Last time I spoke to you I explained my long term plan to retire – eventually – to a veranda and to shout theorems at passers-by. I thought, therefore, that this morning would be a good opportunity to indulge my particular academic passion. You would be forgiven, after listening carefully to months of assemblies, for thinking that this is the poetry of William Butler Yeats but, as I think is well publicised, poetry is a bit of a sideline: a lengthy exercise in academic promiscuity, and that I am, in fact a mathematician. I delight in logic, I see patterns everywhere, I can’t resist a carefully constructed diagram (geometric or statistical) and if you cut me I bleed numbers.
So, welcome to a few of my favourite things: proofs in white dresses with blue satin sashes, theorems and lemmas and conjectures with noodles. Actually that sounds quite good – I just need a tune to go with it.
That veranda though, what theorems would I be shouting? Well, probably not Fermat’s last theorem which states that if you have four positive integers a,b,c and n which obey the rule a to the n+b to the n=c to the n then n<3. i’ve="" nothing="" against="" the="" theorem="" actually-="" it’s="" quite="" pretty:="" easy="" to="" state,="" quite="" counter="" intuitive="" –="" it’s="" a="" strange="" thing="" to="" be="" true,="" right,="" there="" are="" infinitely="" many="" pythagorean="" triples="" –="" solutions="" to="" a="" squared="" +b="" squared="c" squared="" –="" and="" a+b="c" is="" even="" easier="" to="" solve,="" so="" why="" should="" there="" be="" no="" solutions="" whatsoever="" with="" higher="" powers.="" very="" odd.="" no,="" the="" reason="" i="" wouldn’t="" be="" yelling="" it="" is="" that="" i="" wouldn’t="" want="" to="" hear="" the="" proof="" yelled="" back="" –="" it’s="" a="" bit="" of="" a="">3.>
When I was a graduate student I had weekly meetings, supervisions, with Richard Taylor – one of the key players in the proof of Fermat’s Last Theorem and at my first meeting he handed me a sheaf of paper – “here,” he said, “this might help”. When I took it back to the cubby hole I called an office and looked my new treasure (a bit like Gollum, “my precious”), I found that it was a sixty seven page mathematical paper that explained how the two papers that proved Fermat’s last theorem worked. I spent a week trying to understand the first page and then went back to my supervisor to explain what I’d learned. Unfortunately I’d got it all wrong and had to start again. The proof of Fermat’s last theorem is hard.
However, I’m not one to turn down a challenge so I shall attempt to sketch it for you now. It involves a type of shape called an elliptic curve: it’s like a special kind of doughnut shape that is defined by an equation with x,y and z in it. We also need to know about a type of function called a modular form which looks a bit like a crazy set of tiles on a plane. The key idea was that for a long time people had known that there was a way of matching elliptic curves to modular forms but finding out quite how it worked and proving that it always work was hard. In the 80s someone pointed out that the Fermat equation "x to the n+y to the n=z to the n" gave you an elliptic curve which should match to a modular form and that if there was an integer solution then it would be a special point on the doughnut which would match to a special bit of the tiling. Someone else then noticed that such a special bit of tiling was impossible if n was bigger than 2 and so all they needed to do was to fix up the matching of elliptic curves to modular forms. That was what the two papers of proof did and the reason that it takes 67 pages to explain it is that both elliptic curves and modular forms are more complex and complicated than I’ve said and that the matching process is exceptionally devious. I spent three years working with the mathematicians whose favourite thing was figuring out exactly how the matching process works and I still don’t really understand it.
Fermat, by the way, claimed that he had a marvellous proof that didn’t involve elliptic curves or modular forms and was only slightly too long for a margin but Fermat claimed a lot of things including a rather marvellous theorem that all numbers of the form 2^2^n + 1 are prime. He did a bit of checking – if n=0 you get 3 which is prime, if n=1 you get 5, if n=2 you get 17, if n=3 you get 257 and if n=4 you get 65537 which he showed was prime. If n=5 you get 4,294,967,297 which is probably prime and that’s as far as Fermat’s checking got. It was a decent guess but unfortunately for Fermat 4,294,967,297 is not prime and nor are any of the numbers that come afterwards as far as have been checked so far. So I won’t be shouting that theorem either.
When I say none of the numbers that come after I mean numbers of the form 2^2^n+1 – there are, of course, primes bigger than 4,294,967,297. I say of course but Dr Hrasko will tell you that Maths is always obvious when it’s been proved (except for Fermat’s last theorem which I still hold to be entirely not obvious) and so Mathematicians only know obvious things. So how do we know this? Well, there’s a great theorem that says there are infinitely many primes and that’s a good one to shout because the proof is completely shoutable back. Suppose there was a biggest prime – let’s call it 4,294,967,297 for the sake of argument: now think about what you get if you multiply all the numbers up to 4,294,967,297 together – well you get a pretty stupidly large number, that’s what but the key thing is that you do get a number – in theory you could write it down and add one to it and that answer you get when you’ve written it down and added one to it can’t be divisible by 4,294,967,297 – we know for a fact you’d get remainder one. Similarly it can’t be divisible by 2 or 3 or 4 or 5 or any of them numbers less than 4,294,967,297 and so there are two possibilities: either nothing divides into it except itself and 1 and so it’s a prime or something else does divide into it but all of its proper factors must be bigger than 4,294,967,297 and so, since every non-prime has a prime factor (there’s another theorem I could shout if I had time) there must be a prime number bigger than 4,294,967,297 although we don’t know what it is and so there can never be a largest prime number.
I love that “we don’t know what it is” in the proof – some Mathematicians look down their noses at this kind of non-constructive proof: they like a proof where if you’re showing that something exists you at least go to the trouble of finding it but I like the uncertainty. Another of my favourite theorems to shout from the veranda has a nice bit of uncertainty to it. The first thing to say is that some numbers are rational – this does not mean that they are perfect logicians and philosophers but simply that they can be written as fractions: p over q where p and q are integers without a common factor. Some numbers, meanwhile, are irrational: the square root of 2, for example (and that fact has a lovely proof which is certainly veranda-worthy). Now, when we do one number to the power of another sometimes we get rational numbers: like if we do 2 to the power of 3 and sometimes we get irrational numbers: like if we do 2 to the power of a half (hopefully you all remember that would give you the square root of 2). Both of those examples were a rational number to the power of another rational number but we can use irrational numbers too and my question, my theorem, regards what happens when you do an irrational number to the power of another irrational number. That’s a pretty irrational way to behave and sure enough most of the time you get an irrational answer but can you ever get a rational answer? The theorem is that you can and the proof is so deliciously clever that if it were a poem it would be by WH Auden.
First – let’s think about the number root 2 to the power root 2. Don’t – unless it really amuses you – try to work that out, just let it sit there floating in space. Now there are two possibilities. Either it’s a rational number or it isn’t. If it is then we’re done – we’ve found the rational answer and we can go home so let’s suppose that it isn’t. And now for the clever bit – the part of the proof that is as cunning as a fox that’s just been made professor of cunning at Oxford University – take this answer which we’re supposing is irrational and raise it to the power of root 2. Whoah – mind blown right? Well, maybe not, but look at your answer and remember GCSE power rules. We have root 2 to the root 2 in brackets to the root 2 and when you have a power to a power you multiply the powers which means we have root 2 to the power (in brackets) root 2 times root 2 but root 2 times root 2 is 2 and root 2 to the power of 2 is just 2. So that answer we supposed was irrational to the power root 2 is the completely rational and well behaved number 2. So we know one of these paths leads to a rational answer to our hideously irrational question but we don’t know which one.
Other theorems high on my veranda list are Pythagoras’s theorem which has a collection of entertaining proofs and the generalisation of Pythagoras you know as Cosine rule (for which I only know the one). Leading on from root 2 to the root 2 to the root 2 we have the countability of the rationals – which means that there aren’t very many of them even though it seems like there are and the uncountability of the irrationals – there really are a lot of those which makes it even more amazing that you might end up with a rational number when you take one irrational to the power of another – it’s like firing a rifle at a haystack and hitting a needle except the haystack is infinitely far away and the needle is infinitely small.
Theorem and Proof – the echo of mathematics, the lub-dub, the heartbeat: the idea that we don’t really know something until we know why. In the Harris Westminster lexicon we might call this critical thought and the same idea appears in other subjects. Without the question how do we know this History is just the study of fairy stories; without it Science is just alchemy. In Language the question leads you from how French is taught in the classroom to how it’s spoken on the streets of Paris – or Kinshasa. In Literature one expects a degree of internal consistency which is why I’m still wondering about the philosophical status of orcs. One expects, I say, but maybe one has no reason to expect and maybe as, with Art, the role of Literature is to challenge and subvert our expectations but, I’d say, we should be aware of those expectations and know when they are being subverted or the point of the piece will sail blithely over our heads.
Theorem and Proof.
Lub and Dub.
What are we told and how do we know it to be true?