by #LizPublika
“But I don’t want to go among mad people,” Alice remarked.
“Oh, you can’t help that,” said the Cat: “we’re all mad here. I’m mad. You’re mad.”
“How do you know I’m mad?” said Alice.
"You must be,” said the Cat, “or you wouldn’t have come here.”
— by Lewis Carroll, from Alice's Adventures in Wonderland (1865)
Charles Lutwidge Dodgson (1832 — 1898) was the real name of the distinguished mathematician, admired clergyman, pioneering photographer, and popular children’s book author widely known as Lewis Carroll. Born in Cheshire, England, Dodgson became a mathematician and logician at Oxford, then an all-boys school, by his early twenties. Although students disliked his “singularly dry and perfunctory manner” in the classroom, he won them over with his love of games and riddles.
“He could recite the first 71 digits of pi using a series of nonsense couplets as memory aids, and once contrived an algorithm that could give the date of every Easter Sunday until 2499.” Dodgson firmly believed that entertaining mental exercises — such as logic puzzles, brain-teasers, and word-play — improved critical thinking skills, enabled students to effectively analyze different subjects and, most importantly, to detect and unravel logical fallacies in problems and arguments.
He specialized in linear and matrix algebra, geometry, and mathematical logic; and “at a time when non-Euclidean geometries were catching on, he wrote a four-act play stubbornly arguing that Euclid should remain at the centre of the Oxford curriculum.” As such, he was widely regarded as a conservative mathematician. But, because he was a fan of using fun activities to retain the attention of his pupils, mathematician Robin Wilson considers Dodgson to be the grandfather of recreational mathematics.
Although Dodgson was an accomplished and passionate mathematician, he was also someone who craved variety. He started writing as a young man and was fond of making up stories for his brothers and sisters, often with a satirical and humorous flair. And even though he produced works for both children and adults throughout his life, almost no other work of his literature is more synonymous with Dodgson’s alias than the Alice series, which was inspired by the daughter of Henry Liddell (1811 — 1898), the dean of Christ Church at Oxford.
The story that would start the Alice series was conceived during a pleasant outing on a July afternoon in 1862, when Dodgson took the three Liddell sisters out on a stretch of the river between Oxford and Godstow. He expanded the work into Alice’s Adventures in Wonderland (1865) the following year and “Lewis Carroll” became a household name shortly thereafter. He followed it up with a sequel, 1871’s Through the Looking-Glass and What Alice Found There, and a long-form nonsense poem “The Hunting of the Snark” five years later.
Mathematician and historian Helena Pycior from the University of Wisconsin-Milwaukee first linked Dodgson’s trial of the Knave of Hearts with an algebra book by Augustus De Morgan (1806 — 1871) in 1984. Morgan was the first British mathematician to lay out a consistent set of rules for symbolic algebra; his contributions to the study of logic include the formulation of De Morgan’s laws. The Alices, argues Pycior, embody ”Dodgson's mis-givings about symbolical algebra.”
She begins her argument as follows:
“Augustus De Morgan introduced Chapter II or his Trigonometry and Double Algebra of 1849 with a precis of symbolical algebra: ‘With one exception, no word nor sign of arithmetic or algebra has one atom of meaning throughout this chapter, the object of which is symbols, and their laws of combination, giving a symbolic algebra.’ Slightly over fifteen years later, Lewis Carroll’s Alice interrupted the trial of the Knave to state her opinion of the nonsense verse read by the White Rabbit. ‘I don’t believe,” Alice declared, “there is an atom of meaning in it.”
Scholars agree that Dodgson considered Euclid’s Elements, the ancient Greek textbook, as the epitome of mathematical thinking. But, mathematicians were developing all kinds of strange “new algebras, where x times y was not equal to y times x.” So, “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.”
Let’s start with the fact that algebra, as explained by De Morgan in one of his footnotes, “comes from an Arabic phrase he transliterated as ‘al jebr e al mokabala,’ meaning restoration and reduction. He explained that even though algebra had been reduced to a seemingly absurd but logical set of operations, eventually some sort of meaning would be restored.” With this in mind, Alice’s exchange with the Caterpillar early on in the story not only makes a lot more sense, but reveals that the circular conversation is actually a brilliant example of semantic wordplay.
First and foremost, the caterpillar is smoking a hookah, which has Arabic origins like the word algebra. “Restoration was what brought Alice to the mushroom: she was looking for something to eat or drink to ‘grow to my right size again,’ and reduction was what actually happened when she ate some: she shrank so rapidly that her chin hit her foot.” Many attribute the famous scene to drug use, but — according to Melanie Bayley writing for The New York Times — it’s an excellent reference to the absurd world of abstract algebra:
“Alice has slid down from a world governed by the logic of universal arithmetic to one where her size can vary from nine feet to three inches. She thinks this is the root of her problem: ‘Being so many different sizes in a day is very confusing.’ No, it isn’t, replies the Caterpillar, who comes from the mad world of symbolic algebra. He advises Alice to ‘Keep your temper.’
“In Dodgson’s day, intellectuals still understood ‘temper’ to mean the proportions in which qualities were mixed as in ‘tempered steel’ so the Caterpillar is telling Alice not to avoid getting angry but to stay in proportion, even if she can’t ‘keep the same size for 10 minutes together!’ Proportion, rather than absolute length, was what mattered in Alice’s above-ground world of Euclidean geometry.”
Alice swallows a piece of mushroom, which causes her neck to grow until she is able to balance out her shape by biting a piece of the mushroom from the other side. Although Alice needs to act like a Euclidean geometer and keep her ratios constant — even when her size fluctuates — she doesn’t, which leads into the next part of the story and a different mathematical critique. Eventually, Alice finds a way to shrink down to the nine inches she needs to be to enter the house with the Duchess, her baby, the Cook, and the Cheshire Cat.
The Duchess hands the baby over to Alice that somehow turns into a pig. Baylor explains that the scene satirizes the principle of continuity, a concept from projective geometry that was introduced in France during the mid-19th century. Now an important aspect of modern topology, the principle “involves the idea that one shape can bend and stretch into another, provided it retains the same basic properties a circle is the same as an ellipse or a parabola (the curve of the Cheshire cat’s grin).”
The Cheshire cat points Alice toward the Mad Tea-Party, where the Hatter, Hare, Dormouse and Alice proceed to circle around static place settings in a way that strongly resembles modular design. The scene is a references to William Rowan Hamilton (1805 — 1865), an Irish mathematician whose work — he discovered quaternions in 1843 — proved significant for the development of quantum mechanics, which was considered an important milestone in abstract algebra, since it allowed for the algebraic calculation of rotations.
So, “quaternions belong to a number system based on four terms. Hamilton spent years working with three terms — one for each dimension of space — but could only make them rotate in a plane,” writes Melanie Bayley. “When he added the fourth, he got the three-dimensional rotation he was looking for, but he had trouble conceptualizing what this extra term meant.” In the preface to his Lectures on Quaternions of 1853, a footnote states: “It seemed (and still seems) to me natural to connect this extra-spatial unit with the conception of time.”
It helps to think of tea-party as t-party, with t being the mathematical symbol for time. Before Alice (the fourth “term”) joins the mad tea party, time is the absent fourth “term” at the table. The Hatter informs Alice that he quarreled with Time last March, and now Time won’t do anything he asks. As a result, the Hatter, the Hare and the Dormouse (the third “term”) are stuck, forever rotating in a plane around the tea table. The scene ends with the Hatter and the Hare trying to stuff the Dormouse into the teapot, hoping to exist as independent, non-rotating “terms.”
In the story, Alice will go on to meet the Queen of Hearts, whose “keenness to execute everyone comes from a ghastly pun on axes — the plural of axis on a graph.” But, there are a lot more instances of math that could be found within the pages of Dodgson’s Alice series as well as some of his other works. Still, the single most important thing that unifies Dodgson's oeuvre is the wit and color apparent in the manifestations of his wide-ranging mathematical interests, particularly in his beloved geometry and logic.
Note* Third featured image is a page from the original manuscript copy of Alice's Adventures Under Ground, 1864, held in the British Library.