Go Figure

GO FIGURE: Not “an Atom of Meaning in It…”

by Susan Sechrist

“Go Figure” is a new regular feature at Bloom that highlights and celebrates the interdependence and integration of math and literature, and that will “chip away at the cult of youth that surrounds mathematical and scientific thinking.”  Read the inaugural feature here.

 

How did an uptight, conservative, Victorian-era academic write one of the most iconic masterpieces of children’s fantasy literature?

He did the math.

Charles Lutwidge Dodgson was a mathematician who specialized in linear algebra, geometry, and logic. He had an absolute and Euclidean love for his profession – from his book, Curiosa Mathematica: “It may well be doubted whether, in all the range of Science, there is any field so fascinating to the explorer – so rich in hidden treasures – so fruitful in delightful surprises – as that of Pure Mathematics. The charm lies chiefly, I think, in the absolute certainty of its results: for that is what, beyond almost all mental treasures, the human intellect craves for. Let us only be sure of something!” (Dodgson page xv).

Dodgson is known literarily as Lewis Carroll – the creator of a cornerstone in the literary nonsense genre, Alice’s Adventures in Wonderland. The story behind the story is that Carroll often entertained his friend and colleague Henry Liddell’s young children by taking them rowing and telling them tales. Out of this, Alice was born – a protagonist who, when faced with the utter absurdity and terror of an unpredictable underworld, turns – like any educated English child of the time – to her multiplication tables to try to sort it out. However, she discovers in this strange new place, nothing (including nothing itself!) is as it seems:

“How queer everything is to-day! And yesterday things went on just as usual. I wonder if I’ve been changed in the night?… Who in the world am I?… I’m sure I’m not Ada… and I’m sure I can’t be Mabel, for I know all sorts of things, and she, oh! She knows such a very little! Besides, she’s she and I’m I and – oh dear, how puzzling it all is!… Let me see: four times five is twelve and four times six is thirteen and four times seven is – oh dear! …no, that’s all wrong, I’m certain! I must have been changed for Mabel!” (19-20).

Carroll was not just poking fun at the hapless and ignorant Mabel. He was using his skill as a writer and a logician to warn about dangerous ideas emerging in mathematics that would undermine absolute certainty and logic in favor of experimentation for experimentation’s sake.

In the early nineteenth century, mathematicians were grappling with negative and imaginary numbers, and the potential of using symbols (a, b, c, and so on) instead of whole integers (1, 2, 3, and so on) in their equations. Symbolic representation would free them to play with difficult concepts, like the square root of a negative number, without the constraint of producing a numerical answer. Mathematician Augustus De Morgan, a contemporary of Carroll’s and a gentler skeptic of the new math, is the original source of the quote in the headline above: “… 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” (Pycior 149). Carroll would equate the new math with absurdity when he creatively plagiarized De Morgan’s phrase years later: during the Knave of Hearts’ trial, Alice responds to the White Rabbit’s nonsense verse with “‘I don’t believe there’s an atom of meaning in it’” (Pycior 149).

Carroll showed his discomfort, disdain, and derision for this unseemly subversion of pure mathematics through artful word play. Negative numbers aren’t that daunting to the modern user, but the concept of a quantity less than nothing was very troubling to many nineteenth century mathematicians. Carroll tackled the sub-zero with several interesting conversations between Alice and her new friends, including the famous tea party scene:

“‘Take some more tea,’ the March Hare said to Alice, very earnestly.

‘I’ve had nothing yet,’ Alice replied in an offended tone, ‘so I can’t take more.’

‘You mean, you can’t take less,’ said the Hatter, ‘it’s very easy to take more than nothing’” (106).

Negative numbers created a conceptual problem for the literal Victorian thinker: how is one “sure of something” that has no physical analogue in the real world? Representation becomes an even bigger problem when one thing, a symbol, can stand for something else, or worse, everything else. Carroll characterized the trap of multiplicity in several scenes in Wonderland, including the Mouse’s proposition to help all the animals get dry after they fall into the pool of big Alice’s tears:

“This is the driest thing I know. Silence all round, if you please! William the Conqueror favoured by the pope, was soon submitted to by the English… even Stigand, the patriotic archbishop of Canterbury, found it advisable…”

“Found what?” said the Duck.

“Found it,” the Mouse replied rather crossly; “of course you know what ‘it’ means.”

“I know what ‘it’ means well enough, when I find a thing,” said the Duck; “it’s generally a frog or a worm. The question is, what did the archbishop find?” (31).

“It” is a symbol or variable, like an “x”, standing in place of many things. This multiplicity was the hallmark of symbolic algebra, allowing mathematicians to develop their own grammars rather than adhere to a predefined set of rules designed to produce one, definitive result. “It” becomes malleable and easy to manipulate, no longer dependent on an absolute truth, but the truth of the duck or the archbishop, who are now on equal authoritative footing.

*

Carroll tried to literalize what he saw as the madness in symbolic algebra, but it would seem his narrative broke free and got away from him. With Alice, he in fact found a way to make symbolic algebra expressible.

My favorite example of the power of his mathematical word play comes early in the book, when Alice drinks the potion and begins shrinking in size:

“…she waited for few minutes to see if she was going to shrink any further: she felt a little nervous about this, “for it might end, you know,” said Alice to herself, “in my going out altogether, like a candle. I wonder what I should be like then?” And she tried to fancy what the flame of a candle looks like after the candle is blown out, for she could not remember ever having seen such a thing’” (11-12).

This is an existential moment. Alice contemplates what it would mean to “go out altogether,” to become something that is essentially nothing – the dying of the candle flame. But she contemplates this nonexistence, this passing through zero, as an entity of its own and even what this nothing might look like. Her imagination may be thwarted by the inability of her memory to fill the gap, but that doesn’t answer the question or end her questioning.

The habit of musing Carroll gives to his characters builds support for the very thing that he fears and distrusts. We see it later in the book when Alice awakens from the dream and transfers her wondering on to her sister – what would the dream of Wonderland be like in her sister’s head? For that matter, what is it like in our heads, as readers? In many of the same ways mathematicians were playing with symbols in their algebraic equations to change the potential inherent in the mathematical problem, Carroll played with the algorithm of the narrative through character and dialogue and plot, creating consciousnesses that could be plugged in to narrative experience, just as variable symbols could be plugged in to formulas.

At the end of Alice’s Adventures in Wonderland, Alice’s sister acknowledges her weird experiences, but grounds her in reality, telling her it is time for tea. As Alice runs off, her sister begins to reflect on the tale, and finds herself in Alice’s shoes:

“..the whole place around her became alive with the strange creatures of her little sister’s dream… So she sat on, with closed eyes, and half believed herself in Wonderland…” (190-191).

…but she also recognizes how easily she could dispel the dream:

“…though she knew she had but to open them again and all would change to dull reality… the sneeze of the baby, the shriek of the Gryphon and all the other queer noises would change (she knew) to the confused clamour of the busy farm-yard…” (192).

The return to reality is not the defeat of nonsense – rather, the key is now left behind in the lock. Carroll used the supposedly suspect mathematical concepts to wend a way into the absurd, abandoning characters there without hope, and making them long for the authoritative certainty of traditional disciplines. But he also crafted a way through: he created a curious and enlightened protagonist who makes the tale relatable, shareable, and reproducible. He inadvertently equated the dream with reality through the interchangeable symbols of Alice, her sister, and all of us countless readers. Carroll experimented with his own rules and found new “laws of combination.” He tried to satirize the new symbolic algebra to weaken it but instead his intentional, irrevocable act of entropy emerged as a lucid and adventurous way to seek new, complex truths.

[For more on this subject, check out Helen Pycior’s At the Intersection of Mathematics and Humor: Lewis Carroll’s “Alices” and Symbolical Algebra, and Melanie Bayley’s  Algebra in Wonderland.]

References

Carroll, Lewis. Alice’s Adventures in Wonderland. London: Macmillian and Co. 1865. Print (scanned and digitized).

Dodgson, C.L. Curiosa Mathematica: Part I A New Theory of Parallels. London: Macmillian and Co. 1890. Print (scanned and digitized).

Bloom Post End

Susan Sechrist is a freelance technical writer who is striving to better integrate her creative and mathematical sides. She is the Interviews Editor at Bloom and published her first short story, the mathematically-themed A Desirable Middle, both in Bloom and the Journal for Humanistic Mathematics.

Illustrations credit: John Tenniel [Public domain], via Wikimedia Commons.

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