January was the month in which Lewis Carroll was both Born (27th) and Died (the 14th) It seems a fitting time to remember some of my favorite (not all true) stories about him. Carroll was of course the pseudonym by which he wrote, but in his day to day life he was an instructor of mathematics at Oxford by the name of Charles Lutwidge Dodgson (Lewis Carroll is an alteration of the Latinization of Charles into Carroll, and the replacement of Lutwidge with Lewis). He was a good (but not great) mathematician, but in the way of the world, he is remembered most for his children’s stories... and had he not written them, he would probably be remembered for his photography... And if he had avoided that also... Maybe people would know he was a mathematician, but probably not; how many people on the street have heard of Euler?. An interesting, but seemingly false, story circulated about a gift of a book on determinants to the Queen of England by Lewis Carroll. Here is the version as it is told on the Mathworld page.

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Several accounts state that Lewis Carroll (Charles Dodgson ) sent Queen Victoria a copy of one of his mathematical works, in one account, An Elementary Treatise on Determinants. Heath (1974) states, "A well-known story tells how Queen Victoria, charmed by Alice in Wonderland, expressed a desire to receive the author’s next work, and was presented, in due course, with a loyally inscribed copy of An Elementary Treatise on Determinants," while Gattegno (1974) asserts "Queen Victoria, having enjoyed Alice so much, made known her wish to receive the author’s other books, and was sent one of Dodgson’s mathematical works." However, in Symbolic Logic (1896), Carroll stated, "I take this opportunity of giving what publicity I can to my contradiction of a silly story, which has been going the round of the papers, about my having presented certain books to Her Majesty the Queen. It is so constantly repeated, and is such absolute fiction, that I think it worthwhile to state, once for all, that it is utterly false in every particular: nothing even resembling it has occurred" (Mikkelson and Mikkelson).

Ok, but still a neat story...

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I love this letter, written by Dodgson to a young man named Wilton Cox. "Honoured Sir, Understanding you to be a distinguished algebraist (that is, distinguished from other algebraists by different face, different height, etc.), I beg to submit to you a difficulty which distresses me much. If x and y are each equal to 1, it is plain that 2 × (x2 - y2) = 0, and also that 5 × (x - y) = 0. Hence 2 × (x2 - y2) = 5 × (x - y). Now divide each side of this equation by (x - y). Then 2 × (x + y) = 5. But (x + y) = (1 + 1), i.e. = 2. So that 2 × 2 = 5. Ever since this painful fact has been forced upon me, I have not slept more than 8 hours a night, and have not been able to eat more than 3 meals a day. I trust you will pity me and will kindly explain the difficulty to Your obliged, Lewis Carroll." Another interest of Dodgson’s was the analysis of tennis tournaments: "At a lawn tennis tournament where I chanced to be a spectator, the present method of assigning prizes was brought to my notice by the lamentations of one player who had been beaten early in the contest, and who had the mortification of seeing the second prize carried off by a player whom he knew to be quite inferior to himself." Carroll set out the guidelines for a seeding system well before the good folk at Wimbledon ever thought of it. "Good on ya" as they say over here. And from a letter he wrote in 1868 with suggestions for essentials of math instruction

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 Dodgson, 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 Gutenberg 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 (pronounced "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.

A fan post (For Carroll) and a problem challenge in the comments:

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.]

Searching for Snarks

The happy band of mathematical warriors above joined me on Thursday night as we set out to the Center for Mathematical Sciences at Cambridge. Our quest was to discover all that Emeritus Gresham College Professor of Geometry Robin Wilson might know about "Lewis Carroll in Numberland." The lecture, which can only be described as "math-lite" was entertaining none the less, and made even better by the good companions who joined me.

Wilson is also, not by coincidence, the author of a book on the topic that is soon to be released in the US (and can be purchased in advance from Amazon at a healthy discount) entitled, Lewis Carroll in Numberland: His Fantastical Mathematical Logical Life.

Wilson explained that if Dodgson (the real name of Lewis Carroll) had not written the "Alice" stories for which he is so well remembered, he might well be remembered for being one of the pioneering child photographers of the 19th century. And if he had not done either, he might be remembered as an accomplished mathematician and teacher who made contributions in the areas of Logic, algebra, geometry, and the mathematics of elections. Wilson points out that: "Yet another interest of his was the study of voting patterns. Some of his recommendations were adopted in England, such as the rule that allows no results to be announced until all the voting booths have closed. Others, such as his various methods of proportional representation, were not. As the philosopher Sir Michael Dummett later remarked: It is a matter for the deepest regret that Dodgson never completed the book he planned to write on this subject. Such was the lucidity of his exposition and mastery of this topic that it seems possible that, had he published it, the political history of Britain would have been significantly different."

He also credits Carroll with the invention of the modern method of seeding tennis matches: Another interest of Dodgson’s was the analysis of tennis tournaments:"At a lawn tennis tournament where I chanced to be a spectator, the present method of assigning prizes was brought to my notice by the lamentations of one player who had been beaten early in the contest, and who had the mortification of seeing the second prize carried off by a player whom he knew to be quite inferior to himself.Let us take sixteen players, for example, ranked in order of merit, and let us organise a tournament with 1 playing 2, 3 playing 4, and so on. Then the winners of the first round will be 1, 3, 5, and so on; those of the second round will be 1, 5, 9 and 13; the final will then be won by player 1, defeating player 9 who wins the second prize but actually started in the lower half of the ranking. To avoid this difficulty, he managed to devise a method for re-scheduling all the rounds so that the first three prizes go to the best three players, which presaged the present system of seeding.