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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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