Mathematical Structure in Wonderland
Gregory George
In his essay “Alice’s Journey to the End of Night,” Donald Rackin describes Wonderland as “the chaotic land beneath the man-made groundwork of Western thought and convention” where virtually all sense of pattern is absent and chaos is consistent. Rackin claims that “there are the usual modes of thought—ordinary mathematics and logic: in Wonderland they possess absolutely no meaning.” Rackin argues that our traditional view of mathematics as an existing set of facts and rules that are predictable does not hold true in Wonderland.1 However, Rackin’s concept of mathematics is limited—he sees math as simply mathematical operations (multiplication, division, addition, and subtraction), which produce predictable results in our “logical” world. But mathematics also exists as abstract forms of structure, which indeed exist in Wonderland through sequence and measurement. Even though Alice’s Adventures in Wonderland presents a world that appears random and full of nonsense and inconsistency, these mathematical forms are preserved in Wonderland.
Contemporary philosophies of mathematics define the subject as the study of patterns, as opposed to the traditional study of numbers. These patterns exist in many abstract forms, such as numeric patterns, spatial (visual reasoning) patterns, patterns of motion, and patterns of growth or decay. Rackin recognizes the breakdown of one specific pattern, specifically the relationship between factors and products in base ten multiplications. From this evidence, he concludes that mathematics is meaningless in Wonderland, with no defined structure. But Rackin is making this assertion based on a small piece of evidence. Operations are a small part of mathematics, and can work as patterns themselves, just like the ones mentioned above. By looking specifically at number sequences and the use of measurement in the text, Rackin’s point is weakened simply by his narrow conception of mathematics.
The most obvious example in the text where mathematics as we know it is different is when Alice tries to recall basic arithmetic facts when she first falls down the well: “Let me see: four times five is twelve, and four times six is thirteen, and four times seven is—oh dear! I shall never get to twenty at that rate!”2 We, along with Rackin, recognize that the multiplication is incorrect, but is that enough to assert that there is no mathematical structure within the facts that Alice rambles off the top of her head? Martin Gardner, in The Annotated Alice, attempts to make sense of Alice’s multiplication, as quoted above from the text, if she continued forward. “The multiplication table traditionally stops with twelves, so if you continue this nonsense progression…you end with 4 times 12 (the highest she can go) is 19—just one short of twenty.” 3
The placement of this scene in the story raises the question of cause and effect. Does Wonderland have the effect on Alice’s thoughts of creating these falsifications, or are these thoughts pure confusion and nonsense on the part of Alice alone? Maybe Gardner is trying to make sense out of nonsense, and the “progression” is truly nonsense with random numbers to show the reader that Wonderland is a completely different place. But by removing the emphasis from the actual multiplication that is taking place and instead looking at the behavior of the numbers themselves, the mathematical pattern of a sequence still exists. Shifting the emphasis to the underlying structures that may exist challenges Rackin’s assertion that mathematics is meaningless in Wonderland.
A sequence is a number pattern that has a mathematical “rule” for creating each term. If we examine the numbers in these multiplication facts, we see that the first factor is a four in each “fact” that Alice states. This represents a constant sequence, where each term is the same as the preceding. The second factors form an increasing sequence of {5, 6, 7,…} with the next term in the sequence one more than the previous term. The “products” also form an increasing sequence {12, 13,…}. Thus, numeric patterns do exist in this seemingly ridiculous example. By using the contemporary definition of mathematics, these patterns are mathematical by the behavior of the numbers. Rackin fails to recognize this connection of sequence and mathematics. This presence of sequence shows that mathematical structure still exists, even though the calculations cause suspicion.
Sequence also exists in the courtroom scene at the end of Alice in Wonderland. When the King read “Rule Forty-Two. All persons more than a mile high to leave the court,” Alice responded with “that’s not a regular rule: you invented it just now.” The King states that “It’s the oldest rule in the book” and Alice challenges the King’s logic by saying “Then it ought to be Number One.” 4 This shows that Alice still has a sense of number sequence in Wonderland, and the King shows that he does as well through his reaction of turning pale and closing his rulebook very hastily. The King’s reaction alone shows that he agrees with Alice’s logic or else he wouldn’t have reacted so strongly by becoming upset. It is as if Alice “did the math” and made a fool of the King when the correct answer is apparent to everyone. The logic and pattern of the natural counting numbers still exist in Wonderland and the rulebook. This brings forth another important structure and sequence in mathematics—the relating and ordering of numbers, generally called “number sense.” Number sense is a strong foundation of mathematics, and Rackin’s narrowed focus prevents him from acknowledging that number sense exists in Wonderland at all.
Another important pattern in mathematics is spatial (visual) patterns and relationships, namely measurement. In Wonderland, measurement is used just as we use it in our above-ground world with inches as the primary unit. When talking to the Caterpillar, Alice claims that “…three inches high is such a wretched height to be” and before she approaches the Duchess’ house, she nibbles on the mushroom to bring herself to only nine inches in height. The Caterpillar’s initial remark to Alice’s complaint of only being three inches high was “It is a very good height indeed!” 5 This shows that the Caterpillar himself is aware of his own height and his relation to his surroundings. Furthermore, the Caterpillar’s own awareness of his own height is apparent when he asks Alice how tall she would like to be when he shows her how she can use the mushroom to change her size. Even though the Caterpillar does not explicitly speak in terms of sizes and units, he is aware that different sizes are equated with different units of measurement. If he did not have this knowledge, there would be no reason for him to find offense at Alice’s remark about a height of three inches. Thus, this is one character in Wonderland that recognizes the structure of measurement as it is used to describe physical heights of objects with numbers and units.
Further patterns of measurement in Wonderland are revealed later in the text, such as the use of miles as another measurement unit. Another example outside of Alice is again in the courtroom with rule Forty-Two, where “All persons more than a mile high to leave the court.” Alice claimed she was not one mile in height, but the King and Queen asserted that she was, with the Queen claiming that she was “Nearly two miles high.”6 This shows that quantitative measurement exists in Wonderland, and this is shown through the size differentiation among the characters. Even the distinction between miles and inches as units of height are another type of pattern, namely the relationship between inches and miles. The conversion of inches to miles produces a constant relationship between the two units of measurement. Once again, these abstract notions of changes in shape and changes in quantities assigned to these shapes show patterns, which make measurement based on patterns. The explicit use of measurement in Wonderland and the patterns associated with it make it mathematical in structure. Rackin does not recognize these forms of measurement within Wonderland as being mathematical.
Not only do the concepts of sequence and measurement exist in Wonderland independently, but they work together in perhaps the most explicit example of sequence and mathematical structure in Wonderland. It is the scene where the Mock Turtle is describing “lessons” to Alice:
“And how many hours a day did you do lessons?” said Alice…
“Ten hours the first day,” said the Mock Turtle: “nine the next, and so on…That’s why they call them lessons…because they lessen from day to day”….
This was quite a new idea to Alice, and she thought it over a little before she made her next remark. “Then the eleventh day must have been a holiday?”
“Of course it was,” said the Mock Turtle.
“And how did you manage on the twelfth?” Alice went on eagerly.7