# Essentials of Mathematical Thinking

With a title like "Essentials of Mathematical Thinking" one might expect a philosophical treatise, or possibly a research exposition about cognitive processes and math education. But at the top of the cover, you can see that it is announced as a "Textbook in Mathematics". Since that is what it is: a textbook in mathematics, but a rather unconventional one. Several writers of popular science or recreational mathematics have written books in which they collect mathematical topics that are accessible for a general public and that should illustrate that mathematics can be fun and that there are many practical applications in everyday life involving mathematics. The items discussed in these books can involve integers, prime numbers, geometry, probability, counting problems, logic and paradoxes, games, puzzles, etc. But they are mostly "recreational" or at most they can serve as a source of inspiration for math teachers to embellish their courses and candy-coat the theorems and proofs of the actual textbook.

Here however, Steven Krantz uses all these entertaining subjects to use them as an actual textbook to teach mathematical awareness and some skills to students who have not the slightest ambition of using mathematics in their further career. For example if undergraduate students are required to broaden their curriculum with some math course. There is no point in imposing mathematical abstraction on them or to force them to memorize proofs of theorems they will never need in life. So the idea is to use all these entertaining subjects to develop their ability to use logic arguments, to solve problems, and to convince them that mathematics is indeed everywhere, but that it is nothing to be afraid of. They will not become better mathematicians in the narrow sense of the word, but at the end of the journey they should have acquired some skills one could call mathematical and they should be more open minded towards mathematics and mathematicians.

Obviously the book should not have the usual definition-theorem-proof structure and, although formulas have not been abolished completely, there are fewer than in a classical textbook. There are some exercises, but the usual long lists of drilling exercises are absent. Some exercises are meant to drill, some can be more challenging, and chapters are concluded with an open-ended problem. It is like Krantz is telling his story in a stream of consciousness which results in a surprising meandering succession of ideas that will hold the attention of his public or his readers.

In Chapter 2 the breadth of the field is explored covering many different problems. Some examples: the Monty Hall problem, the four colour problem, minimal surfaces, P vs. NP, Bertrand paradox, etc. This sounds impressive as a starter, but these are actually pretexts for introducing the reader to probability, logarithms,... and to modern tools such as proofs by computer, algebraic computer systems, etc.

This is followed by seven relatively short chapters. Now problems are solved. Some are classic (When will be the first time after midnight that the hands of an analogue clock will coincide?) others are less classic (Will new years day fall more frequently on a Saturday than on a Sunday? How many trailing zeros will 100! have?...).

To illustrate how ideas are linked, let us consider a section of Chapter 5 as an example. It starts by telling that Kepler derived his laws for the motion of the planets not by solving equations but by analysing observed data (Newton came later), which leads to the meaning of average and standard deviation, which in turn leads to big data and their analysis such as DNA used in forensics, social studies based on Street View and other big sets of data collected by companies such as Google. A remarkable arc that connects Kepler to Google.

Furthermore many of the classics are passing by on the catwalk: the pigeonhole principle, conditional probability, Benford's law, lottery and roulette problems, Conway's Game of Life, Towers of Hanoi, Buffon's needle problem, Euler's characteristic, sphere packing, Platonic solids, voting systems, interpretation of medical tests, facial recognition, wavelets, prisoner's dilemma, Hilbert's hotel and others. Some of these are worked out and actually solved, others are only mentioned as illustration of what is possible, or what they have been used for.

Up to this point, the text is easily accessible with minimal mathematical background. In the remaining chapters, somewhat more is needed. Chapter 10 is about cryptography (explaining the basics of RSA encryption), the next one gathers some diverse discrete problems (a.o. divergence of the harmonic series, surreal numbers, graphs and the bridges of Königsberg, scheduling problems), and finally a chapter with more advanced problems (Google's Pagerank, needle problem of Kakeya, non-Euclidean geometry, the area of a circle as the limit of the area regular polygons).

Besides mathematical monographs, Steven Krantz has written books on how to write mathematics, and some books that may be considered as introductions to mathematics for a general public and he won several prizes for his writing. He wrote also one on mathematical education before: *How to Teach Mathematics* (3rd ed., AMS, 2015) which is about "how" one has to teach. This one is about "what" to teach to a particular type of students. Whether he has experience teaching the "essentials of mathematical thinking" using the material presented in this textbook, I do not know. It might not be a bad idea for students that are somehow obliged to take a math course but that have no the intention to take subsequent courses. I have no information about experiments with this type of course. It would certainly be interesting to know the results.

The text is typeset in LaTeX with the quality of lecture notes. There are many illustrations, but pictures do not have the resolution of high professional quality, and some are not really necessary (a picture of 3 arbitrary dice is not really helpful in solving a probability problem). There are many line drawings too which are usually quite helpful, but by resizing them to fit properly on the page, sometimes circles are distorted and become ellipses or the text in the figure is stretched and resized out of proportion.

**Submitted by Adhemar Bultheel |

**25 / Nov / 2017