In Pursuit of the Unknown

IN PURSUIT OF THE UNKNOWN

Also by Ian Stewart:

Concepts of Modern Mathematics

Game, Set, and Math

The Problems of Mathematics

Does God Play Dice?

Another Fine Math You've Got Me into

Fearful Symmetry
(with Martin Golubitsky)

Nature's Numbers

From Here to Infinity

The Magical Maze

Life's Other Secret

Flatterland

What Shape Is a Snowflake?

The Annotated Flatland

Math Hysteria

The Mayor of Uglyville's Dilemma

Letters to a Young Mathematician

Why Beauty Is Truth

How to Cut a Cake

Taming the Infinite/The Story of Mathematics

Professor Stewart's Cabinet of Mathematical Curiosities

Professor Stewart's Hoard of Mathematical Treasures

Cows in the Maze

Mathematics of Life

with Terry Pratchett and Jack Cohen

The Science of Discworld

The Science of Discworld II: the Globe

The Science of Discworld III: Darwin's Watch

with Jack Cohen

The Collapse of Chaos

Figments of Reality

Evolving the Alien/What Does a Martian Look Like?

Wheelers
(science fiction)

Heaven
(science fiction)

17 Equations
That Changed the World

IAN STEWART

BASIC BOOKS

A Member of the Perseus Books Group
New York

 

 

 

Copyright © 2012 by Ian Stewart

Published in the United States in 2012 by Basic Books,
A Member of the Perseus Books Group

Published in Great Britain in 2012 by Profile Books

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A CIP catalog record for this book is available from the Library of Congress.

LCCN: 2011944850
ISBN: 978-0-465-02973-0

10 9 8 7 6 5 4 3 2 1

To avoide the tediouse repetition of these woordes: is equalle to: I will sette as I doe often in woorke use, a paire of paralleles, or gemowe lines of one lengthe: =======, bicause

noe .2. thynges, can be moare equalle.

Robert Recorde,
The Whetstone of Witte
, 1557

E
quations are the lifeblood of mathematics, science, and technology. Without them, our world would not exist in its present form. However, equations have a reputation for being scary: Stephen Hawking's publishers told him that every equation would halve the sales of
A Brief History of Time
, but then they ignored their own advice and allowed him to include
E
=
mc
²
when cutting it out would allegedly have sold another 10 million copies. I'm on Hawking's side. Equations are too important to be hidden away. But his publishers had a point too: equations are formal and austere, they look complicated, and even those of us who love equations can be put off if we are bombarded with them.

In this book, I have an excuse. Since it's
about
equations, I can no more avoid including them than I could write a book about mountaineering without using the word ‘mountain'. I want to convince you that equations have played a vital part in creating today's world, from mapmaking to satnav, from music to television, from discovering America to exploring the moons of Jupiter. Fortunately, you don't need to be a rocket scientist to appreciate the poetry and beauty of a good, significant equation.

There are two kinds of equations in mathematics, which on the surface look very similar. One kind presents relations between various mathematical quantities: the task is to prove the equation is true. The other kind provides information about an unknown quantity, and the mathematician's task is to
solve
it – to make the unknown known. The distinction is not clear-cut, because sometimes the same equation can be used in both ways, but it's a useful guideline. You will find both kinds here.

Equations in pure mathematics are generally of the first kind: they reveal deep and beautiful patterns and regularities. They are valid because, given our basic assumptions about the logical structure of mathematics, there is no alternative. Pythagoras's theorem, which is an equation expressed in the language of geometry, is an example. If you accept Euclid's basic assumptions about geometry, then Pythagoras's theorem is
true
.

Equations in applied mathematics and mathematical physics are usually of the second kind. They encode information about the real
world; they express properties of the universe that could in principle have been very different. Newton's law of gravity is a good example. It tells us how the attractive force between two bodies depends on their masses, and how far apart they are. Solving the resulting equations tells us how the planets orbit the Sun, or how to design a trajectory for a space probe. But Newton's law isn't a mathematical theorem; it's true for physical reasons, it fits observations. The law of gravity might have been different. Indeed, it
is
different: Einstein's general theory of relativity improves on Newton by fitting some observations better, while not messing up those where we already know Newton's law does a good job.