Saturday, 11 April 2015


Ritual Random Rabbit Refuge . . . Refmat? Well not quite. But today we're on Fermat and his Last Theorem. Why?  Beacuse 11th April 1953 is the birthday of my near contemporary Andrew Wiles.  Even my Grade 'D' in Pass (i.e. not Honours)  Maths students at the Institute know about Pythagoras' Theorem and the square of the hypotenuse: a2 + b2 = c2 especially such integer solutions as 32 + 42 = 52 which was known to the Egyptians and used to square off the base of their pyramids. Pierre de Fermat (1601-1665) was one of the greatest mathematicians of his day and did solid work on probability and number theory but he is known, even widely outside the world of mathematics, for a throw away line where he broke the two dimensional bounds of Pythagoras and moved on to larger exponents. He tried to find solutions to formulae like an + bn = cn where the exponent n is greater than 2 and a, b and c are greater than 1 and came up with none. Then, with a rush of blood to the head, he proved to himself that there were NO solutions to those equations and wrote a note-to-self in the margin of his copy of Diophantus' Arithmetica: "Cubum autem in duos cubos, aut quadrato-quadratum in duos quadrato-quadratos, et generaliter nullam in infinitum ultra quadratum potestatem in duos eiusdem nominis fas est dividere cuius rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet" Which roughly translates as "It is impossible for a cube to be the sum of two cubes, or a fourth power to be the sum of two fourth powers.  I have discovered a marvellous proof of this idea but the margin here is not big enough to contain it."  Hanc marginis exiguitas non caperet was a gauntlet thrown down to mathematicians for the next 350 years in its implication that, if he had a sheet of clean foolscap paper, Fermat would show everyone the answer to the conundrum.  You could also try to disprove Fermat's proof by finding a single counter-example and many a schoolchild doodled away on that sort of problem - to no avail. Lots of mathematicians rose to the challenge and came up with partial proofs. For example, Sophie Germain (1776-1831) proved that Fermat's Conjecture was true where the exponent was any odd prime less than 100 and others built on that for any odd prime of whatever size. Many a marvellous proofs and intriguing mathematical relationship was revealed along the way.  Many folk were convinced that a wrong-wrong-nearly-right solution declared in the 19thC was what Fermat had discovered but nobody came up with a proof than convinced the mathematical community . . .

. . . until 1993 when British mathematician Andrew Wiles [R with assertion] published his solution to the conundrum. It is too complex to describe here: Wiles' proof runs to 150 pages of closely reasoned arguments, dense with Greek letters and weird symbols but it hinges on making a connexion between problems in geometry and problems in number theory.  The way I visual this is that mathematics had built up two teetering towers of proof upon proof extending from something really solid like Euclid's statement that there is a line, infinitely thin, that connects two points . . .  Mathematicians had followed their noses to show that IF that is true THEN this is also true. They did this unconstrained by any sense of utility; it was just beautiful.  Along the way since 1637, those mathematical ideas have put a man on the moon, put Lara Croft into the pants heart of teenage boys, and developed heart pace-makers. Governments across the world try to fund science that will do good things for the economy or society.  If the pay-off is within the lifetime of that government, then so much the better.  They have no desire to fund ideas that may, in 350 years, bear fruit.  But they are wrong - as many useful ideas are generated from pure science as from applied science - nobody has any idea where the next great leap forward will develop.

What Wiles achieved was finding a one-to-one mapping that completed the circle between the tentacles of two towers of proof that had gone off from the baseline in completely different directions.  The two unstable chains of connected ideas became stronger because of this link and have doubtless served as the jumping off point for many other wonderful mathematical advances. Shortly after Wiles declared his hand, it was shown that there was a glitch in his proof, over which he tore his hair for another 450 days until he was struck by an insight "so indescribably beautiful... so simple and so elegant" that he knew he'd cracked it - and the mathematical community agreed. Another thing that is important in this story is that Wiles had his field of expertise where, after years of cogitation on a series of related ideas, he knew a lot - possibly all there was to know about about that area of mathematics.  One day, he went to a seminar presented by a visiting speaker from another area of mathematics entirely - well outside of Wiles' comfort zone.  At that lecture Wiles had the revelation that he could fulfill his boyhood ambition and solve Fermat's Last Theorem. We should all go to any and all opportunities to listen to new ideas from visiting speakers.

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