Overview:
Mathematics has a very long logical tradition, but it is far more than just logic. Within the logic, geometry, infinity, probability, and statistics, are found applications to real life, science, and the history of math itself. With a psychological undercurrent, the book teaches how decisions are made, how to think, how to reason, the role of perception and beliefs on thoughts and decisions.

Math demands precision, can be founded on imprecision. Math sometimes requires belief in answers devoid of experience and the senses. What can be done mathematically, cannot normally be done in reality. Making math not a perfect representative for reality. But the unrealistic concepts, which themselves are just logical extensions of practical concepts, facilitated in understanding concepts that do impact reality.

Additional differences between reality and math, is that while natural numbers are ordered, objects found in the real world are not ordered. No ordered sequence to be followed. Within things that go on for infinity, becoming is merely appearance. But we do not actually know much beyond the finite. What logic does is spot contradictions between premises and the conclusions. Logic can make statements invulnerable, but logic can also be devoid of meaning. Logic alone cannot persuade.

Caveats?
There are many mathematic puzzles in this book, which can cause the reader to slow down, in order to think about them more deeply. Slowing down can mean breaking the link with the narrative.

As this is primarily a book about math, having an interest and some background in math can help bring more understanding to the topics. Some parts can be difficult to understand, but the book does reveal the beauty in math. ( )

This book is ideal for a smart, but mathematically-untrained adult who wants orientation to some interesting and useful mathematical concepts, as well as a portal to mathematical epistemology. A little high school algebra and geometry, even if it's fairly dusty, is plenty of background to appreciate the explanations here. Author Mazur builds his discussions of deduction, induction, probability, and plausibility into a series of autobiographical and historical anecdotes, within sections titled "Logic," "Infinity," and "Reality." I really appreciated his explanation of transfinite numbers, a modern math concept that has always eluded me. The book has an apparatus with useful directions for further reading.

Some of the little stories in the book were ones I found difficult to credit, but perhaps that was a savvy way of giving the reader a motive to further ponder the notion of plausibility!

"Whimsical" and mathematically "rigorous" are two concepts that seldom apply to the same book, but they do to Joseph Mazur's "Euclid in the Rainforest." ("Goedel, Escher, Bach" comes to mind.)

Mazur develops three separate way of thinking mathematically: the classical logic of the Greek geometers, the weird logic of infinity, and probalistic logic.

He takes us through several proofs of the Pythagorean theorem. He distinguishes between logical induction and mathematical induction and then illustrates the latter with proofs that (1) the square root of 2 is irrational; (2) there is not a largest prime number; (3) between every 2 rational numbers there is an irrational number; (4) between every 2 irrational numbers there is a rational number; and (5) that the cardinality of the irrational numbers is greater than that of the rational numbers.

He also shows Cantor's proof that the real numbers are uncountable. Finally, he adds some counterintuitive examples of probability theory and a discussion of decision making under uncertainty and the normal curve.

Mazur tries to make the rigor painless by illustrating some of the proofs in the context of anecdotes about some of his quirky travel and teaching experiences. That technique can be annoying in the hands of a less skill writer, but I think Mazur pulls it off rather well. All in all, a pleasant read for the non-mathematician.

JAB ( )

Gentlest of intros to mathematical thinking, diluted with personal anecdotes.