The logic of nonsenseAuthor(s): BARBARA ELPERN BUCHALTERSource: The Mathematics Teacher, Vol. 55, No. 5 (MAY 1962), pp. 330-333Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/27956612 .

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The logic of nonsense

barbara elpern buchalter, Catalina High School, Tucson, Arizona.

"You may call it nonsense if you like," she said, "but ve heard nonsense, compared with which that would be

sensible as a dictionary!"

As a participating teacher in the Uni

versity of Illinois Committee on School Mathematics program, I am aware that the formal study of logic is not beyond the

capabilities of the high-school mathe matics student. Indeed, I have found that

good students enjoy logic and through the

study of it are able to formulate logical generalizations from their experiences in

mathematics. Although this emphasis on

the formal structure of mathematics can

be carried to extremes, it does clarify the nature of proof. One way to interest stu dents in the field of logic is to have them reread the books of their childhood?

specifically, the stories of Lewis Carroll. In the light of their new knowledge of

logic they will begin to appreciate the

subtlety of these classics. Critics lavish praise on Lewis Carroll,

author of Alice in Wonderland, Through the Loohing-Glass, Jabberwocky, and other

delightful nonsense stories and poems. The same critics seek an answer to the

puzzle of Lewis Carroll, the nonsense

writer, and Charles L. Dodgson, his real self and a noted logician. There is no

paradox. The magic of Lewis Carroll is a

by-product of the subjects Charles Dodg son taught at Christ Church. Here we

have the subtlest and profoundest of all his parodies, that for the entertainment of his Dean's little daughters, he deliberately travestied mathematics and logic. If

Dodgson and his work were shown as an

organic whole, his "nonsense" would no

longer seem the anomaly which it is usu

ally represented to be. He was not a mere dealer in sentimental whimsy or

drollery, but unique in literature, a poet logician. So that when we stand up for a

toast to Alice and her creator, and

Fill up our glasses of treacle and ink, And everything else that is pleasant to drink. Mix sand with the cider, and wool with the

wine? And welcome Queen Alice with nine times nine.

we should honor along with her and Lewis

Carroll, the Reverend Charles Lutwidge Dodgson, a mathematician.

During his lifetime, Dodgson was stu

dious, religious, and completely engrossed in his role of college don. His mathematics dealt generally with ingenuities rather than profound considerations. He was ad dicted to mathematical puzzles and spor tive syllogisms. Had he taken mathemat ics and logic more seriously, he would never have been able to write the "Alice''

books; but conversely, had he not been a

mathematician and logician, the "Alice" books would never have been written. He was a master of the reductio ad absurdum method. In the White Rabbit's verses to the Court, he constructs a deliberate

absurdity: I gave her one, they gave him two, You gave us three or more; They all returned from him to you, Though they were mine before.

The whimsical, yet mathematical, nature of Charles Dodgson is evident in the fol

lowing quotation. It is an excerpt of a letter written about a friend named Polly. Of her, he says, "She may be limited and

superficial; she may even be without

depth. But she is at least equilateral and

equiangular?in one word, what is she but a Poly-gon."

330 The Mathematics Teacher | May, 1962

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As Lewis Carroll, Charles Dodgson pub lished Alice's Adventures Underground in 1865. This later became Alice's Hour in

Elfland, and finally Alice's Adventures in Wonderland. Other fantasies followed.

Disregarding the obvious literary merits of these stories, let us examine the logic behind them and the mathematical con

cepts embodied in the plots and charac ters.

Lewis Carroll's tales are filled with ex

cellent examples of fallacious reasoning or

travesties of real reasoning. The fallacy of amphiboly concerns the

ambiguous construction of the sentence as a whole. In Through the Looking Glass we have this example: Alice, slightly over come by all the running she has had to do, asks the King, "Would you be good enough to stop a minute to get one's breath?" To which his majesty replies, "I'm good enough, only I'm not strong enough. You see, a minute goes by so fear

fully quick. You might as well try to stop a Bander snatch."

The fallacy of equivocation involves the inconsistent use of words. Lewis Carroll shows this implied ambiguity in such

phrases in The Three Voices. An epicure, in defense of his philosophy urges that

Dinner is Dinner and Tea is Tea.

But the lady of the poem, in reply, over turns his position by taking his statement

literally. She replies, . . . Yet wherefore cease,

Let thy scant knowledge find increase; Say men are men, and geese are geese.

In Through the Looking Glass we find sev

eral obvious uses of this same fallacy. One occurs in a scene between Alice and the

White King.

The White King: Just look down the road and tell me if you see either of my messengers.

Alice : I see nobody on the road. King: I only wish / had such eyes. To be

able to see Nobody and at this distance, too.

Why, it's as much as I can do to see real people in this light. (Messenger arrives and King says :)

Whom did you pass on the road? Messenger: Nobody. King: Quite right?this young lady saw him,

too. So of course, Nobody walks slower than

you. Messenger: I do my best. I'm sure nobody

walks faster than I do. King: He can't do that, or else he'd have

been here first.

In a puzzle Charles Dodgson wrote about two clocks, the unexpectedness of the conclusion from premises freely granted is clearly illustrated :

"Which is better, a clock that is right only once a year or a clock that is right twice a day?" "The latter," you reply, "unquestionably." "Very good, now attend. I have two clocks; one doesn't

go at all and the other loses a minute every day: which would you prefer?" "The losing one," you answer, "without a doubt." "Now observe: the one which loses a minute a day has to lose twelve hours, or 720 minutes, before it is right and it is therefore right about once every two

years, whereas the other is evidently right as often as the time it points to comes around, which happens twice a day. So you've con tradicted yourself once."

It may therefore be granted as a well attested fact about human minds that the conclusion of an argument is not, in gen

eral, known to them when they inspect or

believe the premises, especially if a long chain of inferences is required to reach a

conclusion. But this has nothing to do with the validity of an inference. This il lustrates the distinction between any psy

chological novelty a conclusion may have and any logical novelty it may be sup

posed to have. Logical novelty means the

logical independence of what is said to be the "conclusion" from its "premises." And it is clear that, for this argument to be valid, the conclusion cannot, so long as

it is dependent upon the premises, possess logical novelty.

Everywhere we turn in the writings of Charles Dodgson we find this travesty of

logic. But we also find something else. We find valid arguments for scientific prin ciples. A good argument for the arbitrari ness of names and symbols is found in a scene between Alice and Humpty

Dumpty: "When I use a word," Humpty-Dumpty said

in a rather scornful tone, "it means just what I

choose it to mean?neither more nor less."

"The question is," said Alice, "whether you can make words mean so many different things."

The logic of nonsense 331

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"The question is," said Humpty-Dumpty, "which is to be the master?that's all."

If he meant by this that he had the right to attach a word to whatever object he

pleased, he was correct. This is the very essence of symbolic mathematics where

calculations are carried out in symbols, and it does not matter what we call them

so long as we are consistent. But, if he

meant that by transferring a name of one

object to a second object, this object be

comes identical with the original object, he was merely falling victim to primitive

magic and sorcery. Such magic has occa

sionally been resorted to even in science.

Only a mathematician was capable of

conceiving life in the dimensions beyond the three familiar in our common Euclid

ean world. The turning of time backward, so that the Queen remembers only what

happens next week, and Sylvie and Bruno

are at ease in a universe wherein the clock

runs the other way, is the universe of a

mathematician. In Lewis Carroll's non

Euclidean geometry, he is speculating as

to what would happen if our fundamental

assumptions about the universe did not

hold good?as they do not in the world of

dreams where the White Queen can feel

pain before being pricked and has to keep

running to stay in the same place. As John

Macy foretells, "When the intellectual

history of our era, our general block of real

time, is written, it will be found that

Lewis Carroll discovered relativity before

Einstein was born; certainly it will be un

derstood that relativity and Alice are in

herent in mathematics." We may go fur

ther than John Macy and show that our

theory of mathematics amounts to what

Alice means when she says, "Something unknown is doing we don't know what."

Throughout the Lewis Carroll stories

we can find syllogisms applicable to the

formal study of logic. In Alice in Wonder

land, Alice suddenly grows until her neck

stretches high above the trees. A nearby

pigeon condemns her as a serpent and pro ceeds to baffle her by his syllogistic rea

soning. The principal premises and con

clusions may be stated according to formal

logic. In the first argument, the bird

proves Alice is not a girl. His reasoning

may be summed as

No girl has such a long neck.

You have such a long neck.

Therefore, you are not a girl.

This argument is formally valid, but the

major premise is false. The pigeon bases it

on his own personal experience, i.e., he has

never seen a girl with such a long neck.

This does not mean that none exists. In

his second syllogism the bird proves Alice

is a serpent. He proceeds thus:

All serpents are egg eaters.

You are an egg eater.

Therefore, you are a serpent.

This has the form of the second syllogistic mood:

All is M All S is M

All S is P.

It is in the figure A A A, but is not valid in the second mood, since the quality of the

premises must differ. Alice even tells him

that girls eat eggs, too, so the pigeon main

tains that not only do all serpents eat eggs, but only serpents eat eggs; therefore, if

she is a girl, she is still some kind of ser

pent. Without going into so much detail, the

following are three more examples of this

pseudo-syllogistic reasoning employed in

Alice in Wonderland and Through the

Looking-Glass: Example 1: "You alarm me" said the King. "I

feel faint.?Give me a ham sandwich."

On which the Messenger, to Alice's great

amusement, opened a bag that hung round

his neck, and handed a sandwich to the King, who devoured it greedily.

"Another sandwich," said the King. "There's nothing but hay left now," the

Messenger said, peeping into the bag.

"Hay, then," the King faintly murmured.

Alice was glad to see that it revived him a

good deal. "There's nothing like eating hay when

you're faint," he remarked to her as he

munched away. "I should think throwing cold water over

you would be better," Alice suggested: " . . .

or some sal volatile."

332 The Mathematics Teacher | May, 1962

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"I didn't say there was nothing better"

the King replied, "I said there was nothing like it."

Example 2: Alice didn't dare to argue the point, but went on:

"... and I thought I'd try and find my way to the top of that hill-"

"When you say hill," the Queen inter

rupted, "I could show you hills, in comparison with which you'd call that a valley."

"No, I shouldn't," said Alice, surprised into contradicting her at last: "a hill carit be a valley, you know. That would be nonsense?"

The Red Queen shook her head. "You may call it nonsense, if you like," she said, "but ve heard nonsense, compared with

which that would be sensible as a dictionary!" Example 3: "Cheshire Puss," she began, rather

timidly . . . "would you tell me, please, which

way I ought to go from here?" "That depends a good deal on where you

want to get to," said the Cat. "I don't much care where?" said Alice.

"Then it doesn't matter which way you

go," said the Cat.

"?so long as I get somewhere" Alice added as an explanation.

Anyone who now doubts the logical and, therefore, mathematical background of the "nonsense" stories of Lewis Carroll should read again carefully these fan

tasies. Through the Looking Glass is ac

tually a chess game which, as Lewis Car rolPs own diagram shows, the White

Pawn, Alice, is to play and win in eleven moves. Tangled Tales is an entire mathe matical fantasy, where each tale involves a problem, and the reader is told to un

tangle the knots. As further proof, I shall conclude in the immortal words of Alice:

"365" 1

364.

eeee's

I think that I shall never see A number lengthier than e, Whose hundred-thousandth decimal

Is now revealed to one and all; In whose expression the digits go Across the pages, row on row.

A number, being irrational, And, furthermore, transcendental, Which takes an IBM one day To calculate the electronic way But takes, at our poor human speed, At least a fortnight just to read

And even longer time, by gosh, To evaluate the function cosh.

A number which, when time is late, We call, for short, 2.718. A quantity whose digit Would reach to there and back again. A longer number you'll not spy Unless it should, perchance, be .

Poems are made by fools like me, But mathematicians compute e.

Yet, they compute in vain, you see, For only God knows all of e.

?Robert L. Page, Nasson College, Springvale, Maine.

The logic of nonsense 333

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