Mathematics and the Real World (2 page)

This joke appeals to me because it reflects the widespread attitude of the public to what can be expected from books and lectures on mathematics. We will give the reasons for this attitude later, and here we will just note that even in school we are exposed to indoctrination that causes us to relate differently to texts and lectures on mathematics than we do to other subjects. In school students are expected to solve mathematical exercises to show that they have understood the material. Other subjects such as history, literature, or even biology do not require such exercises. The impression that this creates is that without solving exercises there is no point in listening to mathematics. The development of intuitive understanding of a subject, without practicing what has been learned, is not
accepted as understanding in the case of mathematics. That is so despite the fact that an intuitive grasp of a subject, without needing to put it into practice, is an acceptable objective in other scientific and general disciplines. This is misguided and misleading indoctrination that does an injustice to mathematics. Furthermore, that approach is alien to professional mathematicians too. Of course they must have a deep understanding of the topics they are researching, but an intuitive understanding of other mathematical subjects is sufficient. I will put forward an analogy that I would ask you to keep in mind as you read this book.

I love classical music and regularly attend concerts of the Israel Philharmonic Orchestra, and I greatly enjoy both live performances and recordings. I cannot read music, and I do not know the detailed history of music or the life stories of the different composers. I am confident that those who can read music or are familiar with the history of music enjoy what they hear in a way that is different from my enjoyment. I am not sure if they enjoy it more than I do because, for example, they may be conscious of any note played slightly inaccurately, whereas I would be totally oblivious of it. The experts understand the compositions on different levels from mine, but I enjoy the music immensely; perhaps not from the written notes, but from the tune. Not the trees, but the forest. There are hardly any “notes” in this book, nor trees, mainly a tune, mainly the forest. If one or a few notes appear here or there (at times using a different font, and preceded by a rule line), they can be skipped without breaking the thread of the text.

The different sections of the book are connected, but the ideas are presented in such a way that each section is self-contained and can be read independently of the others. The headings and titles of the sections and chapters indicate the central elements within them. It is advisable to start with
chapter I
, but then the reader can certainly go straight to the chapter on the mathematics of randomness or to the one on the mathematics of human behavior, or even jump to the
last chapter
on teaching mathematics.

Naturally, a book like this could not have been written without information, exchange of views, and help that I received from friends, colleagues, students, those who heard lectures on the topics covered in the book that I delivered in various forums, the translator and the editor, the
publisher's team, and, of course, my family. There are too many people for me to be able to enumerate each one of them here. To all of them, my sincere thanks.

So what is it all about: The books deals with
the mathematics of nature
,
the nature of mathematics
, and their interrelationship. We will describe, by means of a historical review as well as from the aspect of current research, the link between mathematics and the physical world and the social world around us. The discussion in the book also relates to areas of science and society to which mathematics is relevant. We will therefore also present scientific facts and social situations described by mathematics. That presentation is not exhaustive or detailed, as we focus on the mathematical aspects of the various fields. The discussion will be accompanied by the constant presence of the question regarding the extent of the effect of the
evolution of the human race
on the development of mathematics and its applications. We will examine the claim that the manner in which the human brain was fashioned by millions of years of evolution affected humans’ mathematical capabilities and the type of mathematics that is easy for humans to develop and understand. We will also show that, to a large extent, evolution is responsible for the difficulty we have in understanding certain other areas of mathematics. We will try to do all that with a minimum of musical notes but with much pleasing music.

Zvi Artstein

The Weizmann Institute of Science

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 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.