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Mathematics, Philosophy, and the Topic “Pure intellectual stimulation that can be popped into Science Subtopic the [audio or video player] anytime.” & Mathematics Mathematics —Harvard Magazine Mathematics, Philosophy, and the “Real World” “Real the and Philosophy, Mathematics, “Passionate, erudite, living legend lecturers. Academia’s best lecturers are being captured on tape.” Mathematics, Philosophy, —The Los Angeles Times and the “Real World” “A serious force in American education.” —The Wall Street Journal Course Guidebook Professor Judith V. Grabiner Pitzer College Professor Judith V. Grabiner is the Flora Sanborn Pitzer Professor of Mathematics at Pitzer College, where she has taught for more than 20 years. Her acclaimed teaching style, which focuses on the fun in mathematics, has won her numerous awards. These include the Mathematical Association of America’s Deborah and Franklin Tepper Haimo Award for Distinguished College or University Teaching of Mathematics—one of the most prestigious mathematics awards in the United States. THE GREAT COURSES® Corporate Headquarters 4840 Westfields Boulevard, Suite 500 Chantilly, VA 20151-2299 Guidebook USA Phone: 1-800-832-2412 www.thegreatcourses.com Cover Image: © marekuliasz/Shutterstock. Course No. 1440 © 2009 The Teaching Company. PB1440A PUBLISHED BY: THE GREAT COURSES Corporate Headquarters 4840 Westfi elds Boulevard, Suite 500 Chantilly, Virginia 20151-2299 Phone: 1-800-832-2412 Fax: 703-378-3819 www.thegreatcourses.com Copyright © The Teaching Company, 2009 Printed in the United States of America This book is in copyright. All rights reserved. Without limiting the rights under copyright reserved above, no part of this publication may be reproduced, stored in or introduced into a retrieval system, or transmitted, in any form, or by any means (electronic, mechanical, photocopying, recording, or otherwise), without the prior written permission of The Teaching Company. Judith V. Grabiner, Ph.D. Flora Sanborn Pitzer Professor of Mathematics, Pitzer College Judith V. Grabiner is the Flora Sanborn Pitzer Professor of Mathematics at Pitzer College, one of the Claremont Colleges in California, where she has taught since 1985. Previously she was Professor of History at California State University, Dominguez Hills, where she won the institution’s Outstanding Professor Award for 1974–1975. Professor Grabiner received her B.S. in Mathematics, with General Honors, from the University of Chicago, where everybody reads Plato and Aristotle and Kant and thinks they are relevant. She went on to get her Ph.D. in the History of Science from Harvard University. She has taught or been a visiting scholar at a variety of institutions, including the University of California, Santa Barbara; UCLA; the University of Leeds; the University of Cambridge; the University of Copenhagen; the Australian National University; and the University of Edinburgh. Professor Grabiner’s research has focused on the history of mathematics and on its relationship to philosophy, notably on issues involving changing standards of rigor and proof. She is the author of The Origins of Cauchy’s Rigorous Calculus (MIT Press, 1981); The Calculus as Algebra: J.-L. Lagrange, 1736–1813 (Garland, 1990); a forthcoming book, tentatively titled The Calculus as Algebra and Other Selected Writings, to be published by the Mathematical Association of America (MAA); and many articles on the history of mathematics. She has won the MAA’s Carl B. Allendoerfer Award, for the best article in Mathematics Magazine, for three different articles. She has also won the association’s Lester R. Ford Award, for the best article in American Mathematical Monthly, for three different articles. Professor Grabiner won the Distinguished Teaching Award of the Southern California Section of the MAA in 2002, and won the association’s national teaching award, the Franklin and Deborah Tepper Haimo Award for Distinguished College or University Teaching, in 2003. Professor Grabiner’s career reflects the interdisciplinary nature of her field, the history of mathematics. She has long been interested in teaching mathematics to nonmathematicians, wanting to help them see that mathematics is fun, fascinating, and useful. She tries to do this by linking mathematical topics with the students’ own interests, especially philosophy and culture. In turn she has learned a great deal from her students, whose ©2009 The Teaching Company. i knowledge of how mathematics is used in their own specialties—from Leibniz’s philosophy to forensic science, from quilting to baseball—has enhanced her appreciation for the many ways mathematics interacts with the world. Her students have also taught her how to explain things more clearly by their insistence that, now that they are in college, the mathematics has to make sense. ii ©2009 The Teaching Company. Table of Contents Mathematics, Philosophy, and the “Real World” Professor Biography .................................................................................... i Course Scope ............................................................................................... 1 Lecture One What’s It All About?.......................................... 3 Lecture Two You Bet Your Life— Statistics and Medicine ...................................... 8 Lecture Three You Bet Your Life— Cost-Benefit Analysis ...................................... 11 Lecture Four Popular Statistics— Averages and Base Rates ................................. 15 Lecture Five Popular Statistics—Graphs .............................. 18 Lecture Six Popular Statistics—Polling and Sampling ....... 21 Lecture Seven The Birth of Social Statistics............................ 24 Lecture Eight Probability, Multiplication, and Permutations.............................................. 27 Lecture Nine Combinations and Probability Graphs ............. 31 Lecture Ten Probability, Determinism, and Free Will ......... 35 Lecture Eleven Probability Problems for Fun and Profit .......... 39 Lecture Twelve Probability and Modern Science ...................... 42 Lecture Thirteen From Probability to Certainty .......................... 45 Lecture Fourteen Appearance and Reality— Plato’s Divided Line ........................................ 48 Lecture Fifteen Plato’s Cave—The Nature of Learning............ 53 Lecture Sixteen Euclid’s Elements— Background and Structure................................ 56 Lecture Seventeen Euclid’s Elements— A Model of Reasoning ..................................... 59 Lecture Eighteen Logic and Logical Fallacies— Why They Matter............................................. 63 Lecture Nineteen Plato’s Meno—How Learning Is Possible ....... 67 Lecture Twenty Plato’s Meno— Reasoning and Knowledge............................... 70 ©2009 The Teaching Company. iii Table of Contents Mathematics, Philosophy, and the “Real World” Lecture Twenty-One More Euclidean Proofs, Direct and Indirect ........................................... 74 Lecture Twenty-Two Descartes—Method and Mathematics ............. 77 Lecture Twenty-Three Spinoza and Jefferson ...................................... 81 Lecture Twenty-Four Consensus and Optimism in the 18th Century............................................ 85 Lecture Twenty-Five Euclid—Parallels, Without Postulate 5............ 89 Lecture Twenty-Six Euclid—Parallels, Needing Postulate 5 ........... 94 Lecture Twenty-Seven Kant, Causality, and Metaphysics.................... 99 Lecture Twenty-Eight Kant’s Theory of Space and Time ................. 103 Lecture Twenty-Nine Euclidean Space, Perspective, and Art........... 108 Lecture Thirty Non-Euclidean Geometry— History and Examples .................................... 112 Lecture Thirty-One Non-Euclidean Geometries and Relativity..... 116 Lecture Thirty-Two Non-Euclidean Geometry and Philosophy..... 120 Lecture Thirty-Three Art, Philosophy, and Non-Euclidean Geometry........................ 123 Lecture Thirty-Four Culture and Mathematics in Classical China .......................................... 127 Lecture Thirty-Five The Voice of the Critics................................. 131 Lecture Thirty-Six Mathematics and the Modern World.............. 136 Timeline ................................................................................................... 141 Glossary ................................................................................................... 146 Biographical Notes.................................................................................. 154 Bibliography............................................................................................ 164 iv ©2009 The Teaching Company. Mathematics, Philosophy, and the “Real World” Scope: How has mathematics changed the way people look at the world? Surprisingly, throughout the Western tradition, views about human nature; religion; philosophy; truth; space and time; and works of philosophy, art, politics, and literature have been shaped by ideas and practices from mathematics. The most influential mathematical shapers, geometry and statistical thinking, are quite different from one another and reflect the contrasting ideas of certainty and unpredictability. Euclidean geometry was used by philosophers to argue that truth could be achieved by human thought and that nonmaterial objects could in some sense be known. Later, the disciplines of probability and statistics provided a way of understanding—with precision—those events in the physical and social worlds that seemed to come about by chance and follow no laws at all. In these lectures,
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