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Cogwheels of the Mind: The Story of Venn Diagrams

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Some animals are birds. Some animals are black. Some birds are black. Draw the Venn diagram for these propositions. Most high school students have done Venn diagrams; if you can do this picture, you might have drawn a big rectangle, representing animals, containing two overlapping circles, each representing respectively birds and black, and the overlap representing birds that are black. You may be able to do the diagram, but probably you don't know who Venn was, and you probably don't know what strange reaches of mathematics can be achieved simply by the study of the diagrams. _Cogwheels of the Mind: The Story of Venn Diagrams_ (Johns Hopkins University Press) by A. W. F. Edwards was written to bring light on these aspects of the subject, and very well succeeds. A short book, with a fittingly large number of illustrations, it summarizes a wealth of logical and geometric ideas. Some of the math that springs from these initially simple diagrams will be daunting for those who want to read through quickly, but much is basic and well explained, and the lovely diagrams will repay anyone's study.

It is a book that leads in surprising directions, and as befits such a work, Edwards was drawn to the subject almost by chance. He wanted to put a memorial window up to Venn at the University of Cambridge, and the familiar diagram of three overlapping circles suggested itself. In doing research for the simple window, Edwards began to wonder about drawing a Venn Diagram for more than three sets. John Venn had thought about it, too, and found that circles would do for three, not four. Venn did draw a pretty solution using four ellipses, and realized that any number of sets could be diagrammed, but that the shapes would have to be increasingly convoluted and thus decreasingly explicatory. Much of the book covers Edwards's own research and discovery in producing symmetrical seven-set Venn diagrams. There are a total of six versions of such diagrams, and in the color illustrations they look rather like mandalas, with each of the seven regions of exactly the same curvy shape (one looks rather like a fat footprint), rotated around the circle in seven equidistant steps. Another chapter tells about the author's own method of using spherical surfaces on which to plot the diagrams, and then turning them into Mercator projections. The attractive seam of a tennis ball looms large in his investigations.

This is a very personal book about the subject. It is written in the first person, and tells of the author's attempts to hit different problems he had proposed to himself. It is not without humor; at one point in his account of hunting down a symmetrical seven-set diagram (an account which "conveys some of the excitement of the chase"), he says that in 1992 he had drawn a hopeful initial diagram but that "On Monday evening, 16 November, I was bitten by a dog, so on the Tuesday I stayed in [college] in the morning." Thus do worldly concerns impede intellectual pursuits. Those familiar with themes frequently found in recreational mathematics will find friends here, with connections between Venn diagrams and Boolean algebra and Grey codes and hypercubes. It will be no surprise to find Lewis Carroll doing diagrams in his own way, but it might be a surprise to find the diagram, reproduced here, that Winston Churchill drew in 1948 at Hever Castle to show the relations of the sets of the British Empire, United Europe, and the English-Speaking World. It is a vivid illustration of how universal Venn diagrams are, and Edwards's book is a lively description of one researcher's route of enthusiastic discovery.

Cogwheels Of The Mind by A.W. F. Edwards is a pretty and frustrating little book. The text starts out as a history of Venn diagrams, but veers into a trip through Edwards's recent thinking on Venn diagrams [which is the author's prerogative, but does make the subtitle of the book - The Story of Venn Diagrams - a little misleading]. It's all good reading, but given the brevity of the text, I found myself wishing that Edwards had stayed on task a little more. The pictures are stunning! If you're into diagrams like I am, then the pictures will be a feast for your mind. To me the book feels like two well illustrated journal articles or maybe a really small coffee table book. Cogwheels Of The Mind is cool, but flawed.

I was attracted to this book largely because I was amused to see what the well known bio-statistician A.W.F. Edwards was doing with Venn diagrams -- it turns out that it has little to do with the main thrust of his research -- Edwards simply enjoys as a hobby recreational mathematics similar to what used to be presented in Martin Gardner's Scientific American columns.

Well, fair enough. Edwards writes an interesting story dealing with the life of John Venn, various rival presentation schemes, and ends with Edwards' own (successful) quest to generalize Venn diagrams to an arbitrary number of sets. The only problem is it isn't clear for whom Edwards is writing the book. If it's for mathematicians, even amateur ones, some proofs would be in order (none are in the book), and if it is for the general public, more historical detail would be in order. Still, the book is an enjoyable (if short) read.

Few may known about the Venn diagram -- a basic concept of three overlapping circles that intersect to create eight distinct areas, used for business strategy, medicine, computer science and physics alike. Cogwheels Of The Mind: The Story Of Venn Diagrams is expertly written by statistician and geneticist A.W.F. Edwards who covers the history of the Venn diagram, its use, and its presence in a range of disciplines.

A feast for the eyes in its inventive illustrations of the author's expansions of Venn diagrams, resulting in many appealing geometric designs. Although, however, the subtitle is "The Story of Venn Diagrams", the book deals hardly at all with their original intent, which was to validate syllogisms or other logical propositions. There is indeed no instance given of that procedure.

Concerning it and notwithstanding accepted views to the contrary, Venn's diagrams are further not only not an improvement over what are known as Euler diagrams, but they add a superfluous and distractive complication. It is the introduction of all combinations of classes that are part of the propositions at issue. I go into this and other matters more thoroughly in my book, giving here an indication of what is involved.

Venn appears influenced by Boolean algebra and related sets, which concentrate on how sets and their negations can be combined. But this is different from a statement, or a series of them, regarding particular connections only. Thus by injecting all other possible connections, attention from the desired inferences is sidetracked, apart from the additional difficulty of diagrammatically depicting all possible connections when the number of sets increases. Instead, when depicting only connections at issue, their number is of no concern, requiring no more than extended diagrams.

Euler's diagrams, consisting of simple and usually overlapping circles, did not solve all the problems encountered. They can, however, be adequately improved by such as making circles stand for denials when considered. Unlike claimed by even Venn in the book, it is these simple connections on which one can "work step-by-step towards the conclusion", seeing clearly what does and does not follow from the premises.

Accordingly such diagrams can do more than validate. They can serve as visual demonstrations of conceptually sought logical truths.

I wrote and published this review in The Journal of Irreproducible Results, the science humor magazine, vol. 48 #3, September 2004:

This pleasant little book tells a bit more about Venn Diagrams than most scientific people vould vant to know. In addition to the familiar 3-ring version, it revards the reader vit beautiful new examples of many complexly intervoven curves. The matematics, the logic, und the graphics are all beautiful.

The publisher puffed out the easy, short text by doublespacing the lines. The diagrams get generous space. If you don't pause to carefully examine the geometry of each diagram, you can read this book in a couple of hours.

The historical introduction proved useful. Less than a dozen pages in, I had just passed Euler Diagrams, Venn's predecessor, in vich an outer circle encompasses a whole set, und an inner circle marks off a subset. My son broke in to complain that he vus hungry, but his pizza vus burnt und he didn't like having to vait for anutter pizza to bake. I set down the book, und accompanied him to the kitchen. There sat the very image I had just seen in the book. The whole set vus the overbaked pizza. The outer ring vus indeed charred beyond edibility. But the center half vus merely crisp, und still edible. So I cut it out. That subset vus just vutt I had been reading about. Und it vus also dinner.

Binary numbers und Gray codes feature prominently. Lewis Carroll, George Boole, C. S. Peirce, und Martin Gardner each play roles, but John Venn himself remains curiously under-described. Recent und current matematicians continue to advance the field.

But this book is ultimately insufficient, because these are only Venn Diagrams. For more complete aspects, vun must also consult Vutt Diagrams, und Vair Diagrams, und Vye Diagrams.