## Mathematical Adventures in Wonderland

What happens when a Lecturer in Mathematics at Oxford takes delight in wordplay puzzles, wants to entertain children with stories, and does not like the new developments in the mathematics of his age? Why, you get the wonderful Alice’s Adventures in Wonderland, Through the Looking Glass, and other delightful stories and poems, of course. Written by Lewis Carroll and still in print, these are the among the most popular pieces of fiction, bringing delight to countless children. However, there is more to Lewis Carroll than meets the eye.

Lewis Carroll, whose real name was Charles Lutwidge Dodgson, lived in a time when a revolution was taking place in the world of mathematics and logic. The old guard of that time, of which Dodgson considered himself a part of, believed that “mathematics should be based on axioms, the truth of which ought to be self-evident”. In contrast, the new guard recognized “that the most we can hope for is a set of postulates which we choose to accept.” The difference might not seem too important for someone not familiar with mathematics, but this new idea is the basis of almost all of modern mathematics. Though the Alice stories on the surface seem meant to entertain children, there were in fact designed by Dodgson to illustrate to his peers in the field of mathematics and logic what he thought was so inherently wrong with their new ideas.

One of Carroll’s favourite ways of doing this is by taking the idea his peers championed and taking them to their logical limits, where they took up an absurd form. In fact, the whole of the Alice adventures take place in a completely new world that can be seen as Carroll's idea of what the world in which the new mathematical ideas were accepted would look like. Consider, for instance, the scene of Alice’s meeting with the Duchess and her baby. Everything is wrong with this world. The Duchess is a very bad aristocrat and a horrible mother, paying no attention to her crying baby. The Cook, if anything, is of the worst kind, putting too much pepper in the soup and throwing pots and pans at her mistress for no reason at all. In the middle of this the baby is a ‘queer-shaped little creature” (p.62) whose legs stick out in all directions. Alice takes it outside, only to find that it turns into a pig albeit with it’s legs sticking out and small eyes like those of the baby! What’s happening here is that Carroll is mocking the idea of continuous functions in topology (then the field of projective geometry). The idea is that if an object is varied in a set manner within certain limits, some of it’s properties don’t change. This was a new idea in Carroll’s time and in Carroll’s opinion a step in the wrong direction from the rigidity of Euclid’s elements that he valued so highly being of the old guard. Ridiculing this idea is easy enough for Carroll. In topology the objects being transformed are usually geometric figures, but he ignores this for a moment as he takes the idea of objects being transformed, takes it to it’s logical limit and comes up with the absurdity of an ‘ugly’ baby (p.62) turning it into a ‘handsome pig’ (p.62).

Another of Carroll method’s of making his points is through clever word play and puns. For example, when the caterpillar advises Alice to “keep your temper”, if all we do is to find him incredibly rude or at least extremely mysterious we are missing the pun that Carroll intended. In his times, the ‘temper’ was not only used in the sense of anger but also to mean the properties of an object. “Keep your temper.” actually means to for Alice to keep her properties or proportions. In Alice’s tale, this might only mean for her to return to her proper size and proportions, which she is finding very hard to do. However, in the real world it’s a references to symbolic algebra and new mathematical inventions such as imaginary numbers, which as the name suggests have no physical interpretation - or a sense of ‘proportion’.

Later on, in Through the Looking Glass, Alice encounters Humpty Dumpty- the one from the popular nursery rhyme - perched on a very thin wall, and unknown to him very likely to fall. Alice is initially confused when Humpty Dumpty uses words that don’t seem to fit into the context of the sentence in which they are used. When she admits that she doesn’t know what he means, Humpty Dumpty smiles and informs her, “Of course you don't -- till I tell you...”(p.196) This leads up to the following exchange:

'When I use a word,' Humpty Dumpty said in rather a scornful tone, 'it means just what I choose it to mean -- neither more nor less.'

'The question is,' said Alice, 'whether you CAN make words mean so many different things.'

'The question is,' said Humpty Dumpty, 'which is to be master-- that's all.'(p.197)

Alice is much too confused to ask what he means by ‘master’, and instead relents to his viewpoint for a while, asking him what he means by words he uses in the dialogue that follows. But later when she leaves him she declares to herself, “of all the unsatisfactory people I ever met” (p.203), but never finishes the sentence on account of a loud crash that is nothing other than Humpty Dumpty falling off his wall. The message is clear. Humpty Dumpty is none other than new age mathematicians, who choose to give symbols names as they wish, and who unknown to themselves are in a very precarious position and are soon to fall down to their destruction.

Carroll had a penchant not only for including math and logic in his literary works, but also of making his arguments in the field of logic and math in literary style. Most notable among these is What the Tortoise Said to Achilles (1895), in which Carroll borrows the Tortoise and Achilles from the paradoxes of motion of Zeno. In this piece, the tortoise asks Achilles to convince him using logic to accept the conclusions of a simple syllogistic argument based on the given premises. In the highly entertaining dialogue that follows, Carroll sets up a logical paradox that has yet to properly answered. The paradox deals with the metalanguage, the language that describes language itself. For example “my last sentence was about metalanguage” is a statement of metalanguage, for it describes what a statement of language (and not I) talked about. The purpose of the paradox is to note that metalanguage is itself governed by rules similar to those governing language, and so there can exist a meta-metalanguage that describes statements of metalanguage. But then there also exists meta-meta-metalanguage and so on and so forth until we have an infinite number of metalanguages. This is a problem once one realizes that rule of inference which lets us say that “Socrates is a man” and “all men are mortal” and then conclude that “Socrates was mortal”, is correct because metalanguage says so. But then to validate this rule, there needs to be a rule in meta-metalanguage about this rule. But for that we need a rule in meta-meta-language... And so we have a infinite regress that has never been properly resolved.

Carroll alludes to this problem with the field of logic much earlier in Through the Looking Glass. A dialogue between Alice and the White Knight takes place in the following way:

'You are sad,' the Knight said in an anxious tone: 'let me sing you a song to comfort you.'

'Is it very long?' Alice asked, for she had heard a good deal of poetry that day. [...] “The name of the song is called "HADDOCKS' EYES."'

'Oh, that's the name of the song, is it?' Alice said, trying to feel interested.

'No, you don't understand,' the Knight said, looking a little vexed. 'That's what the name is CALLED. The name really IS "THE AGED AGED MAN."'

'Then I ought to have said "That's what the SONG is called"?' Alice corrected herself.

'No, you oughtn't: that's quite another thing! The SONG is called "WAYS AND MEANS": but that's only what it's CALLED, you know!'

'Well, what IS the song, then?' said Alice, who was by this time completely bewildered.

'I was coming to that,' the Knight said. 'The song really IS "A-SITTING ON A GATE": and the tune's my own invention.' (p.224)

A first the reader himself might be quite confused by the semantics of the Knight. But a closer reading reveals that Carroll is talking about metalanguage. The poem belongs to the language level, the name of the song belongs to the level of metalanguage, but what the name of the song is called belongs one level up in meta-metalanguage. Poor Alice fails to keep this distinction, and gets horribly confused by the answers she receives, though all the Knight is doing is keeping his semantics clean and correct.

It was in these ways that Carroll sought to save the mathematics he knew from the birth of new ideas that were trying to change it into something he was not comfortable with it at all. But he was no martyr. The ideas he was trying to repress were the very ideas that were at the centerfold of mathematics in the decades to come, and resulted in unimaginable developments in human knowledge in ways unforeseen. Without those ideas, the world would have been a much poorer place indeed. But he was no devil either. In at least one place, his contribution is still remembered and debated upon. In other places, which were not the subject of this piece, he had many notable contributions and he is well remembered for them. But forgetting the issue of mathematical and logical contributions for a moment, we can remember that Lewis Carroll wrote two wonderful tales, tales that have brought supreme delight to countless people across a century and a half and from the looks of it will keep on doing so for a long long time. That is contribution enough.

Further Reading:

Warren Weaver, “The Mathematical Manuscripts of Lewis Carroll”. Proceedings of the American Philosophical Society, Vol. 98, No. 5 (Oct. 15, 1954), pp. 377-381.

Sophie Marret, “Metalanguage in Lewis Carroll”. SubStance, Vol. 22, No. 2/3, Issue 71/72: Special Issue: Epistémocritique (1993), pp. 217-227, University of Wisconsin Press, http://www.jstor.org/stable/3685282

Carroll, Lewis. "What the Tortoise Said to Achilles". Mind, n.s., 4 (1895), pp. 278–80.

Carroll, Lewis. “The Complete Illustrated Lewis Carroll”. Wordsworth Edition, 1996.

Melanie Bayley, “Alice's adventures in algebra: Wonderland solved”. New Scientist, Magazine issue 2739, 16 December 2009, http://www.newscientist.com/article/mg20427391.600-alices-adventures-in-algebra-wonderland-solved.html