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The Language Of Mathematics: Making the Invisible Visible

Keith Devlin
Publication Date: 
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[Reviewed by
Ioana Mihaila
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Books that take beautiful bits of mathematics and present them to the curious non-mathematician reader are very dear to my heart. I was reading such books long before I decided to pursue a career in mathematics, and I always marveled at the beautiful and accessible logic that they breathed. Among such books, Devlin's The Language of Mathematics stands out for the breadth of its coverage. It spans an array of eight areas of mathematics, each of them neatly encapsulated in a chapter containing a fascinating mixture of history, exposition, and proof.

The author begins by taking us on a journey from arithmetic and number theory to coding theory, and manages to land us in the middle of Fermat's last theorem. While doing this we brush past Pythagorean numbers and their applications in construction and we learn how a proof by induction works. In the next chapter we start by learning logic from Aristotle, and travel all the way to the Zermelo-Frankel axioms, just to find out that we now have the tools to solve such linguistic mysteries as "who wrote the Federalist papers?".

It is all mathematics, everywhere we look around us. Whether we see it or not, usually hidden by a different language, the mathematics is there. Of course, there is motion and the calculus, the author does not miss that. But when we are introduced to projective geometry, we find that even art, perhaps the field considered the farthest from mathematics, has mathematics in it. From the different kinds of geometry we slide into symmetry, and, of course, algebraic structures. We are then stacking oranges in a grocery store, just to find out that at the same time we are uncovering secrets about the structure of crystals.

Then come graph theory and topology. We are told about little amazing things, such as the Mobius strip, but then we build up to talk about knot theory, string theory and the structure of matter. In recognition of the times we live in, there is a chapter dedicated to statistics and probability. We learn about the importance of these fields not only in beating the odds in gambling, but also in running the modern day society, from the stock market, to insurance and medical science.

Towards the end of the book, many of the topics treated before are tied together in a discussion of the mathematics of the universe, focusing especially on relativity theory and spacetime.

Throughout the magnificent ride, the author manages to touch on the main questions and results of mathematics: the most important axiomatic systems are discussed, the Riemann hypothesis and Fermat's last theorem are mentioned, the concepts of relativity theory are outlined, the main types of geometry are described, and their applications in understanding the universe are explored. And most of it is done in a fairly accessible language, in order to assure that the book speaks to the appropriate audience.

As I read the book I was amazed by the amount of mathematics the author had managed to include in his exposition. I also wondered who would be the best audience for this book. The author's love for mathematics, and his desire to share it with the world, is on display throughout the book. This is not surprising: it fits well with all the other things Devlin is involved in, from his column in MAA Online to his part in the PBS Series Life by the Numbers, for which he also wrote a book. For a mathematician, The Language of Mathematics is a lot of fun to read, though most of us will perhaps learn more from the historical portions and from the sections on applications than from the presentation of the mathematics itself. The book, however, doesn't seem aimed at the mathematically illiterate, since there are quite a few (elementary, yes) proofs and a tremendous number of mathematical concepts to absorb.

I dream of the day when ordinary people will pick up a book like this one from a paperback rack, read it, and then say "wow, this is quite interesting!". Unfortunately that day, if it ever comes, seems to be a long way in the future, and I find it hard to imagine that this book can really make much headway with a general audience. There is no question, however, that The Language of Mathematics is a wonderful resource for people involved in mathematics and science education at all levels. Most of all, I think, this is the book that could make a real difference in the hands of a curious high school kid, or first-year college student, who is not afraid of mathematics, but doesn't know what it is good for. Let's see that we put it in those hands.

Ioana Mihaila is assistant professor of mathematics at Coastal Carolina University, SC. Her research area is analysis and she has a special interest in student math contests at all grade levels.

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