The literary works of Lewis Carroll are littered with references to mathematics, the vast majority more subtle and involved than the above. Although unquestionably written for a young audience, his renowned Alice books and his less well known novels all include references and allusions to sophisticated mathematical concepts and problems. This essay will introduce Carroll as a man and author, and attempt to identify examples of these references, analyzing their purpose and effect. Charles Lutwidge Dodgson was born on January 271832 in rural Cheshire, the third of eleven children.

He was apparently of colourful ancestry, Bakewell (1996) claiming he can, with “a certain amount of ingenuity” claim to have been related to Lady Godiva, the Earls of Mercia and Northumbria and even Queen Victoria. Yet the man himself seems to have led a far from colourful life. In 1851 he entered Christ Church, Oxford where he distinguished himself in mathematics, and remained there as librarian, mathematics lecturer, and curator for the next forty-seven years. As a mathematician his texts are widely considered as conservative and never significant, and are read now only because of historical interest.

As a man he has been traditionally regarded as eccentric and at home only in the company of little girls. In fact, only the recent work of Karoline Leach (1996) in destroying the reliability of Dodgson’s early biographers has begun to alter the long held view that his love for young girls was a suppressed sexual passion. Under the pen name of Lewis Carroll, Dodgson produced in Alice’s Adventures in Wonderland, and the follow up Through the Looking Glass and what Alice Found There, two of the best loved children’s books of all.

It is widely known that the Alice tales were born during what Dodgson described in his diary as “that golden afternoon” of the 4th of July 1862 when he entertained the children of the Dean at Christ Church with his stories, whilst on a rowing trip in Oxford. One of these children was of course the six-year-old Alice Liddell, the inspiration for the main character. Some months later Dodgson was encouraged to write the first draft of Alice’s Adventures Under Ground, which by 1865 had developed into Alice’s Adventures in Wonderland. This being well received, he followed up with the equally successful Through the Looking Glass in 1871.

His first publication of humorous and other verses was Phantasmagoria and Other Poems in 1869. The acclaimed nonsense poem The Hunting of the Snark: an Agony in Eight Fits was published in March 1876. Sylvie and Bruno in 1889 and Sylvie and Bruno Concluded four years later represented the biggest effort by Carroll in literature, a mixture of fairy-tale, social novel and collection of ethical discussions, but they never came close to matching the success of his Alice books. His other works are generally not well known, although were often quite interesting.

A Tangled Tale published in 1885 was a series of puzzles and paradoxes and according to Fisher (1972) represented “… Carroll at his playful best in combining tantalising whimsy with straightforward mathematics”. Similar puzzles appeared in Pillow-Problems in 1893. An attempt to introduce formal deductive logic to a kindergarten audience, was The Game of Logic which Carroll published in 1886. Although unsuccessful in so far as it was too challenging for such a young audience, it did introduce some interesting methods and the examples were as quaint as the contents of Wonderland.

In developing the theme for a more adult audience his last book was Symbolic Logic: Part 1, Elementary, interestingly under the authorship of Lewis Carroll rather his real name that he still used for his mathematical publications. Of these none were of great impact, Euclid and his modern rivals is usually pointed to as a work of historical interest if not great mathematical incision. To consider first Alice’s Adventure in Wonderland, mathematical references and hints are present right from the very beginning.

Alice’s fall down the rabbit hole at the start of Wonderland is prompted partly by her dissatisfaction her sister’s book which “… had no pictures or conversations in it, ‘and what is the use of a book,’ thought Alice, ‘without pictures or conversation? ‘”. From a mathematical point of view it is interesting that from within Alice’s system of understanding (that of a seven-year-old girl), her sister’s book has no use in that it contains nothing by which she is able to interpret any meaning, through mechanically applying her rules of inference.

Alice could just as easily have followed the white rabbit directly into the “low long hall” without falling four thousand miles, but it is a device by which the continued theme of distortion of time is introduced; without it the Mad Hatter’s watch could not possibly be two days out. Not only that, but it allows for the introduction of a bizarre mechanics problem, seemingly a passion of Carroll’s. Speculation of the outcome of a fall through a ‘hole’ in the centre of the earth, assuming one to be possible, continued to be rife in Carroll’s day having been previously addressed by Plutarch, Bacon and Voltaire.

That Carroll used such a method to introduce Alice to her Wonderland is typical of the way in which he hints at his mathematical enthusiasms throughout his works.. We can see though his other publications that he was interested in such dynamics problems, that through his literature he was willing to ponder the effects of gravity in similarly unfeasible, yet mathematically interesting conditions. There is for example Mein Herr’s gravity-trains in Sylvie and Bruno Concluded, which are “without any engines-nothing is needed but machinery to stop them with”. ‘Can you explain the process? ‘ said Lady Muriel. ‘Without using that language, which I can’t speak fluently? ‘ ‘Easily,’ said Mein Herr. ‘Each railway is in a long tunnel, perfectly straight: so of course the middle of it is nearer the centre of the globe than the two ends: so every train runs half-way down-hill, and that gives it force enough to run the other half up-hill. ‘” Here we can clearly see the mathematical brain of Carroll, or rather Professor Dodgson, at work.

Perhaps such a system might under specific conditions in theory behave as Mein Herr describes, yet the idea finds itself in such a ‘little fairy-tale’ (as Carroll describes the book in its preface) because as everybody knows in a real world of friction and other external forces the idea is fantasy. Yet, this fantasy situation has been arrived at through following a reasonable concept to an unreasonable length. Meanwhile Alice would presumably neglecting forces due to the earth’s rotation, perform simple harmonic motion about the centre of the earth damped by air resistance until eventually coming to rest at the centre of the earth.

Yet here Carroll is not interested in a discussion of the cleverness of the ideas of such dynamics, but for us to share in Alice’s bewilderment, and so she safely lands “… upon a heap of dry leaves, and the fall was over”. A recurring theme in Carroll’s works, is that of geographic maps. As Alice falls down the rabbit hole she points out “maps and pictures hung upon pegs”. Just as a symbol in algebra is an arbitrary encoding of something, a map is similarly an arbitrary encoding of a portion of geographical terrain.

Carroll turns his attention to maps in a particularly interesting section from Sylvie and Bruno, when the ever knowledgeable Mein Herr explains the development of map-making in his country. “`We very soon got to six yards to the mile. Then we tried a hundred yards to the mile. And then came the grandest idea of all! We actually made a map of the country, on the scale of a mile to the mile! ‘ `Have you used it much? ‘ I enquired. `It has never been spread out, yet,’ said Mein Herr: `the farmers objected: they said it would cover the whole country, and shut out the sunlight!

So we now use the country itself, as its own map, and I assure you it does nearly as well. ” The idea of a map of unit size, such that it would cover exactly the country it is supposed to represent, appears ludicrous to us. Is it obviously ridiculously impractical, and of no real use or interest. However in mathematical terms, Carroll has taken the one-to-one correspondence of the map and terrain it describes, and made it an identity function. This concept is perfectly valid and interesting in the world of mathematics, only when giving it this physical meaning does it appear at all absurd.

In constructing his make-believe world Carroll seems to be strictly applying the notions of abstract mathematics to elements of reality, as a means of creating the dreamlike state his characters inhabit. A similarly strange and useless map appears in The Hunting of the Snark: an Agony in Eight Fits. “He had bought a large map representing the sea, Without the least vestige of land: And the crew were much pleased when they found it to be A map they could all understand. ” The Bellman has, it is revealed, bought “A perfect and absolute blank! “.

Despite the obvious difficulties a blank map will bring, the buying what was defined as a map has given the Bellman and his crew the idea that is will be of use. The map is said to ‘represent’ the sea, and taking any small enough part of the sea where water is represented by blankness, it would be as accurate as any map. Indeed the lines of longitude and latitude are merely arbitrary conventions. Again when looked at with disregard for any practical physical interpretation, but importantly with a strict and correct idea of reason, the idea of such a map is sound.

It is only with the application to a real situation that it appears foolish, the nonsense world of Carroll’s literature is here conceived from a logical base. Lewis Carroll’s restless mind was always eager to invent new games and objects, he apparently invented the original double-sided adhesive tape and designed an early form of Scrabble (Fisher page 12). It is generally considered that Carroll modelled his White Knight character in Through the Looking-Glass upon himself, and they clearly share a passion for inventions and tricks.

One example of this, which incorporates Carroll’s eagerness to include situations where gravity behaves unusually, is the White Knight’s charming invention of “… a new way of getting over a gate”. “Now, first I put my head on the top of the gate-then the head’s high enough-then I stand on my head-then the feet are high enough, you see-then I’m over, you see. ” Here it is the Knight who has the distorted view of gravity. We see earlier with his exclamation “Now the reason hair falls off is because it hangs down-things never fall upwards you know” that he has a basic understanding, but his theory is not complete.

This is clear through his description of his box of clothes and sandwiches; “… I carry it upside-down, so that the rain can’t get in”. Just as he has overlooked the effect of gravity upon himself in his gate invention, he has let his things fall from his box. He seems to believe gravity only has an effect upon things he has actually seen falling. Things never fall upwards but not necessarily everything falls downwards. Carroll deliberately avoids using the word gravity during this chapter, yet it is surely no coincidence that the White Knight was looking “a little grave”, and twice remarks “gravely”.

These puns will probably have been beyond his young readers, and therefore most likely represent Carroll playfully hinting at a vaguely mathematical topic, probably for his own fun caring not whether his readers appreciated it. Lewis Carroll was interested by problems of the nature of time, and consequently his works are full of fanciful and erratic time schemes, which sometimes reveal his reflections upon the subject. A problem that troubled him for over thirty years was that of where, and when the new day begins. His pondering of this subject apparently ‘cast a gloom over many a pleasant party'(Fisher 23).

Carroll wrote of how in tracing the path of the day around the earth either there would be no distinction at all between successive days, or there be a fixed line where the change take place, where he reckoned ‘there would be quarrel every morning as to what the name of the day should be’. Carroll troubled government offices and telegraph companies with this dilemma, even after the establishment of the International Date Line in 1884 he was not settled in his mind. While time behaved in such an illogical way in his eyes in reality, through his books of nonsense he allowed it to behave even more so.

The most celebrated example of this is the striking example of the Red Queen running in Through the Looking Glass. “… the Queen went so fast that it was all she could do to keep up with her: and still the Queen kept crying `Faster! Faster! ‘ but Alice felt she COULD NOT go faster, though she had not breath left to say so. ” The strange thing being of course “… that the trees and the other things round them never changed their places at all: however fast they went, they never seemed to pass anything. ” Mathematicians have of course been keen to read into this queer behaviour all sorts of things.

Taylor (1952) was almost certainly being rather optimistic when he suggests that Carroll “may anticipate Einstein” here. However, what he and other commentators agree on is that this episode surely represents a mathematical trick. The accepted definition of speed in our world is the ratio of distance to time. Lewis Carroll’s Looking Glass world seems to have inverted this concept, redefining speed as the ratio of time to distance. Such inversion was a great obsession of the author, not only in his fiction but in his mathematics and other hobbies.

His fascination with such mirror reversal was apparent not only in his creation of a Looking Glass world, but he would, according to Fisher, “… frequently write letters back to front, not only in mirror-writing but with the last word first and vice-versa”. He also remarks that this passion was extended to mathematics “… where he devised a new multiplication process whereby the multiplier is written backwards and above the number to be multiplied”. As a consequence of this redefinition the higher the speed at which the Queen and Alice run, the smaller the distance they cover.

Thus the Queen’s explanation: “… HERE, you see, it takes all the running YOU can do, to keep in the same place. If you want to get somewhere else, you must run at least twice as fast as that! ” They seem to be running not in a spatial co-ordinate, but in time. This prompts the idea the Queen expresses with her “`Why, we passed it ten minutes ago! Faster! ‘”. Again, an apparently random and absurd happening can be explained clearly and concisely through a mathematical concept, albeit one which has been subtly altered to achieve the desired effect.

The absurdity is there, but is certainly not as confused as is first seems. Further on in Through the Looking Glass the Red Queen explains her own unfamiliar situation regarding time, her “living backwards”. “`It’s a poor sort of memory that only works backwards,’ the Queen remarked. `What sort of things do YOU remember best? ‘ Alice ventured to ask. `Oh, things that happened the week after next,’ the Queen replied in a careless tone. ” This does in fact follow on quite logically from the White Queen’s earlier running in time.

If time can be treated as a co-ordinate through which one moves; if one length of the chess board the characters inhabit is represented by time, then it seems sensible moving in one direction means time must run forwards, the other backwards. Here the Red Queen is following this idea to a logical conclusion, but of course due to the original inversion the results seem far from logical: “`there’s the King’s Messenger. He’s in prison now, being punished: and the trial doesn’t even begin till next Wednesday: and of course the crime comes last of all! ” In Sylvie and Bruno Lewis Carroll , again through the wise Mein Herr, explains another stimulating curiosity regarding time. He tells of how in a country he has visited there exists a remarkable scheme to make use of wasted time.

“By a short and simple process-which I cannot explain to you-they store up the useless hours: and, on some other occasion, when they happen to need extra time, they get them out again. ” When asked why he was unable to explain the process, Mein Herr replies with typically sound Carrollian logic. Mein Herr was ready with a quite unanswerable reason. ‘Because you have no words, in your language, to convey the ideas which are needed. I could explain it in-in-but you would not understand it! ‘” Perhaps indeed such a scheme is possible if only our language allowed for such concepts to be considered. The restriction imposed by the language we have developed perhaps echoes the restrictions upon mathematical thought that is imposed by our entrenched views. Carroll often visits the topic of judicial proceedings in his fictional work.

Perhaps the proceedings of a court of law held interest for Carroll as a mathematician, in that a legal system is effectively an attempt at creating a formal structure by which to define and punish wrongdoing. Legal language represents an attempt to achieve a formal system the like of which only occurs in the sciences, through the refining of natural language. That Carroll is aware such a system must, unlike any mathematical parallel remain imperfect, is perhaps illustrated by the chaotic trial of the Knave of Hearts in chapter eleven of Alice in Wonderland.

Not only does the judge initially forget to allow for the calling of any witnesses, but the jurors are as Alice notices; “Stupid things! “, who have to write down their names “… for fear they should forget them before the end of the trial”. Belmonte (1992) sees this, as well as the mock judicial proceedings in the sixth fit of The Hunting of the Snark as Carroll’s “… indictment of legal finagling and the too-serious application of legal systems”. Interestingly he also points to the fact that Alice identifies the judge “.. ecause of his great wig”, emphasizing the fact that a wig is a symbol signifying judgeship, in the same way that an algebraic symbol can signify a particular property. The issue of what exactly Lewis Carroll the mathematician was playing at when in Alice’s Adventures in Wonderland he lets a confused Alice say “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! “, has been much discussed by those attempting to read into the mathematics within his books.

It seems fair to suggest there is no doubt that a mathematician such as Lewis Carroll could not write such a passage without first it making some sort of sense, probably not on a level to be understood by Alice or his target audience, but for the satisfaction of his own playful mind. A man whose poetry includes the following stanza, part of a type of poem he called the Double Acrostic, the forerunner of the modern crossword puzzle, surely cannot write such mathematics without giving it some precise basis.

“Yet what are such gaieties to me Whose thoughts are full of indices and surds? 2+7x+53 =11/3. ” It is Gardner (1960) who gives the simplest explanation for this problem, one based upon this traditional practice of teaching multiplication tables only as far as twelve. He argues that if 4 times 5 is 12 and 4 times 6 is 13, then continuing this pattern will yield 4 times 7 as 14, 4 times 8 as 15, 4 times 9 as 16 and so on until 4 times 12 is 19, where traditional teaching, including that of Alice will stop, one less than 20. However a more complete explanation, which offers a clue as to why four times five can be twelve to start with, is that of Taylor (1952).

He points out that on a number system on the scale 18 where the numbers progress 1,2, 3, 4, 5, 6, 7, 8, 9, (10) (11) (12) (13) (14) (15 ) (16) (17) 10, 11, 12, and so on, 4 times 5 does in fact give 12. And similarly on the scale 21, 4 times 6 does equal 13. Increasing the scale by three each time gives… 4 times 7 is 14 (on the scale 24), 4 times 8 is 15 (on the scale 27), and so on until one reaches 4 times 12 is 19 (on the scale 39). However, it is with the next step that it is revealed why Alice will never reach 20. times 13 on the scale 42 is not 20, as here the number which follows 19 is not 20 but 1(10) or one-ten, the equivalent to 52, or 4 times 13 on the scale 10. As Taylor states this problem would be of course way beyond Alice Liddell, aged ten, “… but he could use it to amuse and bewilder her-to give the effect of nonsense”. If, as seems likely this was his aim, then this represents a very clear example of how Lewis Carroll’s enthusiasm for mathematics helped him develop the apparent absurdity of his fiction out of a sound logical, and here a sound mathematical base.

Surely Carroll the mathematician would not have allowed himself to write mathematically invalid nonsense, even in such a children’s tale, and so instead sought to “… give the effect of nonsense”, whilst it could remain justifiable and even sensible in his eyes. A more trivial occurrence of number in Carroll’s fiction is the presence of a certain number in his work that had no significance in his day, but has more recently become celebrated through the work of Douglas Adams. Whenever Carroll needed to use an arbitrary number, he seems to have chosen forty-two.

During the trial of the Knave in Alice’s Adventures in Wonderland the judge declares “Rule forty-two. All persons more than a mile high to leave the court”. Similarly in the preface to The Hunting of the Snark “Rule 42 of the code” is that “No one shall speak to the man at the helm”, and in chapter seven of the same book the Baker has “… forty-two boxes, all carefully packed”. It was the number forty-two that Adams suggests is found to be the answer to “life, the universe and everything” and Belmonte (1992) suggests that it “… eems likely that Adams borrowed this particular answer from Lewis Carroll”.

The works of Lewis Carroll frequently show subtle signs that they were written by a mathematician, not only in the manipulation of mathematical ideas as I have detailed above, but also through his precise use of language and phrases that echo mathematical concepts. For those untrained in mathematics these would surely pass unnoticed, but to some one more knowledgeable in the subject the very style of this author is friendly and certain passages hold particular familiarity.

Consider the assertion of the Bellman in the Hunting of the Snark, “What I tell you three times is true”. Obviously an abominable suggestion to a mathematician, perhaps this mocks the less rigorous work of undergraduates whom Carroll taught. Consider an arbitrary, otherwise innocuous extract from the Red Queen in Through the Looking Glass “`when you’ve once said a thing, that fixes it, and you must take the consequences. ‘”. For me, this echoed the act of stating a definition in mathematics. Frequently Alice’s confusion is a result of the precision of the language of the characters with which she converses.

Humpty Dumpty’s “‘How old did you say you were? ‘” does not of course require the same answer as “‘How old are you? ‘”, as he indignantly reminds Alice: “‘If I’d have meant that, I’d have said it'”. To a mathematician this is not pedantic, but quite valid and indeed a necessary distinction. Nonsense would only arise here if the two questions did in fact mean the same thing. A similar theme is explored by the White Knight in Through the Looking Glass, when attempting to explain to Alice the name of his song.

Much to Alice’s confusion the Knight is keen to differentiate between what the song is, what the song is called, the name of the song, and what the what the name of the song is called. To make such distinctions here helps to give an impression of confusion, but distinguishing between these different concepts is necessary to a mathematician. Indeed such problems of identity are of particular interest to computer scientists, making such a distinction within their programs is of fundamental importance.

The examples I have drawn attention to by no means represent a full commentary on the mathematics in Carroll’s fiction, as such a task would be huge. Dozens of example remain untouched for no particular reason other than time. Examples as the play with the concept of size as Alice successively shrinks and grows early on in Wonderland and the logic of Tweedledum and Tweedledee could just as easily been included. Yet what I hope the examples from the texts I have used do show, is that Carroll the mathematician can be clearly seen through his fiction.

The apparently nonsensical ramblings that gives Alice and his other books their unique quality, can in fact often be traced back to a sound logical base that hints at the mathematical enthusiasms of this most curious and playful author. To those of us with a knowledge of mathematics, the best summing up of Carroll’s fiction perhaps is by the Red Queen in Through the Looking Glass: “`You may call it “nonsense” if you like,’ … `but I’VE heard nonsense, compared with which that would be as sensible as a dictionary! ‘”.