# In Our Time: Fermat's Last Theorem

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12:50, Friday, 26 October 2012

Editor's note: In Thursday's programme Melvyn Bragg and his guests discussed Fermat's Last Theorem. As always the programme is available to listen to online or to download and keep - AI

Hello

No sooner had we finished what I thought was a brilliantly clear explanation by the three academics of Fermat's theorem than Marcus du Sautoy said, "the Travelling Salesman Problem is the next one you should tackle".

It sounded like something out of Thomas Hardy, but I still had enough energy to say "what's that?" Marcus explained that the problem was how to find a mathematical formula which would enable you to visit a number of cities in the shortest possible time. This isn't too difficult to do when few cities are involved, but when an enormous number of cities are involved it becomes extremely difficult indeed to find the shortest way. Working it out would also solve another mathematical mystery, the N versus NP problem. The benefits that this could bring to commerce and computing would be enormous, not to mention allowing football supporters (Marcus is, like myself, a disappointed Arsenal fan this morning - we were all over the place, the House of Lords was stuffed with them) to calculate precisely where we'd end up in the Premier League.

No, I mustn't trivialise it. The Travelling Salesman Problem will literally - Marcus again - "change the world". I've spent long enough now with what seem like abstruse or trivial mathematical problems not to believe that the consequences can be colossal for the future workings of the world. So watch out for the Travelling Salesman in a year or two on In Our Time.

Yesterday I tried to wind up for this, for me, difficult feat of Fermat. I went on Hampstead Heath which I like to do as many mornings as I can. It was as if it were painted in mist. Which made it, of course, most mysterious (sorry about that, but I do feel rather euphoric after Fermat). Dog walkers came across the hills with their splay of eight or nine dogs. People seemed to drift along the short horizons to unknown destinations. The further I walked the more I came to the wooded parts of the Heath which are now carpeted in yellow gold, with leaves happily rotting into the earth. When I walked on the streets of London later, the same leaves will have been blown into machines or blown off the top of ponds, or nature being destroyed! Or at least, not to be too dramatic, removed from view.

I saw a sign in Soho the other day which said: 'Waxing, £30, body and half leg'. Which leg? Which half? It really did say that!

Last week I did a very long newsletter which taxed Ingrid a bit too much, so this time I will say little except that the Angel Awards earlier this week, founded by Andrew Lloyd Webber, brought to the Cambridge Theatre in London sixteen groups which had restored utterly derelict churches, or factories, or industrial buildings, or barns, or canals - an amazing array of places in this country brought back to function and to splendour by armies and armies of volunteers.

The judges had a tough job. But in the end the four winners were: a derelict railway station in Tynemouth which is being refurbished and rebuilt to extraordinary standards, as high as you would expect from the engineering skills in the North East; a village church - congregation: 6 - which had restored a beautiful medieval tower and re-thatched the church roof and set in motion a thatching family; Droitwich Canals, at which one man had laboured voluntarily for more than fifty years and hundreds, even thousands, had helped him over that time; and the people who had withstood the flood in Cockermouth, and set their minds not only to restoring the town but to making it even better than it was before. This is the One Nation. This is the Big Society. People just getting on with it and not waiting to be told.

I can feel that I have to stop now.

Best wishes

Melvyn Bragg

• #### Comment number 1.

Not about Fermat's at all I'm afraid, but a chance remark by Evan Davies, on the Today programme this morning, after he chaired a brief debate between a physicist and a philosopher on the problems of detecting a simulated universe, that this is one for 'In Our Time'.

Isn't it time for a head to head between these two camps?

Stephen Hawkins confidently declares that philosophy is dead, (The Grand Design), but yet Quantum Theory has Bohr declaring the Copenhagen Interpretation, that the observer is part of the process of defining what is being observed.

Brian Cox (The Quantum Universe) quotes Born as saying that 'The motion of particles follows probability laws but probability itself propagates according to the law of causality.'

Probability was an obsession of Popper's!

Time, surely, to draw these strands together in a debate for an interested layman?

• #### Comment number 2.

I think this was a good programme but at times got bogged down in the historical perspective as it detailed the progressive attempts by different mathematicians to move the theorem on.We get a litany of ‘one of the greatest mathematicians’ type descriptions until we get to the 20th century,where Fermat’s theorem truly takes off.The solutions to FLT over the centuries have been a series of mistakes that have contributed to overall knowledge of FLT. "Fermat could not have this proven, it's a 20th century proof, there is no way this could have been proven before the 20th century".Andrew Wiles found the intricate way of explaining something that Fermat could explain in a very simple way?WasFermat bluffing?Hiding his theorem under a bushel?Was his proof more direct than Wiles’s? I can see how the Wiles’ proof is indirect with the incorporation of a number of branches of mathematics. It's either right or wrong. What is less well known is that there are many routes though any given problem some using elegant 'simple' maths and others using indirect complicated ideas drawing on many different branches of the subject.Was Fermat a joker,was the generosity of narrow margins a gift?

• #### Comment number 3.

Before Wiles there had been proofs demonstrating the validity of the theorem for certain powers, but no generalization for all powers. Wiles and crew changed all that.. math may be comparable to philosophy which also totally depends on logical evidence. There's a whole mathematical discipline known as statistics which is often linked to probability theory, but, to oversimplify, does not concern itself in general with their accuracy, but rather with their patterns, i.e., drawing general conclusions about them. Also, mathematics (as opposed to arithmetic) does not deal with numbers, but rather concepts. This is especially true of the mathematical discipline known as number theory.I think the TV documentary Fermat’s Last Theorem by Singh was a better medium to convey the human story about what is admittedly a pretty hard subject. You're not hit with too many equations here, rather, one gets a sense of the monumental struggle involved, the tenacity of Wiles, the flashes of brilliance and the beauty of higher ideas.It especially depicts the maths visually,Wiles’ proof of the Tamymura-Shimura conjecture and hence his solution of FLT, his conjecture as to sums greater than squares.The equivalence between elliptic curves and modular forms proved by Wylie made the Tamymura-Shimura fact.”I told you so” as Shimura said when hearing of the proof.

• #### Comment number 4.

Several years ago when I first heard one of your contributors begin her answer with the word "So, " it grated most irritatingly. Now that every contributor follows suit, I'm soon intollerably agitated and cannot attend to their message at all.

Do such rotters get PAID eg. their bus fare to the studio? They should be warned it will be halved for each sentence in which they fail their schooling in this way.

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