Plus, the usual dose of Funbers, and excellent music from Panic at the Disco, Sum 41 and Muse.
With thanks to Alice Taylor for production assistance.
A short review of intern Joe Double’s work with Tom Rocks Maths over the summer of 2018. Written for the OUS East Kent branch who provided funding for the project.
‘First of all, I must thank you again for the grant, and for the warmth and friendliness at your event; it was an absolute delight to give my presentation and talk to your members, as it has been interacting with you in general.
I had the opportunity to work with one of my tutors over the summer to produce pieces for a general audience about complex mathematical topics. Without the help of the OUS East Kent group, I couldn’t have taken up this opportunity – with their grant’s help, I was able to afford to live in Oxford through a large part of the summer, allowing me to work in close contact with my tutor and use his studio for creating the videos and audio pieces I worked on. The OUSEK grant can be put to use far more flexibly than those from bigger schemes (which always have preconditions to meet about how the project will apply to industry, say), so I couldn’t recommend applying more if you have an idea for a project for your time at Oxford which is on the unusual side!’
Pieces I produced during the project:
Why do Bees Build Hexagons? Honeycomb Conjecture explained by Thomas Hales
A video I edited of Tom (my tutor) interviewing Thomas Hales about the mathematics behind beehives.
Would Alien (Non-Euclidean) Geometry Break Our Brains?
My main video, written, filmed and edited by me, about demystifying non-Euclidean geometry.
My main audio piece, where I interview Professor Adrian Moore (also of St Hugh’s) about what philosophy can tell us about how aliens might do maths.
Maths proves that maths isn’t boring
An article about Gödel’s incompleteness theorems, and how they show maths is always risky.
An audio piece I edited about a tattoo Tom got of the Platonic solids.
Alien maths – we’re counting on it
An article about how we use the mathematics of prime numbers to send messages to the stars.
An article about a game theory paper which could amongst other things help stop deforestation.
The original article was published on the OUS East Kent website here.
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
