Ms Sia’s Realm of Maths

16 December 2009, by Melanie Bayley

What would Lewis Carroll’s Alice’s Adventures in Wonderland be without the Cheshire Cat, the trial, the Duchess’s baby or the Mad Hatter’s tea party? Look at the original story that the author told Alice Liddell and her two sisters one day during a boat trip near Oxford, though, and you’ll find that these famous characters and scenes are missing from the text.

As I embarked on my DPhil investigating Victorian literature, I wanted to know what inspired these later additions. The critical literature focused mainly on Freudian interpretations of the book as a wild descent into the dark world of the subconscious. There was no detailed analysis of the added scenes, but from the mass of literary papers, one stood out: in 1984 Helena Pycior of the University of Wisconsin-Milwaukee had linked the trial of the Knave of Hearts with a Victorian book on algebra. Given the author’s day job, it was somewhat surprising to find few other reviews of his work from a mathematical perspective. Carroll was a pseudonym: his real name was Charles Dodgson, and he was a mathematician at Christ Church College, Oxford.

The 19th century was a turbulent time for mathematics, with many new and controversial concepts, like imaginary numbers, becoming widely accepted in the mathematical community. Putting Alice’s Adventures in Wonderland in this context, it becomes clear that Dodgson, a stubbornly conservative mathematician, used some of the missing scenes to satirise these radical new ideas.

Even Dodgson’s keenest admirers would admit he was a cautious mathematician who produced little original work. He was, however, a conscientious tutor, and, above everything, he valued the ancient Greek textbook Euclid’s Elements as the epitome of mathematical thinking. Broadly speaking, it covered the geometry of circles, quadrilaterals, parallel lines and some basic trigonometry. But what’s really striking about Elements is its rigorous reasoning: it starts with a few incontrovertible truths, or axioms, and builds up complex arguments through simple, logical steps. Each proposition is stated, proved and finally signed off with QED.

For centuries, this approach had been seen as the pinnacle of mathematical and logical reasoning. Yet to Dodgson’s dismay, contemporary mathematicians weren’t always as rigorous as Euclid. He dismissed their writing as “semi-colloquial” and even “semi-logical”. Worse still for Dodgson, this new mathematics departed from the physical reality that had grounded Euclid’s works.

By now, scholars had started routinely using seemingly nonsensical concepts such as imaginary numbers – the square root of a negative number – which don’t represent physical quantities in the same way that whole numbers or fractions do. No Victorian embraced these new concepts wholeheartedly, and all struggled to find a philosophical framework that would accommodate them. But they gave mathematicians a freedom to explore new ideas, and some were prepared to go along with these strange concepts as long as they were manipulated using a consistent framework of operations. To Dodgson, though, the new mathematics was absurd, and while he accepted it might be interesting to an advanced mathematician, he believed it would be impossible to teach to an undergraduate.

(Ms Sia’s notes – imaginary numbers are also known as unreal numbers!)

Outgunned in the specialist press, Dodgson took his mathematics to his fiction. Using a technique familiar from Euclid’s proofs, reductio ad absurdum, he picked apart the “semi-logic” of the new abstract mathematics, mocking its weakness by taking these premises to their logical conclusions, with mad results. The outcome is Alice’s Adventures in Wonderland.

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