The algebra of Alice

Alice’s Adventures in Wonderland by Lewis Carroll continues to attract new readers ever since it was told to three sisters on a summer afternoon during a boat ride on the Thames. The apparently whimsical fairy tale charmed its listeners on its first telling but the story was expanded by Carroll into the Alice of today. On the 152nd anniversary of the classic’s publication on November 26, 1865, as a Christmas release in England, let’s consider the book as a mathematical puzzle.
Lewis Carroll in the preface to the work ‘All in the Golden Afternoon’, claimed to have invented the story on demand from Alice Liddell, and her two sisters, daughters of an Oxford don – Carroll himself taught mathematics at Oxford – during the boat ride. However, the profusion of mathematical puzzles, logical paradoxes and innuendoes throughout the body of the text tell a different story. While there is no doubt about the fact that it was created for, and to be told to children and young adults, what 21st century readers read today is a cleverly crafted tale to poke fun at the mathematics in Carroll’s time and its practitioners.
Carroll, a nom de plume of Charles Lutwidge Dodgson, a mathematics tutor at the Christ Church College in Oxford, was actually not a front-ranking mathematician. He swore by Elements, the famous geometry text by Euclid. Carroll waged a long battle with his peers who were revolutionising Victorian mathematics. Projective geometry, imaginary numbers, quaternion were turning the old-world of algebra and geometry upside down. Mathematics was no longer tied to the ground insofar as it was becoming more abstract, and logic that appealed to Carroll and his ilk could not be used to demystify the new avatar. Carroll was a Euclidean geometry orthodox who did throw the gauntlet at the new kids on the mathematics block but lost out. These were the times when Alice Liddell asked the young mathematics tutor to tell a story.
 

THE MISS LIDDELLS


Close to a decade and a half later, in 1879, Carroll, under his real name, published Euclid and his Modern Rivals. Written in the form of a play, it was Carroll’s way of telling the world that Euclid’s Elements is the best textbook for teaching geometry. Carroll’s introduction lays out his purpose and why he went about it the way he did. His words on writing for a non-scientific audience still sound particularly relevant. “It is presented in a dramatic form,” writes Charles Dodgson in the introduction, “partly because it seemed a better way of exhibiting in alteration the arguments on the two sides of the question; partly that I feel myself at liberty to treat it in a rather lighter style than would have suited an essay, and thus to make it a little less tedious and little more acceptable to unscientific readers.” Not many now are even aware of this curious publication but this can be seen as an extension of Carroll’s thought process that started with Alice’s Adventures in Wonderland.

There is, however, no direct evidence that Carroll actually planned such a tale. Martin Gardner notes is his book, The Annotated Alice, the definitive edition, that Reverend Robinson Duckworth, who accompanied Carroll and the Liddell sisters on the boat ride, says in his account of the trip: “…when three Miss Liddells were our passengers, and the story was actually composed and spoken over my shoulder for the benefit of Alice Liddell…I remember turning round and saying, “Dodgson, is this an extempore romance of yours?” And he replied, “Yes, I’m inventing as we go along.”” That story, on the insistence of Alice, was turned into a manuscript and presented to her by the Oxford mathematician.
By now, the content of the story is presented in disguised form with the use of riddles, apparently meaningless poems, puzzles, puns, and a lot more that is ostensibly nonsense. Carroll was surely not the first to use such devices.
Several examples of puns and riddles are found in nursery rhymes, and folk tales for children. The mastery of Carroll over this kind of recreational mathematics and logic takes Alice’s Adventures in Wonderland to a different league – it is not without reason that the story continues to inspire mathematical puzzles and word-game designers even today.
 

THE COMMUNITY OF ALICE


Raymond S Smullyan wrote a delightful little book titled Alice in Puzzle-Land: a Carrollian Tale for Children Under Eighty in which Alice and her friends return for another trip through Wonderland and the Looking-Glass. The book has 88 engaging puzzles, paradoxes, and logic problems. Smullyan’s characters speak and behave like the Carroll creations, and their puzzles abound in typical Carrollian word-play, logic problems, and dark philosophical paradoxes.
The rich tapestry of puzzles and paradoxes in Alice’s Adventures in Wonderland was a lifelong fascination for Carroll that in some way brought his ‘fairy tales’ closer to Austrian-British philosopher Ludwig Wittgenstein. In his 1965 essay “Wittgenstein, Nonsense, and Lewis Carroll,” philosopher George Pitcher’s talks about striking similarities between the philosophical writings of Wittgenstein and the children’s stories of Carroll. According to Pitcher, both were concerned with nonsense and language puzzles. While Wittgenstein was tortured by these things, Carroll appeared to be delighted by them.
Reverend Dodgson had a playful approach to mathematics that he imported into the Alice stories. He was known to use little puzzles in his lessons to make mathematics class more engaging. For instance, here is one of his classics (many versions of this puzzle now can be found all over the web): A cup contains 50 spoonfuls of brandy, and another contains 50 spoonfuls of water. A spoonful of brandy is taken from the first cup and mixed into the second cup. Then a spoonful of the mixture is taken from the second cup and mixed into the first. Is there more or less brandy in the second cup than there is water in the first cup? (If you are scratching your head for an answer, it is equal.)
 

FIGURE IT OUT


In that famous conversation with the Cheshire Cat, who wants to convince Alice that they both are mad, the feline tells her that she “…must be, or you wouldn’t have come here”, but Alice refuses to believe him and in turn asks how the cat knows that he is mad. The next set of conversations that appears in Chapter IV of the book shows how deep is the logic play in this work. Here Carroll has employed the so-called modus ponens, or affirming the antecedent logic.
“To begin with,” said the Cat, “a dog’s not mad. You grant that?”
“I suppose so,” said Alice.



“Well, then,” the Cat went on, “you see a dog growls when it’s angry, and wags its tail when it’s pleased. Now I growl when I’m pleased, and wag my tail when I’m angry. Therefore I’m mad.”

One can read the above dialogue without even realising that one is trapped in a logic web spun by Carroll. Here, the Cheshire Cat’s argument may appear sound but it is invalid. Here is how Carroll constructed the trap.
Suppose P and Q are two sentences; here, P is ‘an animal growls when angry and wags its tail when pleased’ and Q is ‘it is not mad’. Let us see what the cat says: ‘If an animal growls when angry and wags its tail when pleased, it is not mad.’ This means, if sentence P is true, then Q is also true.
‘I growl when pleased, and wag my tail when angry.’
Here the cat is not saying what P says.
‘Therefore, I am mad.’
So if the cat’s statement does not agree with P then how can it say Q is true?
One interesting aspect of Carroll’s work is that in the world of literature, especially literary criticism, a lot of emphasis has been on the psychoanalytic aspects of characters. There have been critiques highlighting Carroll’s own personal psychological and sexuality issues but almost nothing on reading the tale as a mathematical text. In 2009, Melanie Bayley, of the University of Oxford, published an article in the popular science magazine New Scientist titled “Alice’s Adventures in Algebra: Wonderland Solved”.
In the article Bayley says that Carroll added a lot of material to the illustrated manuscript he personally made for Alice before it was sent for publication. It is in these parts that Carroll took on the proponents of new mathematics, ridiculing their methods and questioning their rigour. The Cheshire Cat becoming a grin, according to the Oxford researcher, was Carroll’s way of portraying increasing and damaging abstraction in mathematics. In the Mad Hatter’s tea party, Bayley discovered the writer’s satire on Irish mathematician William Rowan Hamilton’s discovery – the quaternion.
There are other similar discoveries made by the Oxford researcher. In the scene where Alice is troubled by growing taller or shorter and meets the hookah-smoking Caterpillar, the creature tells Alice “keep your temper.” This Alice interprets as keeping cool but here Carroll is using an older meaning of the word ‘temper’ which was used for “the proportion in which qualities are mingled.” Bayley interprets this as the Caterpillar telling Alice irrespective of her body size she should maintain her body in proportion. If that is true, this reflects Carroll’s love of Euclidean geometry. In this geometry, absolute magnitude does not matter, it’s important to know the ratio of one length to another.
For a little more than 155 years after the story was first told to Alice, Lewis Carroll’s bestseller continues to throw new conundrums. No one can be absolutely sure whether Carroll actually plays those devious games with his readers. The reverend who stammered a lot and enjoyed the company of young girls did love his logic and Euclid like a fanatic. He is not remembered for his mathematics but for puzzles, logic games and biting satire. It is therefore not surprising that some of it made its way into his boat-ride story.

Debkumar Mitra

hindustan times