## Mathematics and the Real World: The Remarkable Role of Evolution in the Making of Mathematics (2014)

### Preface

There are many jokes about mathematicians. One of my favorites is about an engineer, an architect, and a mathematician who have been sentenced to be hanged. In the evening before the day set for the execution, the warden asks them for their last requests. The engineer asks to be allowed to present a new machine he has designed that can perform all household chores without any human intervention. The warden promises that the next day, before the hanging, he will have one hour in which to show his machine to the prison staff and his two fellow death-row inmates. The architect asks to be allowed to explain his new concept of residential accommodation, a modern house that keeps cool in the summer and warm in the winter, without expenditure on fuel. Again the warden promises that the next day, before the hanging, he will have one hour in which to present his idea to the prison staff and his two fellow death-row prisoners. The mathematician says that he has recently proved a mathematical theorem that will shake the foundations of mathematics, and he would like to reveal it in a lecture to an intelligent audience. The warden starts agreeing…and the engineer and the architect start shouting: “We want our execution to be brought forward to this evening!”

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