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.
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About the Author
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.
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Mathematics and the Real World
The Remarkable Role of Evolution in the Making of Mathematics
By Zvi Artstein, Alan Hercberg
Prometheus BooksCopyright © 2014 Zvi Artstein
All rights reserved.
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?
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 that enabled it to win the battle for survival.
Darwin's great contribution to the theory of evolution was in identifying the mechanism by which different species of animals and plants changed and developed. Unlike Lamarck, who claimed that every species adapts to its environment and the characteristics it takes on are passed from each generation to the next, Darwin proposed a different mechanism responsible for the changes that every species undergoes. The mechanism consists of two main elements: mutation and selection. In the reproduction process, individuals undergo mutations that cause random and generally minor changes in their characteristics. The individuals with the best-suited characteristics reproduce at the fastest rate, and that constitutes the selection that results in successive generations of each species being better suited to the environmental conditions. The best-suited species among those competing for the same food resources are the ones that survive.
Charles Robert Darwin (1809–1882) was born in the town of Shrewsbury, England, to a well-established family. His father, Robert, was a wealthy physician. His grandfather, Erasmus Darwin, who died before Charles was born but whose writings were available to Charles, was a philosopher and a naturalist who favored the theory of evolution of the leading French naturalist Jean-Baptiste de Monet, Chevalier de Lamarck (1744–1829), known simply as Lamarck. Young Charles was exposed to scientific endeavors but was not a particularly industrious student. Instead of devoting his time to his studies, he preferred to observe nature and to collect various items, particularly beetles of different types. When he was twenty-three he was invited to join an expedition due to sail on a ship called the Beagle as its scientist; the main purpose of the expedition was to chart the shores of Australia and South America for the British Empire. His role was to collect and classify geological, zoological, and botanical specimens. In the course of the voyage Darwin noted that different but similar species could be found living in regions near each other, specifically on the Galapagos Islands. It was there that he conceived the model of evolution consisting of mutations and selection. It should be noted that Darwin, in developing his theory of the evolution of types of flora and fauna, was deeply influenced by the theory of the political philosopher Thomas Malthus (who lived half a century before him) on the demographic and economic development of human societies. Darwin's autobiography shows him to have been a modest and wise man; it is replete with statements indicating a deep understanding of evolution beyond the technique he developed. For example, in his reference to his elderly colleague Leonard Jenyns, with whom he had many discussions about nature, Darwin writes, "At first I disliked him from his somewhat grim and sarcastic expression, and it not often that a first impression is lost, but I was completely mistaken...." The connection between "it is not often that a first impression is lost" and evolution will be discussed further on in the book.
Although Darwin shared his thoughts on evolution and the ample evidence he had found supporting his theory with his scientific colleagues, among them some of the best-known English scientists in those days, he refused to publish his findings. He agreed to publish his theory only after Alfred Wallace, a young naturalist researcher who had undertaken many voyages to South America and the Far East, submitted a paper for publication containing ideas similar to those of Darwin but with only a flimsy basis. Darwin's friends became aware of this and urged him to publish his book, The Origin of Species. As a result, in the first official presentation of the theory, Wallace's article and Darwin's theory appeared at the same time.
There were several reasons for Darwin's long hesitation before publishing his conclusions. Some derived from the possible conflict with the religious belief that different species exist because that is how they were created. Darwin's wife, Emma (née Wedgwood), was deeply religious, and Darwin did not wish to upset her. Another reason, however, no less important, for his hesitation to publish his mechanism of evolution was that despite the wealth of facts he had available supporting his theory of evolution, many aspects of the theory had not yet been demonstrated and lacked a scientific basis. In particular, Darwin could not offer a biological mechanism that would cause mutation. That mechanism was not discovered until the middle of the twentieth century, when the genes encoded in DNA molecules were revealed and were found to mutate randomly in the course of reproduction.
Mutation and selection of genes are the basis of the modern understanding of the process of evolution of plant and animal species. The genes carry the characteristics vital to the survival and development of each species. The ensemble of genes and the way they are expressed, at times as a reaction to the environment, define the characteristics of a species. Changes in the genes are responsible for changes in the species. Nevertheless, much can be learned about evolution without monitoring the changes in the genes themselves. By studying the conditions in which the species developed, survived, and was victorious in the evolutionary struggle, we can learn about the characteristics that are encoded in its genes and are passed on from generation to generation. The reverse of this claim is also correct: the characteristics observable at any given time enable us to learn about the conditions in which each species developed.
The following example shows how we can learn about the link between the conditions in which species developed and their characteristics today. I came across this example in the course of a trip I made to the Galapagos Islands a few years ago. It relates to the mating habits of birds.
The rock cormorant in the picture on the left cannot fly. It lives on cliffs close to the seashore that are exposed to strong winds. Its abilities to find twigs and to build a proper nest are vital to the survival of the species in these tough conditions. In its courtship, the male cormorant demonstrates to his potential mate his ability to gather twigs to build the nest they would share. The courted female will respond to the male's advances only if he can prove he has that capability. The middle picture is of the frigate bird. In its courting display, the male inflates his gular pouch until it becomes an enormous red balloon. He does this to show his intended mate the strength of his lungs and his ability to fly long distances to scoop fish out of the water. The picture on the right, the blue-footed booby, demonstrates entirely different qualities in its courtship display. The male of this species incubates the eggs and protects them by covering them with his large blue feet. He therefore tries to woo his potential mate by flaunting the size and shape of its feet, thus proving his ability to protect the eggs they will produce together against enemies and unsettled or rough weather.
These examples illustrate that the characteristics and behavioral patterns we can discern today indicate the characteristics that were of evolutionary importance, and those show us how each species survived the evolutionary struggle.
The essential characteristics that helped any particular population to win the battle for survival during the formation of the species are etched into its genes, and we may identify them as innate attributes. Cheetahs' speed, eagles' "eagle eyes" and cats' tree-climbing ability are all innate traits. A cheetah cub is born with the ability and basic instinct to run fast. It will need help from its parents to learn what it needs to fear, how to hunt, and even how to run more effectively, but the basic features of speed and hunting are carried in its genes. Similarly, the genes of a cat enable it to learn how to catch mice, and the innate characteristics of an eagle include its keen vision and ability to identify potential prey from a great height. Learning merely refines and improves the innate attributes. The attributes of each species enable us to learn about the conditions in which the species developed; similarly, knowing the conditions in which the species developed enables us to learn about the characteristics that evolved.
It is reasonable to assume that just as physical attributes of animal species are innate features etched into their genes, the same will apply to at least some mental attributes. Mental and social skills also play a role in the battle for survival, so that in these too, the selection process strengthens the features that help the species to overcome its rivals. Specifically, in the reproduction process mental attributes can also be changed and improved by mutation. In the following sections we will examine the mathematical capabilities of the human species from an evolutionary aspect. We will ask whether the abilities to understand and to use mathematics are the results of evolutionary development, or whether they may be by-products of a brain that developed to cope with other needs.
2. MATHEMATICAL ABILITY IN THE ANIMAL WORLD
If mathematical ability made a contribution in the evolutionary struggle that brought the human race to the position it currently occupies among the species, it may be assumed that other living beings would possess a certain degree of mathematical ability. But what does mathematical ability mean? Mathematics encompasses a broad range of topics and conceptual methods. The question to ask, therefore, is which of those mathematical features provide an evolutionary advantage? And the follow-up question is how can we identify these mathematical abilities in animals?
The most basic mathematical ability is counting. It is followed by the understanding of the concept of a number as an abstract object and the ability to perform simple arithmetic operations, such as addition and subtraction. We will start by discussing the existence of these simple elements in adult animals. A mother cat moves her kittens from place to place and generally does not forget a kitten or two, and when she has finished moving them, she does not usually go back again to check whether she has moved all of them. She may remember them individually, but it seems reasonable to state that the mother cat has a sense of quantity. The instinct of quantitative estimation clearly provides an evolutionary advantage, so we should not be surprised that adult animals possess that ability. But does that ability extend to the ability to count and to the possibility of performing arithmetical manipulations?
Before presenting several convincing examples showing that some animal species do have mathematical ability, a warning is in order. The results of experiments in general, and of animal experiments in particular, should be interpreted with great caution. A well-known illustration of this is the case of "Clever Hans." (More details and references concerning this story, and concerning the research mentioned later in this section, can be found in the monographs by Dehaene and by Devlin  listed in the sources.) Toward the end of the nineteenth century a horse known as Clever Hans was exhibited on tour in Germany with its trainer, Wilhelm von Osten. The horse showed remarkable ability in adding, subtracting, finding the squares of numbers, simple division, and so on, all with a very high degree of success. The horse was wrong occasionally, but such errors occurred infrequently. The method by which the horse showed its abilities was that when an exercise was read out or written on a board, it would tap its hoof the number of times corresponding with the right answer. It was suspected that the act was simply a clever deception by which the trainer somehow or other managed to give the right answer to the horse. An official committee was appointed, headed by a psychologist named Carl Stumpf and whose members included the director of the Berlin Zoo. The committee checked, among other things, whether the horse could solve the problems if the trainer was not present, and it found that even then the horse could still give the correct answers. The conclusion was that some animals have a fairly advanced mathematical ability. Subsequently, more detailed examinations in 1907 by another psychologist, named Oscar Pfungst, showed that the horse did not know mathematics. The trainer was indeed reliable and honest, but the horse had learned to distinguish involuntary changes in his facial expressions and in the facial expressions of the audience when the trainer was not present. The horse understood from those facial expressions when it had reached the correct number of taps of its hoof. The presence of the trainer or an audience was essential. Pfungst found that if the trainer looked tense at a wrong answer, the horse answered according to the expression and not the correct answer. The research methods developed by Pfungst as a result of this case are now recognized as a breakthrough in psychological research.
Scientific experiments that were more soundly based have proven that some animals do indeed possess mathematical ability. The German zoologist Otto Koehler (1889–1974) proved as early as in the 1930s that some species of birds can identify a collection with a given number of elements. It is apparently not difficult to train a pigeon to choose every third seed when faced with a row of seeds. A squirrel can be trained so that when faced with boxes containing different quantities of nuts, it will choose the box with exactly five nuts. There is a limit to the numerical-identification ability of these animals. Koehler himself found that even the most capable animals could not identify collections with more than seven elements. The number appears in the literature also as a bound to the number of information units that a human brain can process. We will meet the number seven again later on in similar contexts. Still, these experiments demonstrate the mathematical ability to estimate quantity but do not yet prove an ability to count or to grasp the abstract concept of a number.
Excerpted from Mathematics and the Real World by Zvi Artstein, Alan Hercberg. Copyright © 2014 Zvi Artstein. Excerpted by permission of Prometheus Books.
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Table of Contents
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,
Index of Names, 417,