Essentials of Mathematical Thinking

Steven G. Krantz is a professor of mathematics at Washington University in St. Louis. He has written more than 65 books and more than 175 scholarly papers and is the founding editor of the Journal of Geometric Analysis. An AMS Fellow, Dr. Krantz has been a recipient of the Chauvenet Prize, Beckenbach Book Award, and Kemper Prize. He received a Ph.D from Princeton University.

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.~Adhemar Bultheel, European Mathematical Society 2017

• First Thoughts. Diverse Mathematical Thoughts. Strategy. Focus. Science. Counting. Games. Geometry. Practical Matters. Breaking the Code. Discrete Problems. Advanced Ideas. Concluding Remarks