Continuity (Uniwersytet Śląski)

Tytuł Continuity Podtytuł Eleven sketches from the past of Mathematics Autor Jerzy Mioduszewski Język angielski Wydawnictwo Uniwersytet Śląski Tłumaczenie Abe Shenitzer ISBN 978-83-8012-950-4 Rok wydania 2016 Katowice Wydanie 1 ilość stron 202 Format pdf Spis treści Table of contents From the author / 7 Introduction / 9 Chapter I The flying arrow • Aristotle’s view of the aporia of Zeno • Its influence on the evolution of geometry and on the science of motion • Democritus’s version of this aporia • On mathematical atomism / 15 Chapter II Aporia of the wanderer • The Archimedean postulate • The Eudoxian exhaustion lemma • Non-Archimedean continua • Another Zeno’s difficulty: Stadium / 27 Chapter III Number • On idealism in mathematics • Its two varieties: Pythagoreism and Platonism • Discovery of incommensurable segments • The Euclidean algorithm • On some possible meanings of the proportion of segments / 35 Chapter IV On geometric magnitudes • Comparison of polygons from the point of view of area • Comparison through complementation • Comparison through finite decomposition • The role of Archimedean postulate • On quadratures / 51 Chapter V The Eudoxian theory of proportions • The role in it of the Archimedean postulate • The theorem on interchanging terms in a proportion • On Tales’s theorem • Comparison with Dedekind theory • Inequality of proportions • On the area of a circle • On Greek geometric algebra • The Elements as an attempt to geometrize arithmetic / 63 Chapter VI The Arab Middle Ages • Euclid’s parallel postulate: mistakes and progress • Singular features of the philosophy of nature of the Arab East / 81 Chapter VII The European Middle Ages • Disputes about the structure of the continuum • Oresme and the Calculators on the intensity of changes • The 1: 3: 5: 7: … sequence • The theory of impetus • Quies media • The ballistics of Albert of Saxony and of Tartaglia • Galileo • Supplements / 95 Chapter VIII The method of indivisibles • Three ways of computing the area of a circle • Kepler: the principle of fields and a barrel • The Cavalieri’s principle • The Roberval cycloid • Need we explain it aby undivisibles? • Towards magic thinking • Descartes / 123 Chapter IX Calculus • Derivative of xn • Barrow’s observation as motivation for this computation • The role of the Calculators • The impetum theory as motivation • Philosophiae Naturalis Principia Mathematica and the theory of fluxions • Leibniz • Voltaire on Newton • Hypotheses non fingo? / 141 Chapter X Euler • A century of computations and undisturbed progress • Discussion on the term of arbitrary function • The number e • Infinitesimals and infinities • Lagrange’s objection • Barriers of growth / 167 Chapter XI A debt redeemed • The beginning of the new analysis: Cauchy and Bolzano • Weierstrass • The arithmetization of analysis • Cantor • The role of Dedekind • Set theory did not come into being accidentally • Nothing is ever completely settled / 177 Epilogue / 189 Author’s reminiscences / 195