Tuesday 26 May 2009

Logic Diagrams, A Brief History

I just read a nice blog over at the NUMBER WARRIOR by Jason Dyer about using logic diagrams for graphic organizers to point out relationships between math objects. It is a nice post and I wouldn't (couldn't) improve on it, so check it out. BUT..... (isn't there always a but..) In the article, which was titled Carroll Diagrams, somewhere along the way he pointed out that,"I’ve used something like this before in geometry for sorting triangle types, but I never knew there was a name for it..." [emphasis added].

Well that set off the math historian in me.... Jason is a clever guy who knows lots of math and somehow, when we let kids that clever grow up and teach math and they haven't been introduced to (at least a mini-) history of logic diagrams, something HAS to be done.. so some notes about the history of logic diagrams, as I understand them at this moment in time.

When I grew up these type of logic diagrams were always known as Venn Diagrams, after the Cambridge mathematician John Venn(1834-1923). Venn was a lecturer at Cambridge and worked mainly in logic and probability theory. He used diagrams of circles to represent the unions and intersections of subsets of a Universal set in non-overlapping regions. You can find more about his life at this page from the Electronic Journal of Combinatorics.
It appears that the first person to call these types of diagrams "Venn diagrams" was Clarence Irving in his work, A Survey of Symbolic Logic in 1918.

Then, in a note at the Euler Project web site, maintained by Prof. Ed Sandifer, I found a note suggesting that the diagrams are actually the creation of Euler. "Letters to a German Princess is likely to be the source of much of what people attribute to Euler. For example, I know that what we call Venn diagrams first appear in there. (Venn himself first called them "Eulerian Circles", but then managed to get them called Venn Diagrams later on.)"

"WAIT!" You scream, Venn, Euler, but the title was Carroll Diagrams,... Ok, I'm getting there.... you see, Lewis Carroll, who was in his other reality the Oxford math lecturer Charles Dodgeson, also did some nice work on Logic Diagrams. He approached set diagrams with rectangles. The image at the top shows an example of how Carroll's diagram might look for three sets(above the middle, right or left of middle, and inside or outside the inner rectangle).

Carroll probably was most influential in his use of a finite set for the Universal set, as Venn often simply used the infinite plane outside the boundaries drawn to indicate the set of things not belonging to any group. Carroll's book, The Game of Logic can be found free on the web at the Guttenburg Project. Perhaps one of the great logic statements of all time occurs in the beginning of Carroll's book when he writes, "Besides the nine Counters, it also requires one Player, AT LEAST. I am not aware of any Game that can be played with LESS than this number: while there are several that require MORE: take Cricket, for instance, which requires twenty-two. How much easier it is, when you want to play a Game, to find ONE Player than twenty-two." (Ok, you already knew he was a clever guy.)

So why do I (and many others) persist in calling them Venn Diagrams.. In his book Cogwheels of the Mind, The Story of Venn Diagrams, Professor Anthony Edwards of Cambridge explains that the Venn Diagram were much broader in scope than Euler's, and in a comparing Venn's work to previous, and sometimes similar, work he states, "Venn's own contribution, which fully justifies our attaching his name to the general diagram was the first to see that the diagram could and should be generalized to any number of sets..." Professor Edwards is a Fellow of Gonville and Caius college as was Venn, and played a part in the design of the commemorative glass shown above, which is at the college. The Glass is part of a set of six that are all commemorative of math and science people.

Professor Edwards is an accomplished mathematician, statistician, geneticist in his own right, as well as being the last graduate student of the great R. A. Fisher. He came up with a method of extending set diagrams to any indefinite size by drawing them on a sphere, and sterographically projecting them back onto a plane to create the cogwheels of the title of his book. See his book.

Like Venn before him, Professor Edwards is a Fellow of Gonville and Caius (pronouced "keys" for us Americans), at Cambridge. He is not only very brilliant, but a nice guy to boot. He showed my wife and me around the Great Hall at G&C and let me see the inside view of the stained glass tribute to Venn in the hall there. He also gave me directions to Venn's grave site at the Trumpington Parish Extension cemetery. His grave was so covered with vines that I would have never found it except my very psychic wife stops by a clump of brambles and says, "I think this is it.." Sure enough, after clearing away the vines, we managed to expose the grave site, which includes Venn, his wife, his son, and his daughter-in-law.
Venn's grave and memorials can be found at the "Find a Grave" website

For more information about Venn Diagrams check this Survey of Venn Diagrams from the Dept of Computer Science at the University of Victoria.

I took some of my math students down to the graveside a few years ago thinking it would be nice to plant some flowers so they would always remember they honored a mathematician. We planted three colors of Tulips near his grave so that they would come up in a set of Venn Rings... and the grave that hadn't been mowed for years before I tore back the brambles to expose it, was mowed the next spring.. so I saw three tiny circles of green shoots cut very close to the ground.. oh well, it was a nice drive out in the spring.

SO that's what I know, and if you want to add more, send me a comment, I would love to add your information.

1 comment:

Anonymous said...

I found Carroll's Game of Logic at the library and had so much fun with it that I ended up buying a copy from Dover. I never worked with his diagram and counters long enough to figure out the "Game", but who can help but love logic puzzles like this:

"No shark ever doubts that it is well fitted out.
A fish that cannot dance a minuet is contemptible.
No fish is quite certain that it is well fitted out unless it has three rows of teeth.
All fishes except sharks are kind to children.
No heavy fish can dance a minuet.
A fish with three rows of teeth is not to be despised."
[Arrange the sentences to form a conclusion.]