The subtitle *The Remarkable Role of Evolution in the Making of Mathematics* of this book made me expect an explanation of why mathematics has evolved to its current state, and how this was triggered by a "survival of the fittest" kind of mechanism. That, at a very early stage, you could count while others couldn't, would indeed have given you a strategic advantage. In our current technological society, it might have a slight advantage in certain communities if you show some mathematical skills, but the survival aspect has certainly diminished, if not totally disappeared. It will still come in handy if you can multiply, but you can perfectly survive without knowing whether the Zermelo-Fraenkel system is consistent or not. Hence somewhere in between, the practical necessity of doing mathematics has been lifted to abstract and less intuitive reasoning, and that started as early as with the ancient Greeks. Mathematics were done earlier, but that was utilitarian, e.g., for commercial reasons or to construct buildings etc. The Greeks were the first to be free enough to start pondering on fundamental questions. According to Artstein because they wanted to prove things where their senses failed like in optical illusions.

So, it is the intention of Artstein to shed some light on the discord between the intuitive human attitude towards mathematics and the other, less natural way of doing mathematics that has to be learned and that requires effort and investment. The natural, intuitive, genetic kind of mathematical capabilities that humans are born with are the result of millions of years of evolution. This is unlike the mathematics that are constructed, wrought artificially, and thus are to some extent unnatural. At least, they do not give an obvious evolutionary advantage. However, according to Artstein, it are exactly the latter kind of constructs that dominate the way mathematics have evolved. And this had consequences, not in the least on our way of teaching mathematics. In the successive chapters Artstein sketches in an historical context, how and where mathematics has left the natural, intuitive road of human evolution and he conludes with a plea for a adapted educational system that is more conform our human nature.

There were different competing views of the world among the Greeks. The obsession of the Pythagoreans with integer numbers led to an atomistic view of the world. Plato's view is that mathematics exists and that people have to discover it. His student Aristotle however claimed that mathematics were created from an axiomatic system. They also had ideas about the earth being a sphere and proposals for a geocentric as well as a heliocentric system based on epicyclic or circular (i.e. mathematically perfect) trajectories. All these things were still under discussion during many subsequent centuries and it is often referred to in the subsequent chapters of this book.

So that is where Artstein picks up the evolution of mathematical history. He arranges this by application domain. The central theme in the next two chapters is the evolution of physics and the necessary mathematical tools focussing on an explanation for the place of the earth in our solar and later in a cosmic framework: from Copernicus to string theory. The Aristotelian view "mathematics is a (very) good approximation of nature", is still current today, although an (applied) Platonic vision "nature is a very good approximation of mathematics", seems to gain more ground lately.

The mathematical approach to randomness as well as some other mathematical evolutions are relatively recent. These are the subject of the next chapters. If you are able to deal with randomness and make proper predictions, then this can give you an evolutionary advantage. The Greeks gambled, but it was only in the 17th century that mathematicians (Pascal, Fermat) started to show interest and laid the foundations of probability theory. However the rules that have been deduced (e.g. Bayes' law) have to be applied properly or the wrong conclusions will be drawn, and on the other hand there are some correct results that are definitely counter-intuitive. Mathematics is as yet less embedded in social sciences, although one may spot some recent evolutions: there are different mathematical models for voting systems, game theory is applied to human interaction and negotiation, dating services select an ideal partner, and economists analyse the market. Computers form another recent phenomenon that has greatly influenced not only mathematics (e.g. how conjectures are obtained and theorems are proved) but the whole society, e.g. how people interact with machines and with each other.

Given all these historical elements, Artstein now reflects on the nature of mathematics and prepares for his arguments to be used in the last chapter where he gives his vision on how mathematics should be taught. Since the Greeks, mathematics started from (geometric) axioms. It was only in the 19th century that analysis took over from geometry and then numbers lost their intuitive meaning when all was based on sets and their cardinalities and real numbers became Dedekind cuts. This eventually resulted in a crisis of the system when Gödel proved his incompleteness theorem. At this moment we do not know if there exists a mathematics without contradictions, in which everything is irrefutably true or false.

The next chapter is a rather realistic sketch of what mathematical research mean, i.e., what does a mathematician do all day when he is doing research? First of all: how does he think? Artstein makes the distinction between thinking by comparison (i.e., how do we solve this problem given a previous solution found for a similar problem) and creative thinking (to solve totally new problems or finding solutions unlike any other solution found before, i.e., thinking "outside the box", which may require to reject a century-old belief). Major advances are the result of the latter performed by few geniuses, but they rely on the former performed by many on a daily basis. Results may come erratically. Problems can mature unconsciously and solutions may come when you wake up or when you are on holiday when you least expect it. There are pure and there are applied mathematicians, although all pure mathematics may become eventually applied. Hardy loved number theory because he thought it to be the least applicable part of mathematics, and yet, nowadays almost all our encrypted communication used by billions of people every day relies on it. And then there is the (by mathematicians) much acclaimed beauty and universality of mathematics. Beauty is in some way related to symmetry or patterns. Sometimes patterns can be hidden behind chaotic behaviour like in nonlinear dynamical systems. Perhaps our mathematics is still incomplete because nature has more (hidden) patterns that we, humans, cannot, or at least have not yet, detected.

In the last chapter Artstein is defending his strong and outspoken views on mathematical education. He certainly opposes an axiomatic approach for children in elementary school. They can count when they start, so do not confuse them with sets, cardinality, commutativity, and the likes. In fact his views come close to the Bénézet system, and to some extent also in the Waldorf/Steiner education system. Louis Bénézet pioneered an education experiment in 1929 in Manchester, New Hampshire where no formal mathematics were given. Problems were solved with trial and error as they occurred. Formal symbols or a method were only introduced occasionally and only in connection with that particular problem. Some tests seven years later showed that these children performed better than those that had a traditional mathematical education. Formalism and definition-theorem-proof structures are acceptable in books and perhaps for mathematical professionals, but they will only cause frustration in children. An intuitive approach may take this frustration away.

The historical part is the broadest and clearest, i.e., the most accessible account of mathematical history that I know. It is fully understandable for any mathematically uninitiated reader. Examples, rather than general principles make things clear. If some explanation is inserted that is a tiny bit more technical, then it is set in a special font and these paragraphs can be easily skipped without affecting readability. Artstein is professor at the Weizmann Institute in Israel, and that may be why many of the examples involve Jewish mathematicians (i.e., more than in other similar books). Also the critique in the last chapter on mathematics education involves a.o. the author's experience in Israel when his son was attending elementary school and may differ from the current situation and from the situation in other countries.