Lewis Carroll’s De Morgan Fireplace

In the 1860s, Carroll had his most troubled and inspired years, producing both Alice books as he struggled with how to be a good person and great mathematician and logician, as well as his father’s death and the social isolation he felt at Oxford.  Just after completing Alice in Wonderland, Carroll moved into a suite of rooms in 1868 with the money from his popular book and he lived there the rest of his life.

An excellent piece of evidence that Carroll was staging logical connectives as animal characters can be found in his personal fireplace.  Carroll had the logician Augustus De Morgan’s son, William De Morgan, design tiles for his fireplace, most with animals, several with animals attacking each other, and several pairs of tiles with the same animals in opposite emotional states.  When Carroll was later in charge of the common room in the building, he hired De Morgan’s son yet again to put similar tiles in the fireplace and other furnishings.

In Wonderland, many of the characters act as logical connectives, the sorts of nots, ands, and ors that Boole, De Morgan and Carroll worked on as logicians.  There are many puns about oars, like ors, that involve choices, and Alice in the end of her adventures sits between the White and Red Queens much as Boole’s or sits between his inclusive and and exclusive not.  The Caterpillar acts as a more inclusive but reclusive or, who gives Alice the power to change size and perspective.

The Cheshire Cat acts as an exclusive or, who sees and travels between exclusive perspectives opposed to each other, with dogs insane to cats and cats insane to dogs as he explains to Alice, which is why everyone is crazy to someone in Wonderland.  This is why we don’t see him entirely, most of the time, because just like the elephant to the blind men from India, each perspective, such as Alice’s, only sees part, not the whole.  This is how his head watches others argue about whether or not he can be beheaded, each side of the argument seeing a part of the truth, but not the whole situation, as he grins on them from above, seeing both sides fight.

The Caterpillar is like time, which includes more in our lives as it continues in series, but the Cat is like space, moves through it freely, and grins at the divisions it inevitably puts between people, even after the passage of time and experience.  Carroll is himself from Cheshire, and missed his family there at Oxford. We can see De Morgan’s theorems at work in the plot and characters, and particularly in the moves from the Caterpillar to the Cheshire Cat to the Mad Tea Party, with De Morgan’s second law working one way and then the other back again.

Aristotle, Boole, Carroll & De Morgan

Lewis Carroll, the pen name and popular title of Charles Dodgson, mathematician and logician at Oxford, author of Alice and Wonderland and Through the Looking Glass, was well versed in the logical forms of Aristotle, Boole and De Morgan, and was working on his own understanding of how logical connectives, thoughts and emotions include and exclude members of sets before Frege, Russell and Wittgenstein set formal logic in its current form and founded analytic philosophy.  Carroll was engaged with problems of German idealism and British empiricism found in the work of Locke, Hume, Mill, Boole and De Morgan in the late 1850s and early 1860s, the years that he planned and wrote both of Alice’s adventures.

A careful historical, social and psychological reading of both Wonderland and the Looking Glass show that Carroll used his fantastic characters to lead Alice through the forms of the central figures in the history of logic as he knew it, including Aristotle, De Morgan, Boole and Mill, using both stories to explore how inclusion and exclusion structure our thoughts, feelings, relationships and politics.  The characters of both fantasy worlds engage Alice in gathering and dividing, such that through her experiences she assumes the roles and positions she wanders through, engaging with forms and the problems of philosophy and society that involve them.

The forms of Aristotle, Boole, and particularly De Morgan are structural to the characters and plots of both of Alice’s adventures.  In Wonderland, Alice works her way from inclusive child to exclusive adult, from the White Rabbit, an absurdly inclusive conjunction, to the Queen of Hearts, who threatens others with total exclusion and negation.  In the Looking Glass, Alice works her way from the exclusive Red Queen, much like the Queen of Hearts, and says all ways are hers and not one else’s, to the inclusive White Queen, who accepts impossible and absurd things many times before breakfast, to finally balance in the middle, with the White and Red Queens sitting on either side of her.  Alice is a developed OR, having taken all sides, who must gather and divide empirically as she moves through the adult world between total inclusion and total exclusion.

Aristotle’s syllogisms, Boole’s logical connectives, and De Morgan’s theorems and notation are structural to the absurd humor of the work and reveal much of Carroll’s process as he invented the entertaining tales that are also instructive, both false and true, possibilities, fantasies and dreams, half-truths between non-being and being, between 0 and 1 for Boole.  In the opening poem before the story, Carroll says, The dream-child (like the fantastic hybrid White Rabbit) moving through a land of wonders wild and new, In friendly chat with bird or beast- and half believe it true.”  Aristotle’s syllogisms and the corresponding four corners of the square of opposition can be found in Wonderland in the glass table with the golden key, the reasoning of the pigeon,the Mad Tea Party and several key characters, and in the Looking Glass it is displayed in the Red and White queens and kings, with the queens standing for ALL, the kings for SOME, the white for inclusive positive and the red for exclusive negative.  The syllogisms apply to the characters in complex and interesting ways, but simple enough to identify.

De Morgan’s theorems can be found in the structures of the stories as well.  After Alice meets the Caterpillar, a very inclusive but reclusive Or who tells her little, questions her much and gives her the ability to grow or shrink in perspective and size, introducing Alice to transformations, she argues with a pigeon who fears for her egg, and then goes to the Duchess’ house, where the Duchess is beating her baby and the cook is hurling everything he can at her.  The pigeon is negative to alice but conjoined with her egg, and then Alice wanders into an abusive exclusive OR from a conjunction that is also a mother-child relationship, moving along De Morgan’s first law, from ~(A ^ B), pigeon protecting Alice from egg, to ~A v ~B, with dueling Duchess and baby, as well as Duchess and cook, as Alice has moved into the house and inside the brackets, finding division within.

Immediately after this, the Cheshire Cat, who sits as a grinning exclusive OR inside the divided house, wise to how every position is opposed to another, tells Alice everyone is exclusively in their own dream and mad, opposed to anyone against them, so everyone is insane.  The Cat tells Alice he takes the shortcut, just as De Morgan’s theorems are shortcuts for logic, and says you can go one way to the March Hare or the other way to the Mad Hatter, ~A v ~B, but when Alice goes one way she ends up going both ways and finds the two at a table, and they protest there is no room for her, excluding her, which reverses the course of De Morgan’s first law immediately after going the first direction, from  ~A v ~B, the choice of one crazy person or another, opposed to Alice’s perspective either way, to ~(A ^ B), where the two are opposed to Alice at the table together, and say there is no room inside for her. The Tea Party resembles the absurdity of Boolean abstract logical algebra, where time doesn’t move unless you ask it to, and the Mad Hatter schools Alice on how IF-THEN works in syllogistic exercises.

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