Charles Dodgson, aka Lewis Carroll, was a professor of mathematics at Oxford University for most of his life. The Alice books provide ample evidence for his great love of logic puzzles and word games. And there are several moments in chapters 5, 6, and 7 of Alice which make most sense when thought of as a Mathematical Joke. Here are some examples:
1) "One side will make you grow taller, and the other side will make you grow shorter."
"One side of what? The other side of what?" thought Alice to herself.
"Of the mushroom," said the Caterpillar, just as if she had asked it aloud; and in another moment it was out of sight.
Alice remained looking thoughtfully at the mushroom for a minute, trying to make out which were the two sides of it; and, as it was perfectly round, she found this a very difficult question...
The joke: A circle is a simple shape of Euclidean geometry consisting of those points in a plane which are equidistant from a given point (Wikipedia). As such, a circle has no “sides,” or rather it has an infinite number of sides. Thus Alice might rightly be confused when asked to find the two sides of a perfectly round object.
2) "I've seen a good many little girls in my time, but never one with such a neck as that! No, no! You're a serpent, and there's no use denying it. I suppose you'll be telling me next that you never tasted an egg!"
"I have tasted eggs, certainly," said Alice, who was a very truthful child, "but little girls eat eggs quite as much as serpents do, you know."
"I don't believe it," said the Pigeon; "but if they do, why, then they're a kind of serpent; that's all I can say."
The joke: In this case the pigeon uses the mathematical property of exclusivity to “prove” that little girls are a type of serpent. According to the pigeon, having a long neck and eating eggs are not merely properties of a serpent, they are exclusive properties of a serpent. This means that only things called “serpents” have both long necks and an affinity for eating eggs and that if a creature has those two properties it MUST be a serpent. Thus, because Alice has a long neck, has eaten eggs and claims to be a little girl, the pigeon “logically” concludes that little girls must be a kind of serpent.
1) "There's no sort of use in knocking," said the Footman, "and that for two reasons. First, because I'm on the same side of the door as you are…”
"There might be some sense in your knocking," the Footman went on, without attending to her, "if we had the door between us. For instance, if you were inside, you might knock, and I could let you out, you know."
The joke: In this case the Footman is explaining (somewhat unorthodoxly) a rule of geometry, namely that the line joining two points on the same side of a line will not intersect the line. (See Figure).
2) "How am I to get in?" asked Alice again, in a louder tone.
"Are you to get in at all?" said the Footman. "That's the first question, you know."
It was, no doubt: only Alice did not like to be told so.
The joke: The Frog Footman points out that Alice’s assumption that she can get into the house, is not necessarily true. This is actually a type of assumption that mathematicians must make all the time, that the problem they are trying to solve can be solved. And though it is important to recognize that this is an assumption, most mathematicians don’t like having it pointed out to them either.
3) "And how do you know that you're mad?"
"To begin with," said the Cat, "a dog's not mad. You grant that?"
"I suppose so," said Alice.
"Well, then," the Cat went on, "you see a dog growls when it's angry, and wags its tail when it's pleased. Now I growl when I'm pleased, and wag my tail when I'm angry. Therefore I'm mad."
- We agree that Dogs are not mad
- The properties of a Dog (and therefore of “not mad”-ness):
- 1) Growl when angry
- 2) Wag tail when pleased
- Properties of a Cat
- 1) Wag tail when angry
- 2) Growl when pleased
- Therefore Cats are not dogs and therefore not “not mad”
- Thus, Cats are mad.
1) "Then you should say what you mean," the March Hare went on.
"I do," Alice hastily replied; "at last--at least I mean what I say--that's the same thing, you know."
"Not the same thing a bit!" said the Hatter. "Why, you might just as well say that 'I see what I eat' is the same thing as 'I eat what I see'!"
"You might just as well say," added the March Hare, "that 'I like what I get' is the same thing as 'I get what I like'!"
"You might just as well say," added the Dormouse, which seemed to be talking in its sleep, "that 'I breathe when I sleep' is the same thing as 'I sleep when I breathe'!"
"It is the same thing with you," said the Hatter, and here the conversation dropped, and the party sat silent for a minute…"
The joke: In this case Alice makes the mistake of applying a mathematical principle to language. The commutative properties of addition and multiplication in algebra is defined as the property which allows numbers to be added or multiplied in any order and still give the same result. Thus 5 + 7 = 7 + 5 and 5 * 7 = 7 * 5. Alice tries to the apply the commutative property to her language asserting that “I mean what I say” = “I say what I mean,” which the Hatter, Hare, and Dormouse contradict with their counter-examples.
2) "What a funny watch!" she remarked. "It tells the day of the month, and doesn't tell what o'clock it is!"
"Why should it?" muttered the Hatter. "Does your watch tell you what year it is?"
"Of course not," Alice replied very readily: "but that's because it stays the same year for such a long time together."
"Which is just the case with mine," said the Hatter.
Alice felt dreadfully puzzled.
“It's always six o'clock now.”
The joke: Because the hour and minute at the Mad Tea Party never change, the Hatter’s watch turns to show the day of the month. He doesn’t need to reference the time “because it stays the same for such a long time together.” Martin Gardner points out that “one is reminded also of an earlier piece by Carroll in which he proves that a stopped clock is more accurate than one that loses a minute a day. The first clock is exactly right twice every twenty-four hours, whereas the other clock is exactly right only once in two years” (96-97).
3) "Take some more tea," the March Hare said to Alice, very earnestly.
"I've had nothing yet," Alice replied in an offended tone: "so I can't take more."
"You mean you can't take less," said the Hatter: "it's very easy to take more than nothing."
The joke: In this instance, the Hatter both points out the ambiguity of the term “more” and draws our attention to the paradox of negative numbers, which describe a quantity “less than nothing.” Helen Pycior argues that Carroll took “the concept literally, and forced his readers to consider less tea than that contained in an empty cup and fewer hours of study than none. In contrast to such mathematicians as De Morgan, who sought viable analogues of the negative numbers in such concrete objects as financial debts and lines drawn backwards from a zero point, Carroll presented physical situations in which "quantity less than nothing" was nonsensical” (164).
4) A bright idea came into Alice's head. "Is that the reason so many tea-things are put out here?" she asked.
"Yes, that's it," said the Hatter with a sigh: "it's always tea-time, and we've no time to wash the things between whiles."
"Then you keep moving round, I suppose?" said Alice.
"Exactly so," said the Hatter: "as the things get used up."
"But what happens when you come to the beginning again?" Alice ventured to ask.
"Suppose we change the subject…"
The joke: Here the joke is that Alice wants to know whether or not the movement around the table operates on modular arithmetic. Modular arithmetic “counts” by cycling through a set of numbers an infinite number of times. A clock, for instance, counts time by counting the hours 1 through 12 over and over and over and over again. Alice wants to know if this is how the mad tea party works. Will the Hatter, Hare and Dormouse continue around and around and around the table ad infinitem? Or is there a “stop” rule? Unfortunately (for Alice’s curiosity and ours) the March Hare interrupts at this point to change the subject.
Slide Presentation on Math Elements in Chapters 4-6 of Alice
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