A HISTORY OF MATHEMATICS: FROM MESOPOTAMIA TO MODERNITY
CONTENTS INTRODUCTION WHY THIS BOOK? ON TEXTS, AND ON HISTORY EXAMPLES HISTORICISM AND ‘PRESENTISM’ REVOLUTIONS, PARADIGMS, AND ALL THAT EXTERNAL VERSUS INTERNAL EUROCENTRISM 1 BABYLONIAN MATHEMATICS 1 ON BEGINNINGS 2 SOURCES AND SELECTIONS 3 DISCUSSION OF THE EXAMPLE 4 THE IMPORTANCE OF NUMBER-WRITING 5 ABSTRACTION AND USELESSNESS 6 WHAT WENT BEFORE 7 SOME CONCLUSIONS APPENDIX A SOLUTION OF THE QUADRATIC PROBLEM SOLUTIONS TO EXERCISES 2 GREEKS AND ‘ORIGINS’ 1 PLATO AND THE MENO 2 LITERATURE 3 AN EXAMPLE 4 THE PROBLEM OF MATERIAL 5 THE GREEK MIRACLE 6 TWO REVOLUTIONS? 7 DROWNING IN THE SEA OF NON-IDENTITY 8 ON MODERNIZATION AND RECONSTRUCTION 9 ON RATIOS APPENDIX A FROM THE MENO APPENDIX B ON PENTAGONS, GOLDEN SECTIONS, AND IRRATIONALS 3 GREEKS, PRACTICAL AND THEORETICAL 1 INTRODUCTION, AND AN EXAMPLE 2 ARCHIMEDES 3 HERON OR HERO 4 ASTRONOMY, AND PTOLEMY IN PARTICULAR 5 ON THE UNCULTURED ROMANS 6 HYPATIA APPENDIX A FROM HERON’S METRICS APPENDIX B FROM PTOLEMY’S ALMAGEST 4 CHINESE MATHEMATICS 1 INTRODUCTION 2 SOURCES 3 AN INSTANT HISTORY OF EARLY CHINA 4 THE NINE CHAPTERS 5 COUNTING RODS—WHO NEEDS THEM? 6 MATRICES 7 THE SONG DYNASTY AND QIN JIUSHAO 8 ON ‘TRANSFERS’—WHEN, AND HOW? 9 THE LATER PERIOD 5 ISLAM, NEGLECT AND DISCOVERY 2 ON ACCESS TO THE LITERATURE 3 TWO TEXTS 4 THE GOLDEN AGE 5 ALGEBRA—THE ORIGINS 6 ALGEBRA—THE NEXT STEPS 7 AL-SAMAW’AL AND AL-K¯ASH¯I 8 THE USES OF RELIGION APPENDIX A FROM AL-KHW¯ARIZM¯I’S ALGEBRA APPENDIX B TH¯ABIT IBN QURRA APPENDIX C FROM AL-K¯ASH¯I, THE CALCULATOR’S KEY, BOOK 4, CHAPTER 6 UNDERSTANDING THE ‘SCIENTI?C REVOLUTION’ 3 SCHOLASTICS AND SCHOLASTICISM 4 ORESME AND SERIES 5 THE CALCULATING TRADITION 6 TARTAGLIA AND HIS FRIENDS 7 ON AUTHORITY 8 DESCARTES 9 IN?NITIES 10 GALILEO APPENDIX A APPENDIX B APPENDIX C APPENDIX D 7 THE CALCULUS 3 THE PRIORITY DISPUTE 4 THE KERALA CONNECTION 5 NEWTON, AN UNKNOWN WORK 6 LEIBNIZ, A CONFUSING PUBLICATION 7 THE PRINCIPIA AND ITS PROBLEMS 8 THE ARRIVAL OF THE CALCULUS 9 THE CALCULUS IN PRACTICE 10 AFTERWORD APPENDIX A NEWTON APPENDIX B LEIBNIZ APPENDIX C FROM THE PRINCIPIA 8 GEOMETRIES AND SPACE 2 FIRST PROBLEM: THE POSTULATE 3 SPACE AND IN?NITY 4 SPHERICAL GEOMETRY 5 THE NEW GEOMETRIES 6 THE ‘TIME-LAG’ QUESTION 7 WHAT REVOLUTION? APPENDIX B THE FORMULAE OF SPHERICAL AND HYPERBOLIC TRIGONOMETRY APPENDIX C FROM HELMHOLTZ’S 1876 PAPER 9 MODERNITY AND ITS ANXIETIES 3 NEW OBJECTS IN MATHEMATICS 4 CRISIS—WHAT CRISIS? 5 HILBERT 6 TOPOLOGY 7 OUTSIDERS APPENDIX A THE CUT DE?NITION APPENDIX B INTUITIONISM APPENDIX C HILBERT’S PROGRAMME 10 A CHAOTIC END? 3 THE SECOND WORLD WAR 4 ABSTRACTION AND ‘BOURBAKI’ 5 THE COMPUTER 6 CHAOS: THE LESS YOU KNOW, THE MORE YOU GET 7 FROM TOPOLOGY TO CATEGORIES 8 PHYSICS 9 FERMAT’S LAST THEOREM APPENDIX A FROM BOURBAKI, ‘ALGEBRA’, INTRODUCTION APPENDIX B TURING ON COMPUTABLE NUMBERS CONCLUSION