Cogwheels of the Mind: The Story of Venn Diagrams

Rating:
Summary: Lavish and enjoyable
Review: 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.


Rating:
Summary: Interesting, but who's the audience?
Review: 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.

Rating:
Summary: Subsets of Logic
Review: 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.