“Oh, I’ve had such a curious dream!”
Minos presiding, in one of William Blake’s illustrations for The Divine Comedy, circa 1824–7.
A HOUSE OF CARDS Wonderland’s final chapter begins with Alice being called to provide evidence in the trial of the Knave of Hearts. Surprised at being called as a witness, Alice jumps up and—forgetting she has grown so large—“tip[s] over the jury-box with the edge of her skirt … reminding her very much of a globe of goldfish she had accidentally upset a week before.” The exactly worded account and the fact that “the accident of the goldfish kept running in her head” suggests that here Carroll is teasing Alice Liddell (possibly to the amusement of her sisters) by introducing a mildly embarrassing real-life accident with a fish bowl in the Deanery.
Of course, in Wonderland, this will not be the last time Alice literally upsets the court. It foreshadows much of what is to come, and the matter of her great size—physically and metaphorically—suggests her growing power and influence in Wonderland.
“Here!” cried Alice, quite forgetting in the flurry of the moment how large she had grown in the last few minutes, and she jumped up in such a hurry that she tipped over the jury-box with the edge of her skirt, upsetting all the jurymen on to the heads of the crowd below, and there they lay sprawling about, reminding her very much of a globe of goldfish she had accidentally upset the week before.
In Lewis Carroll: A Portrait with Background, Donald Thomas draws a number of comparisons between Wonderland and the Greco-Roman underworld kingdom of Hades. Thomas theorizes on the influence of Virgil when it comes to legal matters: “That Dodgson intended a parallel or was conscious of being influenced by his reading [of the Aeneid] is beyond proof. He certainly used figures from Virgil’s account in Euclid and his Modern Rivals (1879), when two of the judges from the courts of Hades in the Aeneid, Minos and Rhadamanthus, act as mathematical examiners in a dream of contemporary Oxford.”
Thomas finds particular evidence of Virgil in the courtroom scenes and procedures of Wonderland. “The Queen of Hearts would have been peculiarly at home in Virgil’s underworld. Minos and Rhadamanthus preside over the courts of the dead, but hand over the guilty to Tisiphone, Queen of the Furies, for punishment. Virgil describes the procedure of the court of Rhadamanthus. ‘Castigatque auditque dolos,’ he chastises them and then listens to the account of their crimes.”
GEORGE BOOLE (1815–1864), a British mathematician, philosopher and logician, was the author of An Investigation of the Laws of Thought, on Which Are Founded the Mathematical Theories of Logic and Probabilities. The development of Boolean logic coincided exactly with Lewis Carroll’s academic career and seems to have influenced every aspect of his intellectual and imaginative life. In a mathematician’s Wonderland, George Boole would be the most obvious candidate for the King of Hearts.
Before the publication of Boole’s first work on symbolic logic, The Mathematical Analysis of Logic, in 1847, logic had advanced little since Aristotle’s time. However, in that work—and in his Laws of Thought in 1854—Boole showed for the first time how algebraic formulae could be used in logic to reveal (in his own words) “those universal laws of thought which are the basis of all reasoning” and “to give expression to them in the symbolic language of a Calculus.” The early twentieth-century mathematician-philosopher Bertrand Russell believed it to be a major event in the history of mathematics: “Pure mathematics was discovered by Boole, in a work which he called The Laws of Thought.”
A contemporary of Lewis Carroll’s was John Venn (1834–1923), the Cambridge logician who created a simplified notation system of Boolean logic involving interlocking circles known as Venn diagrams. As a measure of Carroll’s enthusiasm for Boolean algebra, he published what he believed was a better notation system, using interlocking squares.
George Boole: A major influence on mathematics and Carroll.
“Beyond six letters Mr. Venn does not go.”
CHARLES DODGSON
Carroll’s contemporary John Conington, the first professor of Latin at Oxford and editor of The Works of Virgil in three volumes, remarked that “this legal procedure of Rhadamanthus … was ‘hysteron proteron,’ that is to say putting the second thing first.” Thomas points out that this is the same procedure employed when “the Queen of Hearts insisted, ‘Sentence first—verdict afterwards.’ Dodgson’s Wonderland and Virgil’s underworld have strikingly similar judicial systems.”
Dodgson knew Conington well, and in his alphabetical squib “Examination Statute” wrote: “C is for [Conington], constant to Horace.” Thomas observes that Conington published his edition of Book VI of the Aeneid, with his comment on the justice of the underworld, “including this judicial dictum of the Queen of Hearts,” a few years in advance of Dodgson’s publication of Wonderland. He adds that perhaps not coincidentally, “another Oxford classicist, Arthur Sidgwick, a younger friend of Dodgson’s, remarked that Virgil’s was ‘a famous line from its inversion of the natural order of justice.’ ”
However, the trial of the Knave of Hearts is clearly the product of the mind of Charles Dodgson the mathematician and logician. Dodgson’s most extensive mathematical work was Symbolic Logic. It was dedicated “to the memory of Aristotle,” the father of logic, and states its primary mission was to give its readers “the power to detect fallacies, and to tear to pieces flimsy illogical arguments.” And although Symbolic Logic was not written for children, the teaching of logic to children was a hobby horse Dodgson rode for his entire life. He frequently gave talks at girls’ schools on this discipline. And two decades after Wonderland, he published The Game of Logic: rules for a board game for children played with a set of counters in which logic is expressed in terms of symbols, syllogisms and sorites.
The trial of the Knave has elements of a logical game played in accordance with certain rules or axioms and employing a specific formal language. The challenge for Alice is to discover the rules and the nature of the game. This is a difficult task, as the formalist’s concerns are not with everyday truths but rather with formal proofs that may be totally independent of reality and meaning in terms of everyday language.
Game theory: The board and counters of Carroll’s Logic.
Or, as Dodgson explains in the preface to his The Game of Logic: “It isn’t of the slightest consequence to us, as Logicians, whether our Premises are true or false: all we have to make out is whether they lead to the Conclusion, so that if they were true, it would be true also.” Why, we may ask, does Carroll set up this game in a courtroom as a trial? The answer is that Aristotelian logic had its origin in the education of lawyers and politicians with the practical aim of sorting out valid from invalid arguments. Since Aristotle, logicians have tried to formulate rules that, when followed, will ensure that only true conclusions are drawn from true premises. These are called “the rules of true argument.”
Although Dodgson had great respect for the classical tradition of Aristotelian logic, he did recognize its limits, and was much excited by the dramatic discovery of the algebraic formulation of logic by the British mathematician George Boole. This discovery of Boolean logic was made during Dodgson’s student years and had a profound effect on his work throughout his life. He was much taken up with this application of algebraic notations and principles to ancient Aristotelian logical problems. In fact, his Symbolic Logic was a work almost entirely concerned with the application of algebra to logic.
The Game in action: In Carroll’s Symbolic Logic.
In logic—as in law—there is a profound difference between evidence and proof, and in both, evidence must be rigorously tested. This is particularly true in Boolean logic, in which “soundness” and “completeness” are the two most critical properties in the construction of sentences and the validation of evidence.
When the King of Hearts attempts to decide whether Alice’s evidence is “ ‘important—unimportant—unimportant—important’—as if he were trying which word sounded best,” the regent is not being frivolous. The King is quite properly evaluating each word (or sentence) for soundness, or what mathematicians call a “well-formed formula” or a wff. The King’s judgment is based on the “sound” structure of the sentence, not on its meaning in ordinary speech.
The sentence (wff) must be sound and complete before any conclusion (or verdict) can be reached. This is what the Queen of Hearts loudly insists upon—in her final dispute with Alice—when she says, “Sentence first—verdict afterwards.”
The Queen is not being perverse: she is simply attempting to enforce the strict rules of Boolean logic in a formal system known as sentential calculus (what today is called propositional calculus). This system requires the Queen’s ruthless application of axes, by which Carroll means (and repeatedly puns) axioms. Just as traditional logicians have tried to set the true rules of argument since Aristotle’s time, Boole’s new algebraic system defines a valid argument as sets of logically progressive propositions.
This is why there are so many arguments rather than polite conversations in Wonderland. Alice is unknowingly entering into arguments that employ the formal language of sentential calculus. The Gryphon, for instance, constantly uses double negatives in his speech—this is in fact a sentential axiom of double negation. The Queen’s attempt to behead the body-less Cheshire Cat is a bizarre demonstration of what is known as the axiom of the excluded middle.
In the dispute over the validity of the so-called Knave’s letter, Alice says, “I don’t believe there’s an atom of meaning in it,” and the King replies, “If there is no meaning in it, that saves a world of trouble, you know, as we needn’t try to find any.” In Boolean terms, a zero value would be a valid and significant conclusion.
DE MORGAN’S LAWS George Boole’s pioneering work on the calculus of propositions was carried forward after his death by his colleague AUGUSTUS DE MORGAN (1806–1871), author of many mathematical works, including Formal Logic: or, The Calculus of Inference, Necessary and Probable (1847). He formulated De Morgan’s laws and the duality principle. If we accept George Boole as our mathematician’s King of Hearts, De Morgan would certainly be our mathematician’s Knave of Hearts. De Morgan was the intellectual heir of George Boole, just as (presumably) the Knave of Hearts was the heir to the King of Hearts.
A passage from De Morgan’s Trigonometry and Double Algebra (1849) is quoted by Helena M. Pycior in her “At the Intersection of Mathematics and Humour: Lewis Carroll’s ‘Alices’ and Symbolic Algebra” (1984). In his précis to symbolic algebra, De Morgan explains: “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.”
Ms. Pycior compares De Morgan’s statement with Alice’s declaration on the Knave’s letter: “I don’t believe there is an atom of meaning in it.” She then firmly concludes: “The coincidence of language in De Morgan’s algebraic text book and Carroll’s Alice’s Adventures in Wonderland is not accidental.”
Coincidentally, two decades after the publication of Wonderland, Carroll came to know De Morgan’s son, a noted artist and ceramicist. Carroll wrote in his diary in March 1887, “Called on Mr. William De Morgan and chose a set of red tiles for the large fire-place.” It appears the artist knew Carroll’s work as well, as the chosen tiles were decorated with figures from Wonderland and Looking-Glass: the Dodo, the Lory, the Fawn, the Eaglet and the Gryphon.
Carroll also became familiar with the paintings of William De Morgan’s wife, the Pre-Raphaelite artist, Evelyn De Morgan. The De Morgans, like Carroll, were deeply interested in psychic phenomena and spiritualism. Evelyn De Morgan’s paintings often depicted scenes from classical myths concerning life after death, and include her Demeter Mourning for Persephone.
“And yet I don’t know,” says the King, as he continues to examine the evidence. He then begins to deconstruct the evidence in the Knave’s letter by reducing it to what logicians would call atomic sentences, or in Alice’s terms an “atom of meaning.” In Boolean terms, the King is required to reduce everything to atomic units and arbitrarily attribute true or false values to each, although he must not define them—that is, the King will not ask the identity of “he or she or it.”
In Boole’s own words, the atomic sentences “admit indifferently of the values 0 and 1, and of these values alone.” In this way, all propositions are either true (with value 1) or false (with value 0). Astonishingly, Boole’s absurd-sounding system of binary logical on-off switches became the basis for all modern computer operating systems.
The trial of the Knave of Hearts is full of legal, as well as logical and mathematical, puns. In his “Metaphysics” Aristotle explains: “Those who use the language of proof must be cross-examined.” This was mirrored by the White Rabbit’s reminder that the King must “cross examine this witness,” although Carroll makes a punning joke of this by having the King interpreting this literally: staring at the witness with such a cross expression that he complains to the Queen, “It quite makes my forehead ache!”
Augustus De Morgan: Makes a symbolic appearance.
It has been suggested that Carroll is making a droll legal joke when the table of tarts are placed before the court as evidence: a delicious example of the corpus delicti, or body of proof. However, more significantly, the table of tarts suggests a pun on a table of torts (or tortious liability). Torts are essentially tables of laws of precedent that litigating lawyers must learn and cite during trials involving restitution.
Then too, it is possible to extend the punning from tarts to torts to tauts, as in tautology. Tautology is a key concept in propositional logic; it is a formula that is always true and can be confirmed, or proved, by use of a “truth table.” So we have typical Carrollian serial, or linked, puns: a table of tarts becomes a table of torts, then transforms into a table of tauts before arriving at a truth table.
A truth table is a mathematical table used in logic—specifically in connection with Boolean algebra and propositional calculus—to determine whether a proposition is true, and—as his twentieth-century editor William Warren Bartley discovered—Carroll had employed in his Symbolic Logic II. That is why, at the end of the trial, our Boolean King of Hearts “went on muttering … to himself: ‘ “We know it to be true—’ that’s the jury, of course,’ ” then confirms their judgment upon examining the table of tarts (or truth table): “ ‘Why, there they are!’ said the King triumphantly, pointing to the tarts on the table. ‘Nothing can be clearer than that.’ ”
Much of the humour in Wonderland is generated by the absurd obviousness of tautological statements when delivered in ordinary speech. They are logically circular; for example, all tautologies are necessarily true because they are tautologies. However, there are twenty-one tautologies (or axioms) that are essential rules in sentential, or propositional, logic. Alice is bewildered by many arguments that are essentially demonstrations of axioms. She is certainly unfamiliar with their Latin names: modus ponens or method of affirming, modus tollens or method of denying, modus tollendo ponens or method of affirming and denying, reductio ad absurdum or reducing to the absurd, and so on.
Two of these tautologies or axioms, known as De Morgan’s laws, appear to be employed by the Knave of Hearts in his absurd defence against the equally absurd charges made against him. First: since two things are false, it is also false that either of them is true. Second: since it is false that two things both are true, at least one of them must be false. These laws are by their nature quite obvious, and so simple even a simpleton like the Knave of Hearts would be capable of raising them.
In his diary of 1858, Dodgson notes that he has purchased and placed on his reading list “De Morgan on Chances.” He was certainly familiar with the work of the noted Cambridge mathematician and logician Augustus De Morgan, as probability was a major field of study for Carroll throughout his life.
De Morgan was also credited with what is today known as the duality principle. This is employed in the translation of concepts, theorems and mathematical structures into other concepts, theorems and structures, often by involution. Simply put: if a theorem is true, its dual is true. It is an important general principle that has application in every area of mathematics.
Lewis Carroll applied the duality principle to literature. He reasoned that if the underlying structure of his writing was mathematically logical, the linguistic structure would retain its logical integrity—and the results would be “as sensible as a dictionary” (as he wrote in Through the Looking-Glass). However, although grammatically logical, its message is usually absurd and comic. For, as Carroll the author—and Dodgson the logician—knew as well as any comic writer, the great secret of nonsense literature is that it is extremely sensible. That is, nonsense is humorous only if it works within a logical framework. Without logic, nonsense makes no sense.
In this courtroom of the King and Queen of Hearts, Alice has unknowingly entered into what the philosopher, logician and mathematician Bertrand Russell described as “the realm of pure mathematics”: “an ordered cosmos, where pure thought can dwell.” It is a heartless and frightening place—and the monstrous Queen of Hearts is well suited to be its ruler.
Logicians like the Queen of Hearts are interested not in the content of an argument but in the features that make an argument valid or invalid. It is a place governed by rules and procedures and form; there is nothing whatever in the rules about the value of emotions, morals or character, nor anything to do with content or substance.
The Queen of Hearts has to be the ruthless executioner or Wonderland could not exist at all. “Axioms cannot tolerate contradictions,” Carroll wrote in his Symbolic Logic; nor can the Queen of Wonderland. Contradiction in any system of logic or mathematics leads to chaos and collapse of the entire system.
Alice invokes the ultimate contradiction and rejects the Queen’s authority. The King, Queen, Knave and all the cards are simply “made entirely of cardboard.” The laws of this heartless court have no power over Alice. She challenges the Queen with “Who cares for you? You’re nothing but a pack of cards!” For Alice, human values and the concerns of the human heart ultimately must trump this heartless tyranny of abstract mathematics. Once confronted with “Alice’s Evidence,” the house of cards collapses in a heap, and the dreamer awakens in the real world.
Frederic Leighton’s The Return of Persephone (1891) is the Victorian era’s most famous and iconic painting of the myth. It shows her ascending from the underworld into the waiting arms of her sister-mother, Demeter—a scene that mirrors Alice’s awakening from the underground dream world of Wonderland and returning to the lap of her sister Lorina (who bears the same name as their mother).
This portrayal is similar as well to that of the last tableau of the Eleusinian Mysteries of the great goddess. Like a pilgrim emerging from the Mysteries, Alice must learn to rescue herself before she can emerge from the underworld and back into the world of the living. She must do this by applying all she has learned, finally taking control in this last trial by claiming the power of the goddess within.
Victorian icon: The Return of Persephone.
Donald Thomas observes this mythological motif of “the return of the dreamer” in Wonderland and compares it to Virgil’s Aeneid: “Alice, like Aeneas, emerges unscathed from the dream, he by the gate of horn and she to the Oxford river bank. The horrors and predictions which Virgil’s hero encountered were implacable and unalterable. But Alice triumphs. However cruel their humour or authoritarian their manner, the figures of tyranny are, at last, ‘nothing but a pack of cards.’ ”
FORTY-TWO RULES Why does Alice’s dream of Wonderland end when the trial of the Knave of Hearts suddenly and dramatically collapses like a house of cards? The reason for the downfall of Wonderland is identical to the answer to the meaning of “life, the universe and everything” in Douglas Adams’s The Hitchhiker’s Guide to the Galaxy. For both Adams and Carroll, the answer is the number 42.
Many have observed that Lewis Carroll had an obsession with the number 42, but nobody seems to know why. In the early poem Phantasmagoria, a ghost haunts “a man of forty-two.” In The Hunting of the Snark—which Carroll wrote at age forty-two—we find that the Baker’s luggage consists of “forty-two boxes, all carefully packed” and that “Rule 42 of the Code” sealed the Snark’s fate. In Sylvie and Bruno Concluded, we have a gravity-operated train that passes through a long tunnel nearer the earth’s centre than either end. This rapid gravity driven train journey takes 42 minutes. According to Martin Gardner, 42 minutes is “exactly the same time that it would take an object to fall through the centre of the earth … regardless of the tunnel’s length.”
In Alice’s Adventures in Wonderland, the number 42 runs amok. It begins on the title page with “Forty-two illustrations by John Tenniel.” After descending into Wonderland, Alice encounters an angry Pigeon who protects her nest “night and day” and hasn’t “had a wink of sleep these three weeks.” This gives her egg a hatching period of 21 days + 21 nights = 42, or a unit value of (3 × 7 × 2) = 42.
Similarly, we have the suppression of two guinea pigs in the trial scene. A guinea in English currency has a value of 21 shillings; consequently, the two guinea pigs (or piggy banks) would have a total value of 42 shillings.
In the Queen’s rose garden, Alice encounters three gardeners who are animated numbered playing cards. If we add up the card numbers (2 + 5 + 7 = 14), then multiply that by the number of cards (14 × 3), once again we get 42.
This episode is followed by the grand royal procession of cards. There are normally fifty-two cards in a deck; however, Carroll has been careful to leave the gardeners (the ten numbered spade cards) out of the procession, with the result that there are exactly 52 – 10 = 42 cards.
This may have something to do with the King invoking “Rule Forty-two,” which, he claims, is the “oldest rule in the book.” Indeed, the King’s Rule Forty-two has wider and deeper implications relating to the mathematical structure of Wonderland.
It could be argued that the Wonderland adventure begins and ends with 42. Under deep cover, the number can be found at the beginning of Alice’s adventures, where, in Wonderland’s great hall, she recites the multiplication table. As we have seen, it is a system that is suddenly foreign to her: “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!”
As Alice says, “the Multiplication Table doesn’t signify.” But as observed earlier, it signifies a great deal and reveals that this table essentially presents us with a problem based on scales of notation. The Wonderland multiplication table is sound up to the 12-times level in base 39, however, once we progress to the 13 times level, to maintain the rule of this system, we must employ base 42. This proves to be fatal and the entire system thereafter collapses. It is an object lesson in what may result from any mathematical system that does not submit to rigorous testing and toward absolute proof.
As we have demonstrated in chapter 2, because of number 42 (as a base number in the Wonderland multiplication system), Alice is right to declare that she will “never get to twenty at that rate.” And neither will the King of Hearts: “ ‘Let the jury consider their verdict,’ the King said, for about the twentieth time that day.” But like Alice, the King never gets to twenty either. For here we find the fatal number 42 looms up once more, and brings all in Wonderland to a cataclysmic end.
It is by the authority of Rule Forty-two that the King attempts to expel Alice from the court. Alice disputes this, however, objecting that if Rule Forty-two is the “oldest rule in the book” as the King claims, “then it ought to be Number One.” And with this peculiar logic, she suddenly finds herself capable of overruling the King and Queen of Hearts.
How is this possible? And why, besides the King and Queen, is Alice the only one not ordered executed? Once again, Carroll is playing a word game, this time the word-within-the-word game that he often played in letters to his child friends. In one example, he suggests that although one may find ink in a drink, it is not possible to find a drink in ink. In another, he explains that one may find love in a glove, but none outside of it. Consequently, Alice is ultimately able to overrule the King and Queen of Hearts when she discovers her true rank in this game: hidden within the word Alice there is an Ace.
From Snark: Baker has 42 pieces of luggage.
According to the rules of Carroll’s card game Court Circular, in which hearts are trumps, “the Ace may be reckoned either with King, Queen, or with Two, Three.” We are told the numbered heart cards in Wonderland are “the royal children,” which would seem to explain why the King of Hearts initially informs the Queen that Alice “is only a child”—in fact, the youngest child. However, as an ace, she can choose to switch from the lowest-ranking heart to the highest. When she claims her power as the highest-ranked card in the deck—the Ace of Hearts—her role in Wonderland suddenly shifts from the virtually powerless to the most powerful.
Alice has finally discovered Wonderland’s “rule of processions.” In the ranking of Wonderland’s forty-two-card deck, Alice has become the highest-ranked heart. She has become the fatal number 42 that in the Wonderland multiplication table wrecks the mathematical structure upon which Wonderland is constructed. She overrules the rulers, and claims the power to end her dream. In waking, Alice brings the whole of Wonderland down like a house of cards.
According to Pausanias (in his Description of Greece—c. AD 160) and others who had undergone the sacred rites of the Mysteries of the Great Goddess, after ascending from the underworld, the initiate returned to the world “clothed with the radiance of things seen and remembered.” So that each initiate’s experiences might be recorded while still fresh in the memory, each of the “newly born” was required “to dedicate a tablet on which is written all that each has heard or seen.”
Lewis Carroll was very much concerned with the “mystic memory” of the ancients, and certainly alludes to it in the prelude to the fairy tale. Since ancient times these pilgrims wore wreaths and garlands of white flowers just like the ones Dodgson made Alice Liddell wear in one of his photographs.
In the end, Alice returns to her dreaming body by her sister’s side under a tree on the riverbank, and to her everyday life. Now, though, Alice has a wise old soul and retains the memory of her dream world, and she recounts her experience to her sister.
Then too, her sister Lorina “began dreaming after a fashion,” picturing herself passing on Alice’s dream to other children; while the author records them in a dedicated book in “which is written all that each has heard or seen,” so the story of the adventure and the revealed mystery of Wonderland might enter the minds and imaginations of children throughout the world.
Of course this is all clearly foreshadowed in the last stanza of Carroll’s prelude poem:
And with a gentle hand
Lay it where Childhood’s dreams are twined
In Memory’s mystic band,
Like pilgrim’s wither’d wreath of flowers
Pluck’d in a far-off land
Curiously enough, Lewis Carroll was not alone in linking Alice Liddell to classical Greek goddesses. The great Victorian photographer Julia Margaret Cameron dressed a twenty-year-old Alice in classical costumes so she might pose for no fewer than three versions of Persephone-Demeter. These variations on the theme of the Great Goddess were: Ceres the Roman goddess of the harvest, Aletheia the goddess of truth and justice and Pomona, the Roman goddess of fruitfulness.
Alice as Ceres-Demeter, by Julia Margaret Cameron, 1872.
Alice as Aletheia, Greek goddess of truth.
Alice as Pomona, Roman goddess of fruitfulness.
