Stewart - Significant figures: lives and works of trailblazing mathematicians

Significant Figures

Ian Stewart is Mathematics Professor Emeritus at the University of Warwick. His recent books include Incredible Numbers, Seventeen Equations that Changed the World, Professor Stewarts Casebook of Mathematical Mysteries and Calculating the Cosmos (all published by Profile). He is a Fellow of the Royal Society.

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Incredible Numbers

Significant Figures

Lives and Works of Trailblazing Mathematicians

Ian Stewart

First published in Great Britain in 2017 by

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To John Davey, editor and friend
(19 April 194521 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 todays 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. Aristotles 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. Its still correct today. We express the result symbolically, but the reasoning is identical. Theres 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 didnt use algebraic symbols, and he didnt think of a specific number 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 todays 4/3r3.

To be sure, a few ancient discoveries outside mathematics have been similarly long-lived. Archimedess 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 its closer to the rule. Euclids 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 dont 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 theyre 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, theyve not only moved on; theyve 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 Euclids 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 doesnt arise in a vacuum: its 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.