God Created The Integers: The Mathematical Breakthroughs that Changed History

1641 Pages·2007·28.11 MB·English

by  Stephen Hawking

GOD CREATED
THE INTEGERS

GOD CREATED THE INTEGERS

THE MATHEMATICAL BREAKTHROUGHS THAT CHANGED HISTORY

EDITED, WITH COMMENTARY, BY

STEPHEN
HAWKING

RUNNING PRESS

PHILADELPHIA • LONDON

© 2007 by Stephen Hawking

All rights reserved under the Pan-American

and International Copyright Conventions This book may not be reproduced in

whole or in part, in any form or by any means, electronic or mechanical,

including photocopying, recording, or by any information storage and retrieval
system now known or hereafter invented, without written permission from the

publisher.

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Digit on the right indicates the number of this printing Library of Congress

Control Number: 2007920542
ISBN-13
978-0-7624-3272-1

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A NOTE ON THE TEXTS

The texts in this book are based on translations of the original, printed editions.
We have made no attempt to modernize the authors’ own distinct usage, spelling
or punctuation, or to make the texts consistent with each other in this regard.

The Editors

TEXT AND PICTURE CREDITS

Text Permissions:
Selections from Euclid’s Elements, by Thomas L. Heath, courtesy of Dover
Publications.
Selections from The Works of Archimedes, by Thomas L. Heath, courtesy of
Dover Publications.
Selections from Diophantus of Alexandria, A Study in the History of Greek
Algebra, by Thomas L. Heath, reprinted with permission of Cambridge
University Press.
The Geometry of Rene Descartes, trans. David E. Smith and Marcia L. Latham
courtesy of Dover Publications.
Selections from Isaac Newton’s Principia, notes by David Eugene Smith,
courtesy of New York: Daniel Adee, © 1848.
English translation of Leonhard Euler’s On the sums of series of reciprocals (De
summis serierum reciprocarum) courtesy of Jordan Bell.
Leonhard Euler’s The Seven Bridges of Konigsberg and Proof that Every Integer
is A Sum of Four Squares courtesy of Dover Publications.
Pierre Simon Laplace’s A Philosophical Essay on Probabilities, introductory
note by E.T. Bell, courtesy of Dover Publications.
Selection from Jean Baptiste Joseph Fourier’s The Analytical Theory of Heat
courtesy of Dover Publications.
Selections from Carl Friedrich Gauss’s Disquisitiones Arithmeticae courtesy of
Yale University Press.
Selections from Oeuvres complètes d’Augustin Cauchy reprinted from 1882

version published by Gauthier-Villars, Paris.
Nikolai Ivanovich Lobachevsky’s Geometrical Researches on the Theory of
Parallels, trans. Dr. George Bruce Halstead, courtesy of Dover Publications.
János Bolyai’s The Science of Absolute Space, trans. Dr. George Bruce Halstead,
courtesy of Dover Publications.
English translation of Evariste Galois’ On the conditions that an equation be
soluble by radicals and Of the primitive equations which are soluble by radicals,
courtesy of John Anders.
Evariste Galois’ On Groups and Equations and Abelian Integrals, courtesy of
Dover publications.
George Boole’s An Investigation of the Laws of Thought courtesy of Dover
Publications.
Georg Friedrich Bernhard Riemann’s Ueber die Darstellbarkeit einer Function
durch einer trigonometrische Reihe, Ueber die Anzahl der Primzahlen unter
einer gegebenen Grösse, and Ueber die Hypothesen welche der Geometrie zu
Grunde liegen, courtesy of Dover Publications.
Karl Weierstrass’ Ausgewählte Kapitel aus der Funktionenlehre: Vorlesung
gehalten in Berlin 1886, mit der akademischen Antrittsrede, Berlin 1857, und
drei weiteren Originalarbeiten . . . aus den Jahren 1870 bis 1880/86 / K.
Weierstrass; herausgegeben, kommentiert und mit einem Anhang versehen von
R. Siegmund-Schultze reprinted from R. Siegmund-Schultze (Ed.), Teubner,
Leipzig, 1886 (Springer, Berlin), 1998.
Richard Dedekind’s Essays on the Theory of Numbers, trans. Wooster W.
Beman, courtesy of Dover Publications.
Selections from Georg Cantor’s Contributions to the Founding of the Theory of
Transfinite Numbers, trans. Philip E.B. Jourdain, courtesy of Dover Publications.
Selections from Henri Lebesgue’s Integrale, Longeur, Aire reprinted from Annali
di Matematica, Pura ed Applicata, 1902, Ser. 3, vol. 7, pp. 231–359.
Kurt Gödel’s On Formally Undecidable Propositions of Principia Mathematica
and Related Systems, trans. B. Meltzer, courtesy of Dover Publications.
Alan Turing’s On computable numbers with an application to the
Entscheidungsproblem, Proceedings of the London Mathematical Society
courtesy of the London Mathematical Society.

Picture Credits:
Euclid: Getty Images.

Archimedes: Getty Images.
Diophantus: Title page of Diophanti Alexandrini Arthimeticorum libri sex. . . .,
1621: Library of Congress, call number QA31.D5, Rare Book/Special
Collections Reading Room, (Jefferson LJ239).
Rene Descartes: Getty Images.
Isaac Newton: Time Life Pictures/Getty Images.
Leonhard Euler: Getty Images.
Pierre Simon de Laplace: Getty Images.
Jean Baptiste Joseph Fourier: Science and Society Picture Library, London.
Carl Friedrich Gauss: Getty Images.
Augustin-Louis Cauchy: © Bettmann/CORBIS.
Évariste Galois: © Bettmann/CORBIS.
George Boole: © CORBIS.
Georg Friedrich Bernhard Riemann: © Bettmann/CORBIS.
Karl Weierstrass: © Bettmann/CORBIS.
Richard Dedekind: Frontispiece from Richard Dedekind Gesammelte
mathematische Werke. Reprint by Chelsea Publishing Company, Bronx, NY,
1969, of the edition published in Brunswick by F. Vieweg, 1930–32. Reprinted
by permission of the Chelsea Publishing Company. Photograph provided by the
Library of Congress, call number QA3.D42.
Georg Cantor: © CORBIS.
Henri Lebesgue: Frontispiece from Henri Lebesgue Oeuvres Scientifiques,
volume I. Reproduced by permission of L’Enseignement Mathématique,
Universite De Geneve, Switzerland. Photograph provided by the Library of
Congress, call number QA3.L27, vol. 1, copy 1.
Kurt Gödel: Time Life Pictures/Getty Images.
Alan Turing: Photo provided by King’s College Archive Centre, Cambridge,
UK, AMT/K/7/9. Contact the Archive Centre for copyright information.

CONTENTS

Introduction

EUCLID

(C. 325BC–265BC)
His Life and Work

Selections from Euclid’s Elements

Book I: Basic Geometry—Definitions, Postulates, Common Notions; and

Proposition 47, (leading up to the Pythagorean Theorem)

Book V: The Eudoxian Theory of Proportion—Definitions & Propositions
Book VII: Elementary Number Theory—Definitions & Propositions
Book IX: Proposition 20: The Infinitude of Prime Numbers
Book IX: Proposition 36: Even Perfect Numbers
Book X: Commensurable and Incommensurable Magnitudes

ARCHIMEDES

(287BC–212BC)

His Life and Work

Selections from The Works of Archimedes

On the Sphere and Cylinder, Books I and II
Measurement of a Circle
The Sand Reckoner
The Methods

DIOPHANTUS

(C. 200–284)

His Life and Work

Selections from Diophantus of Alexandria, A Study in the History of Greek
Algebra

Book II Problems 8–35

Book III Problems 5–21
Book V Problems 1–29

RENÉ DESCARTES

(1596–1650)

His Life and Work

The Geometry of Rene Descartes

ISAAC NEWTON

(1642–1727)

His Life and Work

Selections from Principia

On First and Last Ratios of Quantities

LEONHARD EULER

(1707–1783)

His Life and Work

On the sums of series of reciprocals (De summis serierum reciprocarum)
The Seven Bridges of Konigsberg
Proof that Every Integer is A Sum of Four Squares

PIERRE SIMON LAPLACE

(1749–1827)

His Life and Work

A Philosophical Essay on Probabilities

JEAN BAPTISTE JOSEPH FOURIER

(1768–1830)

His Life and Work

Selection from The Analytical Theory of Heat

Chapter III: Propagation of Heat in an Infinite Rectangular Solid (The

Fourier series)

CARL FRIEDRICH GAUSS

(1777–1855)

His Life and Work

Selections from Disquisitiones Arithmeticae (Arithmetic Disquisitions)

Section III Residues of Powers
Section IV Congruences of the Second Degree

AUGUSTIN-LOUIS CAUCHY

(1789–1857)

His Life and Work

Selections from Oeuvres complètes d’Augustin Cauchy Résumé des leçons
données à l’École Royale Polytechnique sur le calcul infinitésimal (1823), series
2, vol. 4

Lessons 3–4 on differential calculus
Lessons 21–24 on the integral

NIKOLAI IVANOVICH LOBACHEVSKY

(1792–1856)

His Life and Work

Geometrical Researches on the Theory of Parallels

JÁNOS BOLYAI

(1802–1860)

His Life and Work

The Science of Absolute Space

ÉVARISTE GALOIS

(1811–1832)

His Life and Work

On the conditions that an equation be soluble by radicals
Of the primitive equations which are soluble by radicals
On Groups and Equations and Abelian Integrals

GEORGE BOOLE

(1815–1864)

His Life and Work

An Investigation of the Laws of Thought

BERNHARD RIEMANN

(1826–1866)

His Life and Work

On the Representability of a Function by Means of a Trigonometric Series
(Ueber die Darstellbarkeit eine Function durch einer trigonometrische Reihe)
On the Hypotheses which lie at the Bases of Geometry (Ueber die Hypothesen,
welche der Geometrie zu Grunde liegen)
On the Number of Prime Numbers Less than a Given Quantity (Ueber die
Anzahl der Primzahlen unter einer gegebenen Grösse)

KARL WEIERSTRASS

(1815–1897)

His Life and Work

Selected Chapters on the Theory of Functions, Lecture Given in Berlin in 1886,
with the Inaugural Academic Speech, Berlin 1857

§ 7 Gleichmässige Stetigkeit (Uniform Continuity)

RICHARD DEDEKIND

(1831–1916)

His Life and Work

Essays on the Theory of Numbers

GEORG CANTOR

(1848–1918)

His Life and Work

Selections from Contributions to the Founding of the Theory of Transfinite
Numbers

Articles I and II

HENRI LEBESGUE

(1875–1941)

His Life and Work

Selections from Integrale, Longeur, Aire (Intergral, Length, Area)

Preliminaries and Integral

KURT GÖDEL

(1906–1978)

His Life and Work

On Formally Undecidable Propositions of Principia Mathematica and Related
Systems

ALAN TURING

(1912–1954)

His Life and Work

On computable numbers with an application to the Entscheidungsproblem,
Proceedings of the London Mathematical Society

S

INTRODUCTION

WE ARE LUCKY TO LIVE IN AN AGE LN WHICH WE ARE

STILL MAKING DISCOVERIES. IT IS LIKE THE DISCOVERY

OF AMERICA-YOU ONLY DISCOVER IT ONCE. THE AGE IN
WHICH WE LIVE IS THE AGE IN WHICH WE ARE

DISCOVERING THE FUNDAMENTAL LAWS OF NATURE . . .

—AMERICAN PHYSICIST RICHARD FEYNMAN, SPOKEN

IN 1964

howcasing excerpts from thirty-one of the most important works in the
history of mathematics (four of which have been translated into English
for the very first time), this book is a celebration of the mathematicians

who helped move us forward in our understanding of the world and who paved
the way for our current age of science and technology.

Over the centuries, the efforts of these mathematicians have helped the

human race to achieve great insight into nature, such as the realization that the
earth is round, that the same force that causes an apple to fall here on earth is
also responsible for the motions of the heavenly bodies, that space is finite and
not eternal, that time and space are intertwined and warped by matter and energy,
and that the future can only be determined probabilistically. Such revolutions in
the way we perceive the world have always gone hand in hand with revolutions
in mathematical thought. Isaac Newton could never have formulated his laws
without the analytic geometry of René Descartes and Newton’s own invention of
calculus. It is hard to imagine the development of either electrodynamics or
quantum theory without the methods of Jean Baptiste Joseph Fourier or the work
on calculus and the theory of complex functions pioneered by Carl Friedrich
Gauss and Augustin-Louis Cauchy—and it was Henri Lebesgue’s work on the
theory of measure that enabled John von Neumann to formulate the rigorous
understanding of quantum theory that we have today. Albert Einstein could not
have completed his general theory of relativity had it not been for the geometric
ideas of Bernhard Riemann. And practically all of modern science would be far
less potent (if it existed at all) without the concepts of probability and statistics

pioneered by Pierre Simon Laplace.

All through the ages, no intellectual endeavor has been more important to

those studying physical science than has the field of mathematics. But
mathematics is more than a tool and language for science. It is also an end in
itself, and as such, it has, over the centuries, affected our worldview in its own
right. Karl Weierstrass provided a new idea of what it means for a function to be
continuous, and Georg Cantors work revolutionized peoples idea of infinity.
George Boole’s Laws of Thought revealed logic as a system of processes subject
to laws identical to the laws of algebra, thus illuminating the very nature of
thought and eventually enabling to some degree its mechanization, that is,
modern digital computing. Alan Turing illuminated the power and the limits of
digital computing, long before sophisticated computations were even possible.
Kurt Gödel proved a theorem troubling to many philosophers, as well as anyone
else believing in absolute truth: that in any sufficiently complex logical system
(such as arithmetic) there must exist statements that can neither be proven nor
disproven. And if that weren’t bad enough, he also showed that the question of
whether the system itself is logically consistent cannot be proven within the
system.

This fascinating volume presents all these and other groundbreaking

developments, the central ideas in twenty-five centuries of mathematics,
employing the original texts to trace the evolution, and sometimes revolution, in
mathematical thinking from its beginnings to today.

Though the first work presented here is that of Euclid, C.300 B.C., the

Egyptians and Babylonians had developed an impressive ability to perform
mathematical calculations as early as 3,500 B.C. The Egyptians employed this
skill to build the great pyramids and to accomplish other impressive ends, but
their computations lacked one quality considered essential to mathematics ever
since: rigor. For example, the ancient Egyptians equated the area of a circle to
the area of a square whose sides were 8/9 the diameter of the circle. This method
amounts to employing a value of the mathematical constant pi that is equal to
256/81. In one sense this is impressive—it is only about one half of one percent
off of the exact answer. But in another sense it is completely wrong. Why worry
about an error of one half of one percent? Because the Egyptian approximation
overlooks one of the deep and fundamental mathematical properties of the true
number π: that it cannot be written as any fraction. That is a matter of principle,
unrelated to any issue of mere quantitative accuracy. Though the irrationality of
π wasn’t proved until the late eighteenth century, the early Greeks did discover

that numbers existed which could not be written as fractions, and this was both
puzzling and shocking to them. This was the brilliance of the Greeks: to
recognize the importance of principle plura in mathematics, and that in its
essence mathematics is a subject in which one begins with a set of concepts and
rules and then rigorously works out their precise consequences.

Euclid detailed the Greek understanding of geometry in his Elements, in

Alexandria, around 300 B.C. In the ensuing centuries the Greeks made great
strides in both algebra and geometry. Archimedes, the greatest mathematician of
antiquity, studied the properties of geometric shapes and created ingenious
methods of finding areas and volumes and new approximations for π. Another
Alexandrian, Diophantus, looking over the clutter of words and numbers in
algebraic problems, saw that an abstraction could be a great simplification. And
so, Diophantus took the first step toward introducing symbolism into algebra.
Over a millennium later, Frenchman René Descartes united the two fields:
geometry and algebra, with his creation of analytic geometry. His work paved
the way for Isaac Newton to invent calculus, and with it, a new way of doing
science. Since Newton’s day, the pace of mathematical innovation has been
almost frenetic, as the fundamental mathematical fields of algebra, geometry,
and calculus (or function theory) have fed on and in turn nourished one another,
yielding insights into applications as diverse as probability, numbers, and the
theory of heat. And as mathematics matured, so did the range of questions it
addresses: Kurt Gödel and Alan Turing, the last two thinkers represented in this
volume, address perhaps the deepest issue—the question of what is knowable.
Like those of the past, future developments in mathematics are sure to affect,
directly or indirectly, our ways of living and thinking. The wonders of the
ancient world were physical, like the pyramids in Egypt. As this volume
illustrates, the greatest wonder of the modern world is our own understanding.