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First Edition Britannica Educational Publishing Michael I. Levy: Executive Editor J. Luebering: Senior Manager Marilyn L. Barton: Senior Coordinator, Production Control Steven Bosco: Director, Editorial Technologies Lisa S.
Braucher: Senior Producer and Data Editor Yvette Charboneau: Senior Copy Editor Kathy Nakamura: Manager, Media Acquisition Erik Gregersen: Associate Editor, Astronomy and Space Exploration Rosen Educational Services Hope Lourie Killcoyne: Senior Editor and Project Manager Bethany Bryan: Editor Nelson Sá: Art Director Cindy Reiman: Photography Manager Matthew Cauli: Designer, Cover Design Introduction by John Strazzabosco Library of Congress Cataloging-in-Publication Data The Britannica guide to the history of mathematics / edited by Erik Gregersen.—(Math explained) “In association with Britannica Educational Publishing, Rosen Educational Services.” Includes bibliographical references and index. Title: History of mathematics.9—dc22 2010008356 On the cover: Hands with abacus, an old-fashioned counting device. Jed Share/Photodisc/ Getty Images On page 12: Illustrating Pythagoras’s theorem, this diagram comes from a mid-19th-century edition of the Elements of Euclid, a seminal multi-book series incorporating the findings of both mathematicians. SSPL via Getty Images On page 20: A page from Newton’s annotated copy of Elements, Euclid’s treatise on geom- etry.
Hulton Archive/Getty Images On pages 21, 84, 182, 217, 256, 282, 285, 294: This diagram from Newton’s Principia Mathematica concerns hourly variations of the lunar orbit. SSPL via Getty Images www.com Contents 33 Introduction 12 Chapter 1: Ancient Western Mathematics 21 Ancient Mathematical Sources 22 Mathematics in Ancient Mesopotamia 23 The Numeral System and Arithmetic Operations 24 Geometric and Algebraic Problems 27 Mathematical Astronomy Mathematics in Ancient Egypt 29 31 53 The Numeral System and Arithmetic Operations 32 Geometry 36 Assessment of Egyptian Mathematics 38 Greek Mathematics 39 The Development of Pure Mathematics 39 The Pre-Euclidean Period 40 The Elements 46 61 The Three Classical Problems 49 Geometry in the 3rd Century BCE 51 Archimedes 52 Apollonius 54 Applied Geometry 60 Later Trends in Geometry and Arithmetic 66 Greek Trigonometry and Mensuration 67 Number Theory 67 Survival and Influence of Greek Mathematics 70 www.com Mathematics in the Islamic World (8th–15th Century) Origins 72 85 72 Mathematics in the 9th Century 74 Mathematics in the 10th Century 76 Omar Khayyam 79 Islamic Mathematics to the 15th Century 81 Chapter 2: European Mathematics Since the Middle Ages 84 European Mathematics During the Middle Ages and Renaissance 84 The Transmission of Greek and Arabic Learning 86 The Universities 87 The Renaissance 89 Mathematics in the 17th and 18th Centuries 91 The 17th Century 91 Institutional Background 92 Numerical Calculation 92 Analytic Geometry The Calculus 95 101 90 The 18th Century 112 Institutional Background 112 Analysis and Mechanics 114 History of Analysis 116 Other Developments 120 Theory of Equations 120 Foundations of Geometry 122 Mathematics in the 19th and 20th Centuries 125 Projective Geometry 126 Making the Calculus Rigorous Fourier Series 129 133 129 www.com Elliptic Functions 134 The Theory of Numbers The Theory of Equations 136 140 148 Gauss 143 Non-Euclidean Geometry 144 Riemann 147 Riemann’s Influence 151 Differential Equations 154 Linear Algebra 156 The Foundations of Geometry 158 The Foundations of Mathematics 161 Cantor 162 Mathematical Physics 165 Algebraic Topology 169 Developments in Pure Mathematics 173 Mathematical Physics and the Theory of Groups 177 Chapter 3: South and East Asian Mathematics 182 Ancient Traces 182 Vedic Number Words and Geometry 183 166 The Post-Vedic Context 184 Indian Numerals and the Decimal Place-Value System 185 The “Classical” Period 186 The Role of Astronomy and Astrology 187 Classical Mathematical Literature 189 The Changing Structure of Mathematical Knowledge 191 Mahavira and Bhaskara II 192 Teachers and Learners The School of Madhava in Kerala 194 193 172 www.com Exchanges with Islamic and Western Mathematics 195 Mathematics in China 195 The Textual Sources 196 The Great Early Period, 1st–7th Centuries 198 The Nine Chapters 198 The Commentary of Liu Hui 204 The “Ten Classics” 206 Scholarly Revival, 11th–13th Centuries 207 199 Theory of Root Extraction and Equations 208 The Method of the Celestial Unknown 209 220 Chinese Remainder Theorem 211 Fall into Oblivion, 14th–16th Centuries 211 Mathematics in Japan 213 The Introduction of Chinese Books 213 The Elaboration of Chinese Methods 214 Chapter 4: The Foundations of Mathematics 217 Ancient Greece to the Enlightenment 217 Arithmetic or Geometry 217 Being Versus Becoming 218 Universals 221 The Axiomatic Method 222 Number Systems 223 The Reexamination of Infinity 224 Calculus Reopens Foundational Questions 225 221 www.com Non-Euclidean Geometries 226 Elliptic and Hyperbolic Geometries 227 227 Riemannian Geometry 228 Cantor 229 The Quest for Rigour 230 Formal Foundations 230 Set Theoretic Beginnings 230 Foundational Logic 232 Impredicative Constructions 233 Nonconstructive Arguments 234 Intuitionistic Logic 235 Other Logics 237 Formalism 237 Gödel 238 Recursive Definitions 241 Computers and Proof 243 Category Theory 244 Abstraction in Mathematics 244 Isomorphic Structures 246 Topos Theory 247 Intuitionistic Type Theories 248 Internal Language 249 Gödel and Category Theory 250 The Search for a Distinguished Model 251 Boolean Local Topoi 252 One Distinguished Model or Many Models 254 Chapter 5: The Philosophy of Mathematics 256 Mathematical Platonism 258 Traditional Platonism Nontraditional Versions 258 260 240 www.com Mathematical Anti-Platonism 263 Realistic Anti-Platonism 263 Nominalism 266 Logicism, Intuitionism, and Formalism 270 Mathematical Platonism: For and Against 272 The Fregean Argument for Platonism 273 The Epistemological Argument Against Platonism 277 Ongoing Impasse 280 Glossary 282 Bibliography Index 285 294 265 www.com 7 Introduction 7 I t seems impossible to believe that at one point in ancient time, human beings had absolutely no formal mathematics—that from scratch, the ideas for numbers and numeration were begun, applications found, and inventions pursued, one layered upon another, creating the very foundation of everyday life. So dependent are we upon this mathematic base—wherein we can do every- thing from predict space flight to forecast the outcomes of elections to review a simple grocery bill—that to imag- ine a world with no mathematical concepts is quite a difficult thought to entertain. In this volume we encounter the humble beginnings of the ancient mathematicians and various developments over thousands of years, as well as modern intellectual battles fought today between, for example, the logicians who either support the mathematic philosophy of Platonism or promote its aptly named rival, anti-Platonism.
We explore worldwide math contributions from 4000 BCE through today. Topics presented from the old world include mathe- matical astronomy, Greek trigonometry and mensuration, and the ideas of Omar Khayyam. Contemporary topics include isomorphic structures, topos theory, and comput- ers and proof. We also find that mathematic discovery was not always easy for the discoverers, who perhaps fled for their lives from Nazi threats, or created brilliant mathematical inno- vation while beleaguered by serious mental problems, or who pursued a mathematic topic for many years only to have another mathematician suddenly and quite conclu- sively prove that what had been attempted was all wrong, effectively quashing years of painstaking work.
For the creative mathematician, as for those who engage in other loves or conflicts, heartbreak or disaster might be encoun- tered. The lesson learned is one in courage and the pure 13 www.com 7 The Britannica Guide to the History of Mathematics 7 guts of those willing to take a chance—even when most of the world said no. Entering into math history is a bit like trying to sort through a closet full of favourite old possessions. We pick up an item, prepared to toss it if necessary, and suddenly a second and third look at the thing reminds us that this is fascinating stuff.
First thing we know, a half hour has passed and we are still wondering how, for instance, the Babylonians (c. 2000 BCE) managed to write a table of numbers quite close to Pythagorean Triples more than 1,000 years before Pythagoras himself (c. 500 BCE) sup- posedly discovered them. The modern-day math student lives and breathes with her math teacher’s voice ringing in her ear, say- ing, “Memorize these Pythagorean triples for the quiz on Friday.” Babylonian students might have heard the same request.
Their triples were approximated by the formula of the day, a2 + b2/2a, which gives values close to Pythagoras’s more accurate a2 + b2 = c2. Consider that such pre-Pythagorean triples were written by ancient scribes in cuneiform and sexagesimal (that’s base 60). One such sexagesimal line of triples from an ancient clay tablet of the time translates to read as follows: 2, 1 59, 2 49. In base 10 this line of triples would be 120, 119, 169.
The reader is invited for old time’s sake to plug these base 10 numbers into the Pythagorean Formula a2 + b2 = c2 to verify the ancient set of Pythagorean triples that appeared more than 1,000 years before Pythagoras him- self appeared. An equally compelling example of credit for discov- ery falling upon someone other than the discoverer is found in a quite familiar geometrically appearing set of 14 www.com 7 Introduction 7 numbers. Most math students recognize the beautiful Pascal’s Triangle and can even reproduce it, given pencil and paper. The triangle yields at a glance the coefficients of a binomial expansion, among many other bits of use- ful mathematics information.
As proud as Blaise Pascal (1600s) must have been over his Pascal’s triangle, imag- ine that of Zhu Shijie (a. Chu Shih-Chieh), who first published the triangle in his book, Precious Mirror of Four Elements (1303). Zhu probably did not give credit to Pascal, as Pascal would not be born for another 320 years. Zhu’s book has a gentle kind of title that suggests the generous sort of person Zhu might have been.
Indeed, he gave full credit for the aforementioned triangle to his predecessor, Yang Hui (1300), who in turn probably lifted the triangle from Jia Xian (c. In fact, despite sig- nificant contributions to math theory of his times, Zhu unselfishly referred to methods in his book as the old way of doing things, thus praising the work of those who came before him. We dig deeper into our closet of mathematic treasures and imagine mathematician Kurt Gödel (1906–1978). His eyes were said to be piercing, perhaps even haunting.
Like a teacher of our past, could Mr. Gödel pointedly be asking about a little something we omitted from our homework, perhaps? We probably have all been confronted at one time or another for turning in an assignment that was incomplete. Gödel, however, made a career out of incom- pleteness, literally throwing the whole world into a tizzy with his incompleteness theorem. Paranoid and mentally unstable, his tormented mind could nonetheless uncover what other great minds could not.
It was 1931, a year after his doctoral thesis first announced to the world that a young mathematics great had arrived. Later an Austrian escapee of the Nazis, Gödel with his incompleteness theorem proved to be brilliant and 15 www.com 7 The Britannica Guide to the History of Mathematics 7 on target, but also bad news for heavyweight mathema- ticians Bertrand Russell, David Hilbert, Gottlob Frege, and Alfred North Whitehead. These four giants in the math world had spent significant portions of their careers trying to construct axiom systems that could be used to prove all mathematical truths. Gödel’s incompleteness theorem ended those pursuits, trashing years of math- ematical work.
Russell, Hilbert, Frege, and Whitehead all made their marks in other areas of math. How would they have taken this shocking news of enormous rejection? Let’s try to imagine.