Alice’s Adventures in Wonderland was first published in 1865 and written by Charles Dodgson under the pseudonym, Lewis Carroll. The story was an expansion of Alice's Adventures Under Ground, a handwritten book produced by Carroll for Alice Liddell, a young daughter of Henry Liddell, Vice-Chancellor of Oxford University and Dean of Christ Church, a college at the university. Dodgson was a mathematician who taught at Christ Church. A sequel followed in 1871, entitled, Through the Looking-Glass, and What Alice Found There.
Much has been written over the years about the influence of both Oxford and mathematics on Dodgson’s children’s works in general and Alice’s Adventures in Wonderland in particular. In this blog we will consider two main aspects. First, we will seek to bring the sense of wonderment in the book to the fore. This is plainly present in the text, but can get overlooked in the drive to ‘explain’ things by reference to Dodgson’s background and circumstances. Secondly, we will discuss the supposed mathematical allusions in the text.
The theme of wonder in Alice in Wonderland is firmly rooted in Alice herself and this is emphasized in the opening chapter where seven times we find Alice wondering. The wonderment begins as she falls down the well, for she had time, ‘to wonder what was going to happen next’. And later, still falling, she poses several ‘I wonder’ questions: ‘I wonder what Latitude or Longitude I ’ve got to ...?’; ‘I wonder how many miles I ’ve fallen by this time ?’; ‘I wonder if I shall fall right through the earth !’; and, ‘But do cats eat bats, I wonder ?’. Then, having landed and walked into a hall, we find her, ‘wondering how she was ever to get out again’. And finally, as she shrunk and shrunk after drinking the potion, she imagined herself disappearing like a candle going out and asked: ‘I wonder what I should be like then?’ This sense of wondering was not present in the character immediately, for there is another occurrence of the verb wonder in the chapter where, reflecting afterwards about the speaking rabbit, ‘it occurred to her that she ought to have wondered at this, but at the time it all seemed quite natural’.
The use of the word wonder and its related forms occurs on several other occasions in the succeeding chapters, but the first chapter has by far the most occurrences. The place to which Alice goes is never referred to her as Wonderland by her or by any of its inhabitants. The term is only used in the opening poem and by the narrator at the end of the story. It is a privileged piece of information between the author and reader which neither Alice nor her sister apparently knew. But the frequent use of wonder, especially in the opening chapter, shows that Wonderland is not a trivial choice of term, but captures the attitude of the curious Alice towards the land she had fallen into. But what kind of wonderland was it? Was it a mathematical wonderland?
Mathematics is clearly present in the text. When Alice tries multiplication in Chapter 2, she says, ‘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!’. In The Annotated Alice - The Definitive Edition, Martin Gardner gives two possible explanations for these, ‘wrong’, sums.1 The first is that children would normally only learn multiplication tables up to twelve, and 4 x 12, following the sequence of answers, 12,13,14 ..., would give 19. To get beyond 20 would require 4 x 13, which a child would not normally learn. The second suggestion, which he ascribes to A.L. Taylor, is more ingenious, but also quite neat, not least because it shows that Alice’s answers are, in a sense, correct. It proposes that the calculations are actually using bases other than base-10. A detailed description of this is given in an appendix at the end of the blog. This is an unequivocal example of mathematics in the story. But others have proposed that the connection between the story and mathematics runs much deeper.
Melanie Bayley, in an article in the New Scientist, argued that Dodgson, a conservative mathematician in outlook, was a making a thinly disguised attack in the story against the more esoteric areas of mathematics which were developing at the time. In particular, she says that this accounts for some of the incidents added to the story for publication which were not in the original handwritten story. Take for example, the Mad Hatter’s tea party. There are three characters present: the Hatter, March Hare and Doormouse. But Time, personified as a character in the story, is missing. As a consequence, time is stuck at six o’clock. Bayley proposes this is a parody of quarternions, a feature discovered by the mathematician William Hamilton, which involved four terms: the three spatial dimensions and time.
There is no doubt that the story provides a useful resource for illustrations of various aspects of mathematics and related disciplines. For example, the numerous occasions where Alice changes in size, and even the instance of a baby changing into a pig, reflect aspects of geometrical continuity and transformation. Yet, without explicit evidence, it is difficult to prove that Dodgson intended the story to be a parody of, or an attack on, nineteenth century mathematics. But is this biographical approach to critiquing books, whereby the author’s life history and personality are drawn on to ‘explain’ the story, the right approach?
The author, Sebastian Faulks, in his book, Faulks on Fiction, warns against the overuse of the biographical approach to critiquing books.2 He argues that this detracts from the simple truth that what has been written is a work of imagination. This surely also applies to Dodgson. Rather than scour Oxford and nineteenth century mathematics for explanations of his fiction, maybe we should give much more emphasis to his imagination per se. Writing of adult fiction, Faulks says that the obsession with a biographical approach has meant that, ‘the element of wonder has somehow been lost’. It would be sad if this also happened to children’s books, especially Alice’s Adventures in Wonderland.
4 x 5 = 20.
Here, ‘20’ is in base-10, the base we commonly use, where ‘2’ means two tens. But Alice says:
4 x 5 = 12.
How can this be right? Well, we can assume that the ‘1’ does not mean one ten, but one eighteen, which, in base-10, gives 18+2 = 20. In other words, 4 x 5 = 12 if we assume that the numbers are in base-18.
4 x 6 = 13?
This is ‘correct’ if we assume that a base-21 is being used. That is, ‘1’ means one twenty-one, which gives, in base-10, 21 + 3 = 24.
So we have a pattern developing. The answers are increasing by one (12, 13 ...), but the bases are increasing by three (18, 21 ....)
Let’s continue this pattern and see what happens ...
4 x 7 = 14 (Base-24). In base-10, 24 + 4 = 28.
4 x 8 = 15 (Base-27). In base-10, 27 + 5 = 32.
4 x 9 = 16 (Base-30). In base-10, 30 + 6 = 36.
4 x 10 = 17 (Base-33). In base-10, 33 + 7 = 40.
4 x 11 = 18 (Base-36). In base-10, 36 + 8 = 44.
4 x 12 = 19 (Base-39). In base-10, 39 + 9 = 48.
So far so good. But can we get to 20?
Well, in base-10, 4 x 13 = 52. But Alice would have said, 4 x 13 = 20. But 20 in base-42 would be 84 in base-10 (i.e. two forty-twos). Clearly this is not the right answer. The problem is that there is no symbol for 10. Let’s make up a symbol for ten; let’s call it ‘a’. Then we have:
4 x 13 = 1a (Base 42). In base-10, 42 + 10 = 52.
Does this explain Alice’s calculations? We can only wonder ...