Alice's Adventures in Wonderland (commonly shortened to Alice in Wonderland) is an 1865 novel written by English author, mathematician, logician, Anglican deacon and photographer Charles Lutwidge Dodgson under the pseudonym Lewis Carroll.
Much has been said of the mathematical undercurrent in Alice’s adventures in wonderland. Before reflecting upon the math it’s useful to look at the times, and backdrop of mathematics in the latter half of the nineteenth century.
How Dodgson came to write his novel has been much documented – in the way of a very brief summary, the story goes that in 1862, Dodgson, together with the Reverend Robinson Duckworth, rowed in a boat up the River Thames with three young girls, Lorina Charlotte Liddell, aged 13, Alice Pleasance Liddell, aged 10, and Edith Mary Liddell, aged 8, the daughters of Henry George Liddell, the Vice-Chancellor of Oxford University and Dean of Christ Church College, as well as headmaster of the nearby, private, Westminster School.
As they rowed, Dodgson made up and told the girls a story about a bored little girl named Alice who goes looking for an adventure. The three girls loved it, and Alice Liddell asked Dodgson to write it down for her. Two years later he did just that, and in November 1864 he gave Alice the handwritten manuscript of what he then called "Alice's Adventures Under Ground," illustrated by his own drawings.
The math
In the latter half of the nineteenth century, it was a turbulent period for mathematicians, with the subject becoming more and more abstract. Non-Euclidean geometry, Symbolic algebra and the growing use of imaginary numbers were just some of the developments that challenged classical mathematics.
Martin Gardner, whose The Annotated Alice was published in 1960, Helena Pycior of the University of Wisconsin-Milwaukee, and more recently Melanie Bayley, of the University of Oxford in England, have examined linkages between mathematics and the experiences of the characters in the novel
According to Bayley, Dodgson was dismayed by what he saw as the declining standards of mathematical rigor. She suggests Dodgson additions for publication was a wicked satire on those new developments.
What of the links with mathematics - Here are just a few examples
In the chapter "Advice from a caterpillar." Alice has fallen down the rabbit hole and eaten a cake that has shrunk her to a height of just 3 inches. The Caterpillar enters, smoking a hookah pipe, and shows Alice a mushroom that can restore her to her proper size. But one side of the mushroom stretches her neck, while another shrinks her torso, so she must eat exactly the right balance to regain her proper size and proportions. Bayley believes this expresses Dodgson's view of the absurdity of symbolic algebra.
Bayley suggests that the overall madness of Wonderland reflects Dodgson's views on the dangers of this new symbolic algebra. Alice has moved from a rational world to a land where even numbers behave erratically.
In another scene, the hallway, Alice tries to remember her multiplication tables, but they have slipped out of the base-10 number system she is used to. Yet another, perhaps, unsubtle but pointed remark.
In the caterpillar scene, Alice's height fluctuates between 9 feet and 3 inches. Alice, bound by conventional arithmetic where a quantity such as size should be constant, finds this troubling: "Being so many different sizes in a day is very confusing," she complains.
To survive in Wonderland, Alice must act like a Euclidean geometer, keeping her ratios constant, even if her size changes.
The baby's discomfort with the whole process, and the Duchess's unconcealed violence, signpost Dodgson's virulent mistrust of "modern" projective geometry, Bayley says. Everyone in the pig and pepper scene is bad at doing their job.
