Last summer during Mathfest I found myself one day at a Boulder coffeeshop working on my research. This particular day, working on my research consisted of drawing Venn diagrams and putting dots in the various sections (representing the overlapping branch loci of various hyperelliptic curves if you must know, but that's not important to this review). Page after page was filled with three overlapping circles and dots. At one point, while refilling my coffee, I walked by a man who also had in front of him page after page of drawings of three overlapping circles with various dots in them. Intrigued, I asked him about his doodling, and it turned out he was a complex analyst also working on his research. There we were, two mathematicians working in very different fields, both of whom were using Venn diagrams as an integral part of their research.
I would wager that — whether explicitly or implicitly — we all use Venn diagrams in our thinking. They give a pictorial illustration and concreteness to the set theory and logical constructions which we all use constantly, whether we realize it or not. From my own experience, I know that ever since I was first introduced to the idea as a child I have used Venn diagrams to figure out or record any number of things, and I doubt that I am in the minority. Yet I never had any idea where they came from, or even who Venn was. Therefore, in this age where books — and bestselling books at that — are written about the history of everything from the number zero to salt to the color mauve, it is not surprising that someone would write a book about the history of the Venn diagram.
A. W. F. Edwards has done just that with Cogwheels of the Mind. It is a short book, clocking in at just over 100 pages including a copious number of illustrations, which is a quick, enjoyable, and informative read. It is the kind of book that I can imagine giving to a wide range of readers: any junior high student would be able to follow the mathematics, and most professors would find it interesting. Edwards is a fellow of Gonville and Caius College at Cambridge, and he was originally attracted to learning about Venn after working on a stained glass window at the college dedicated to Venn and to the biologist and statistician Ronald Fisher, both of whom were presidents of the college.
The first few chapters of the book discuss the history of Venn diagrams. In particular, while Euler had developed diagrams which appear similar much earlier, Venn found these to be too restrictive to deal with the logical propositions he wished to consider. So he created a new way of graphically representing what amounts to a Boolean algebra. As Edwards writes, "the novelty of [Venn's] diagram lies in its ability to represent propositions and not merely sets." This is why, in a July 1880 article in The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, Venn introduced to the world the diagrams that now carry his name. Personally, I would have liked more detail in the historical portions of the book, and the mathematical context of the time which Venn was working in. But Edwards chooses not to linger and to move on to the more mathematical content.
The later chapters of the book are essays and ruminations on different properties of Venn diagrams. The text is written in the first person, and walk the reader through the author's thought processes as he considers the intimate connection between Venn diagrams and Boolean algebra, and the implications this has on using Venn diagrams to work with Grey codes, binomial coefficients, hypercubes, and numerous other topics. A large amount of space is dedicated to the problem of constructing general Venn diagrams — we all know how to draw two or three set Venn diagrams in our sleep, and while you cannot do it with circles it is nonetheless not to difficult to use ovals to create four set Venn diagrams. But the question of creating Venn diagrams with arbitrary numbers of sets was not settled until a 1988 paper by Edwards himself, despite the fact that many others, including Lewis Carroll, had written about most of the important ideas that Edwards uses.
There are two somewhat more technical appendices. The first deals with a topic, Metric Venn Diagrams, which ties Venn's work in with Fisher's work in genetics. Typically one does not attempt to draw Venn diagrams so that the size of the regions conveys anything about the frequency of the events (that is, we draw the circle representing "redheads" to be the same size as that representing "right handed", despite the fact that the latter is far more frequent). However, it is often desirable to attempt to break this convention, and Edwards discusses some results in this direction.
The book is full of nice illustrations, ranging from flags and stained glass windows, to many drawings illustrating the key ideas. It even includes a copy of a Venn diagram drawn by Winston Churchill depicting the three sets of Europe, the British Empire, and the English-speaking world, with the United Kingdom the only thing in the intersection of all three. The interior of the book looks very nice, despite a cover that one would do best not to judge the book by. Cogwheels of the Mind is a slow meandering walk through many different ideas related to Venn diagrams, and as such I imagine it would frustrate some readers. However, I found it to be a nice — if light — read, and it is well worth a look.
Darren Glass is an Assistant Professor of Mathematics at Columbia University. His research interests include number theory, algebraic geometry, and cryptography. He can be reached at firstname.lastname@example.org