Review of The Best Writing on Mathematics 2012


In describing the essays in this volume as the “best” writing on mathematics, the word “best” can’t be taken literally. For one thing a mathematician would naturally point out that there is no simple, obvious linear ordering on the set of writings about mathematics that reflects “quality”. There is certainly no metric to quantify quality for this type (or any other type) of writing. Specifying a particular audience that might have useful opinions on quality would help, but still be inadequate. That said, let’s assume the audience is at least people who understand some mathematics and who appreciate it and value it. That’s still a number of audiences, since it includes professional mathematicians, teachers of mathematics at all levels, and users of mathematics in fields like physics, statistics, economics, etc.

There’s a little something in this volume for many of these audiences, but probably not enough to satisfy most of them. The editor of this collection is a specialist in mathematics education, so that category is over-represented. People who apply mathematics in their own special fields of expertise as well as research mathematicians will probably find the material on mathematics education to be of minor interest.

Many of the other topics tend to be treated in a way that would appeal to that fairly small part of the “general public” that actually has any interest at all in mathematics. Check out the table of contents to see if any topic of that sort interests you. Since this is a personal evaluation, I’ll just mention articles that seemed interesting to me.

Perhaps unsurprisingly, the two best articles in the volume were written by research mathematicians who’ve won Fields Medals: Terrence Tao and Timothy Gowers. Tao writes about a topic he knows extremely well (of which there are quite a few): the distribution of prime numbers. The question is whether there are any patterns in how prime numbers are distributed, or whether the distribution is essentially random. Gowers has two contributions, but the interesting one is the question of whether mathematical truths are “invented” or “discovered”. This can be treated as either an empirical question or a philosophical one. Gowers focuses on the empirical question, and the answer is “both”.

There are several good articles on the history of mathematics. One (by Peter Rowlett) offers a number of examples where solutions to practical problems were enabled by theoretical work that may have occurred centuries earlier. An article by John Baez and John Huerta discusses unfamiliar sorts of “numbers”, including quaternions and octonions, and how they are relevant to symmetry, especially in string theory (as in theoretical physics). Charlotte Simmons writes about the influence of the logician Augustus De Morgan on other notables, such as William Hamilton (discoverer of quaternions) and George Boole (inventor/discoverer of Boolean algebra). Fernando Gouvea shows how the modern theory of real numbers and of sets arose from interactions over many years between George Cantor and his teacher, Richard Dedekind. A satisfactory theory was discovered only after various promising but false starts.

Only three other articles seem worthy of mention. Brian Hayes explains the remarkable topological fact that the volume of a unit n-ball (n-dimensional solid sphere of radius 1) approaches 0 as n tends to infinity. Richard Elwes writes about the very esoteric theory of infinite cardinal numbers (as originally developed by George Cantor). Some of the most recent work has been done by Hugh Woodin, and it suggests that even Kurt Gödel and Paul Cohen didn’t have the last word on Cantor’s Continuum Hypothesis.

Lastly, philosopher Ian Hacking asks the question, “Why is there a philosophy of mathematics at all?” As a philosopher, he attempts to present several plausible justifications. This discussion concerns mathematical Platonism and the “invented” vs. “discovered” issue. Professional mathematicians, though, aren’t that concerned with the philosophy, since for them the really important thing is proving theorems, while letting philosophers worry about what that “means”.

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