In this accessible and illuminating study of how the science of mathematics developed, a veteran math researcher and educator looks at the ways in which our evolutionary makeup is both a help and a hindrance to the study of math.Artstein chronicles the discovery of important mathematical connections between mathematics and the real world from ancient times to the present. The author then describes some of the contemporary applications of mathematics-in probability theory, in the study of human behavior, and in combination with computers, which give mathematics unprecedented power.The author concludes with an insightful discussion of why mathematics, for most people, is so frustrating. He argues that the rigorous logical structure of math goes against the grain of our predisposed ways of thinking as shaped by evolution, presumably because the talent needed to cope with logical mathematics gave the human race as a whole no evolutionary advantage. With this in mind, he offers ways to overcome these innate impediments in the teaching of math. Zvi Artstein is the Hettie H. Heineman Professor of Mathematics at The Weizmann Institute of Science, where he has worked for over thirty-eight years as a scientist, a teacher, and an administrator. He is the author of more than 120 scientific articles published in peer-reviewed journals. Mathematics and the Real World The Remarkable Role of Evolution in the Making of Mathematics By Zvi Artstein, Alan Hercberg Prometheus Books Copyright © 2014 Zvi Artstein All rights reserved. ISBN: 978-1-61614-091-5 Contents Preface, 11, CHAPTER I. EVOLUTION, MATHEMATICS, AND THE EVOLUTION OF MATHEMATICS, 15, CHAPTER II. MATHEMATICS AND THE GREEKS' VIEW OF THE WORLD, 67, CHAPTER III. MATHEMATICS AND THE VIEW OF THE WORLD IN EARLY MODERN TIMES, 91, CHAPTER IV. MATHEMATICS AND THE MODERN VIEW OF THE WORLD, 129, CHAPTER V. THE MATHEMATICS OF RANDOMNESS, 181, CHAPTER VI. THE MATHEMATICS OF HUMAN BEHAVIOR, 227, CHAPTER VII. COMPUTATIONS AND COMPUTERS, 275, CHAPTER VIII. IS THERE REALLY NO DOUBT?, 315, CHAPTER IX. THE NATURE OF RESEARCH IN MATHEMATICS, 343, CHAPTER X. WHY IS TEACHING AND LEARNING MATHEMATICS SO HARD?, 381, Afterword, 411, Sources, 413, Index of Names, 417, CHAPTER 1 EVOLUTION, MATHEMATICS, AND THE EVOLUTION OF MATHEMATICS Could evolution have affected mathematics? • Can horses perform calculations? • Can rats count? • Do infants solve problems of addition and subtraction? • Which rectangles please us? • Why are clowns scary? • What color are the sheep in Ireland? • What number comes next in the sequence 4, 14, 23, 34, 42, 50, 59, ...? • Why square the circle? • How have optical illusions contributed to science? 1. EVOLUTION The theory of evolution is attributed to Charles Darwin, but it was not Darwin who initiated the study of evolution. King Solomon stated, "There is nothing new under the sun," (Ecclesiastes 1:9), a philosophical statement alluding to the observation that the world is in a constant state of flux. At any given time we see the current situation around us, but we also follow changes that take place in our lifetime, and we are aware of changes occurring over periods of time that we are unable to observe directly. The evidence regarding changes that took place in the past often enables us to infer what caused those changes. That applies both to the physical world, such as rocks, flora, and fauna, and to society, including modes of behavior, fashion, literature, medical practices, and technology. The changes take place according to their own mechanism. Sometimes it is clear to us what survives, what is modified, and what becomes extinct, but it is not always easy to identify the mechanism. Let us take as an example the Earth's surface. Some rocks exist for many years, while others are weathered and eroded by the wind almost as we watch. What causes the difference? Clearly it is the different rock textures that determine the differences in their ability to survive. Basalt will last, while limestone will crumble. There are no sand dunes on the tops of mountains because they would be blown away by the wind. We could say that the strong triumph, the fittest survive. We can deduce that being made of basalt is an advantage in the battle for survival on a mountain peak. That statement is trivial in the realm of rocks, and we do not usually examine rocks in terms of the competition for survival. But the conclusion that whatever is most suited to its environment survives is correct regarding rocks as well as human society. Historians, in discussing human history, try to understand why a particular society survived and another disappeared. Their conclusions generally refer to the advantages that the victors had over the vanquished. We can learn about the conditions under which a society or species developed from its specific characteristics. Likewise, from the conditions in which it developed, we can learn about the advantages th