Alexandre Borovik is on a quest to understand how we think about mathematics. And by “we” I mean not only mathematicians but people more generally. Why is it that some mathematics is hard and some is easy? How is it that small mathematical activities build on themselves to form much larger mathematical activities, and to what extent are the thought processes the same? Why is it that our brain parses certain mathematical expressions in the way that it does? What are the ways that mathematics is shaped by the way that our brain works? These might sound like questions that a neuroscientist or cognitive psychologist would be better suited to answer than a mathematician, but Borovik feels that he — and mathematicians more generally — have something of substance to add to the conversation and that if we don’t get involved in the conversation soon then our voices will be crowded out. He blogs about this topic frequently at Mathematics Under The Microscope, and has written a book by that title which has been recently published by the American Mathematical Society.
Other authors have thought about the underlying thought processes behind mathematics: Keith Devlin’s The Math Gene and Lakoff & Nunez’s Where Mathematics Comes From are two examples from recent years. However, Borovik feels that these works concentrate too heavily on the cognitive processes related to counting and enumeration rather than those related to problem solving and higher mathematics, and hopes that his book will address mathematics in its full scope as “the study of mental objects with reproducible properties”, a definition that he attributes to Davis & Hersh and which he uses repeatedly.
In roughly 300 pages, Borovik touches on a wide range of topics. He begins by discussing what he claims is the simplest possible example of a nonlinear function, y = x. He shows how even this example can be used to construct extremely complex examples, which leads into a discussion of the difference between the “switch” mode of computation used by discrete mathematicians and the “flow” mode of computation used by others. This leads Borovik to a study of computational complexity and polynomial time algorithms, which then leads to the idea of “choiceless” algorithms. From there, Borovik moves on to think about analytic functions and the cubic formula and the history of algebra. Somehow, this flows into a discussion of tropical geometry and its applications, before looking at what Borovik refers to as the “emotional side” of mathematics and the question of why some repetitive activities, such as Sudoku puzzles, are pleasurable while others, such as GaussJordan elimination, are not. Finally, he discusses bubble wrap, and some studies by neuroscientists about why it is so darn addictive.
When I say “finally”, I should point out that I mean “finally” in the first chapter, as all of these topics are covered in less than 20 pages. The rest of the book continues to go through topics at a breakneck speed, as a quick tour through some of the titles of the sections shows. A sampling includes:

Image Processing in Humans

Coxeter’s proof of Euler’s Theorem

How to draw an icosehedron on a blackboard

What about music?

Cognitive nature of cryptomorphism

How humans manage to lose to insects in mind games

Can one live without actual infinity?

Hedy Lamarr

Kolmogorov’s 5/3 law

The Post Office Conjecture

Keystones, arches, and cupolas

Mathematics and origami

The vivisection of the Cheshire Cat

Explication of (in)explicitness
After hearing all of this, it will probably come as no surprise to learn that Borovik’s book is extremely disjointed, and at times feels very sloppy. It feels much more like a collection of blog posts than it does a cohesive book, and perhaps would have benefitted from some more serious editing. In particular, Borovik spends more time telling the reader what he is going to tell them than actually telling them, which is possibly a metacomment on cognition, but felt more like roughness around the edges. Unlike in most books of mathematics, Borovik has a rather strong point of view and is willing to share it. In fact, he insists upon it. Whether it is a point of view you are interested in hearing is hard for this reviewer to judge, but despite its flaws I really enjoyed the few evenings I spent digesting this book, and I imagine that some of the ideas will stick with me for quite some time. There is no question that there are many interesting ideas contained in the book, and with apologies to any grammarians who cringe at the phrase, Mathematics Under The Microscope is a very unique book. As he writes on his blog, “Only my readers can judge whether my books are good or bad. But they have no analogues … otherwise what was the point of writing them?”
Darren Glass is an associate professor of mathematics at Gettysburg College, where he thinks more about algebraic geometry and cryptography than he thinks about thinking about those things — or thinking about thinking about thinking about them. He can be reached at dglass@gettysburg.edu