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If you would like permission to use material from the book (other than for review purposes), please contact Contents Cover Title Page Copyright Dedication Introduction 1 Do Not Disturb My Circles • Archimedes 2 Master of the Way • Liu Hui 3 Dixit Algorismi • Muhammad al-Khwarizmi 4 Innovator of the Infinite • Madhava of Sangamagrama 5 The Gambling Astrologer • Girolamo Cardano 6 The Last Theorem • Pierre de Fermat 7 System of the World • Isaac Newton 8 Master of Us All • Leonhard Euler 9 The Heat Operator • Joseph Fourier 10 Invisible Scaffolding • Carl Friedrich Gauss 11 Bending the Rules • Nikolai Ivanovich Lobachevsky 12 Radicals and Revolutionaries • Évariste Galois 13 Enchantress of Number • Augusta Ada King 14 The Laws of Thought • George Boole 15 Musician of the Primes • Bernhard Riemann 16 Cardinal of the Continuum • Georg Cantor 17 The First Great Lady • Sofia Kovalevskaia 18 Ideas Rose in Crowds • Henri Poincaré 19 We Must Know, We Shall Know • David Hilbert 20 Overthrowing Academic Order • Emmy Noether 21 The Formula Man • Srinivasa Ramanujan 22 Incomplete and Undecidable • Kurt Gödel 23 The Machine Stops • Alan Turing 24 Father of Fractals • Benoit Mandelbrot 25 Outside In • William Thurston About the author By the same author More advance praise for Significant Figures Mathematical People Further Reading Notes Index To John Davey, editor and friend (19 April 1945–21 April 2017) Introduction ALL BRANCHES OF SCIENCE can trace their origins far back into the mists of history, but in most subjects the history is qualified by ‘we now know this was wrong’ or ‘this was along the right lines, but today’s view is different’. For example, the Greek philosopher Aristotle thought that a trotting horse can never be entirely off the ground, which Eadweard Muybridge disproved in 1878 using a line of cameras linked to tripwires. Aristotle’s theories of motion were completely overturned by Galileo Galilei and Isaac Newton, and his theories of the mind bear no useful relation to modern neuroscience and psychology. Mathematics is different. It endures. When the ancient Babylonians worked out how to solve quadratic equations – probably around 2000 BC, although the earliest tangible evidence dates from 1500 BC – their result never became obsolete. It was correct, and they knew why. It’s still correct today. We express the result symbolically, but the reasoning is identical. There’s an unbroken line of mathematical thought that goes all the way back from tomorrow to Babylon. When Archimedes worked out the volume of a sphere, he didn’t use algebraic symbols, and he didn’t think of a specific number π as we now do. He expressed the result geometrically, in terms of proportions, as was Greek practice then. Nevertheless, his answer is instantly recognisable as being equivalent to today’s 3 πr . To be sure, a few ancient discoveries outside mathematics have been similarly long-lived. Archimedes’s Principle that an object displaces its own weight of liquid is one, and his law of the lever is another. Some parts of Greek physics and engineering live on too. But in those subjects, longevity is the exception, whereas in mathematics it’s closer to the rule. Euclid’s Elements, laying out a logical basis for geometry, still repays close examination. Its theorems remain true, and many remain useful. In mathematics, we move on, but we don’t discard our history. Before you all start to think that mathematics is burying its head in the past, I need to point out two things. One is that the perceived importance of a method or a theorem can change. Entire areas of mathematics have gone out of fashion, or become obsolete as the frontiers shifted or new techniques took over. But they’re still true, and from time to time an obsolete area has undergone a revival, usually because of a newly discovered connection with another area, a new application, or a breakthrough in methodology. The second is that as mathematicians have developed their subject, they’ve not only moved on; they’ve also devised a gigantic amount of new, important beautiful, and useful mathematics. That said, the basic point remains unchallenged: once a mathematical theorem has been correctly proved, it becomes something that we can build on – forever. Even though our concept of proof has tightened up considerably since Euclid’s day, to get rid of unstated assumptions, we can fill in what we now see as gaps, and the results still stand. Significant Figures investigates the almost mystical process that brings new mathematics into being. Mathematics doesn’t arise in a vacuum: it’s created by people. Among them are some with astonishing originality and clarity of mind, the people we associate with great breakthroughs – the pioneers, the trailblazers, the significant figures. Historians rightly explain that the work of the greats depended on a vast supporting cast, contributing tiny bits and pieces to the overall puzzle. Important or fruitful questions can be stated by relative unknowns; major ideas can be dimly perceived by people who lack the technical ability to turn them into powerful new methods and viewpoints. Newton remarked that he ‘stood on the shoulders of giants’. He was to some extent being sarcastic; several of those giants (notably Robert Hooke) were complaining that Newton was not so much standing on their shoulders as treading on their toes, by not giving them fair credit, or by taking the credit in public despite citing their contributions in his writings. However, Newton spoke truly: his great syntheses of motion, gravity, and light depended on a huge number of insights from his intellectual predecessors. Nor were they exclusively giants. Ordinary people played a significant part too. Nevertheless, the giants stand out, leading the way while the rest of us follow. Through the lives and works of a selection of significant figures, we can gain insight into how new mathematics is created, who created it, and how they lived. I think of them not just as pioneers who showed the rest of us the way, but as trailblazers who hacked traversable paths through the tangled undergrowth of the sprawling jungle of mathematical thought. They spent much of their time struggling through thorn bushes and swamps, but from time to time they came across a Lost City of the Elephants or an El Dorado, uncovering precious jewels hidden among the undergrowth. They penetrated regions of thought previously unknown to humankind. Indeed, they created those regions. The mathematical jungle isn’t like the Amazon Rainforest or the African Congo. The mathematical trailblazer isn’t a David Livingstone, hacking a route along the Zambezi or hunting for the source of the Nile. Livingstone was ‘discovering’ things that were already there. Indeed, the local inhabitants knew they were there. But in those days, Europeans interpreted ‘discovery’ as ‘Europeans bringing things to the attention of other Europeans.’ Mathematical trailblazers don’t merely explore a pre-existing jungle. There’s a sense in which they create the jungle as they proceed; as if new plants are springing to life in their footsteps, rapidly becoming saplings, then mighty trees. However, it feels as if there’s a pre-existing jungle, because you don’t get to choose which plants spring to life. You choose where to tread, but you can’t decide to ‘discover’ a clump of mahogany trees if what actually turns up there is a mangrove swamp. This, I think, is the source of the still popular Platonist view of mathematical ideas: that mathematical truths ‘really’ exist, but they do so in an ideal form in some sort of parallel reality, which has always existed and always will. In this view, when we prove a new theorem we just find out what has been there all along. I don’t think that Platonism makes literal sense, but it accurately describes the process of mathematical research. You don’t get to choose: all you can do is shake the bushes and see if anything drops out. In What is Mathematics, Really? Reuben Hersh offers a more realistic view of mathematics: it’s a shared human mental construct. In this respect it’s much like money. Money isn’t ‘really’ lumps of metal or pieces of paper or numbers in a computer; it’s a shared set of conventions about how we exchange lumps of metal, pieces of paper, and numbers in a computer, for each other or for goods. Hersh outraged some mathematicians, who zoomed in on ‘human construct’ and complained that mathematics is by no means arbitrary. Social relativism doesn’t hack it. This is true, but Hersh explained perfectly clearly that mathematics isn’t any human construct. We choose to tackle Fermat’s Last Theorem, but we don’t get to choose whether it’s true or false. The human construct that is mathematics is subject to a stringent system of logical