Fermat’s Last Theorem & the Four Colour Map Problem are two seemingly simple but for a long time intractable problems in mathematics. Proofs were finally found for both of them in the last quarter of the C20th.
In 1976 Kenneth Appel and Wolfgang Haken used a computer to prove that four colours are indeed sufficient to colour any map, for example an atlas of the countries of Europe, so that no two countries sharing a border have the same colour.
By the end of 1994 Andrew Wiles had finally proved Fermat’s Last Theorem.
My problem with both of these is that both proofs are so complicated that only real mathematicians can understand them – to an ignoramus like me they lack the true elegance of simplicity.
And Wiles’ proof certainly cannot be the one which Fermat described as ‘a truly marvellous demonstration of this proposition that this margin is too narrow to contain’, though to be fair even Fermat probably came to realise that he was wrong about that.
One day I expect someone will work out a way of explaining to us laymen how these proofs work. It is natural to be curious about the world & we would really like to know how & why we need only four colours for a map, or how we know that you cannot have integer solutions to a simple formula such as A4 = B4 +C4
In 1993, the UK Science Minister William Waldegrave challenged physicists to produce an answer that would fit on one page to the question 'What is the Higgs boson, and why do we want to find it?' There was quite an indignant reaction from some quarters – how can we explain something so complicated in such a short space, and to people who are not even scientists.
Well as government ministers & civil servants knew well, most prime ministers (at least from Winston Churchill to Margaret Thatcher) usually demanded that a submission be set out on no more than one side of paper. There might in some cases be copious briefing material attached, but someone who has to consider as many issues in a day as a prime minister needs something succinct to tell them what it is all about & if they really need to bother to go into it any more deeply.
William Waldegrave had every need to understand something about Higgs boson at the time because he was being asked to sign the cheques for the UK contribution to CERN. The physicists needed to rise to the challenge out of self-interest if not courtesy.
And after all, as Richard Feynman said, ‘If you really understand something you can explain it to anyone.’
My favourite explanation was that provided by David Miller of University College London, who used the example of the effect that the arrival of Mrs Thatcher would have on a roomful of party workers.
Until then, not only did I not know what Higg’s boson* is, I actually thought it was called Higg’s bosun, after that fine Old English seafarer the bosun, an officer in a ship who has charge of the sails, rigging, etc., and whose duty it is to summon the men to their duties with a whistle.
Which, when you think about it, is not all that far from an appropriate metaphor.
*It was actually christened by Paul Dirac after Satyendranath Bose who first studied the new statistics for particles for which only symmetrical states occur in nature bosons.
