What the Tortoise Said to Achilles
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Reprinted from Lewis Carroll, "What the Tortoise Said to Achilles",
Mind 4, No. 14 (April 1895): 278-280
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Achilles had overtaken the Tortoise,
and had seated himself comfortably on its back.
"So you've got to the end of our race-course?" said the Tortoise.
"Even though it does consist of an infinite series of distances?
I thought some wiseacre or other had proved that the thing couldn't
be done?"
"It can be done," said Achilles. "It HAS been done! Solvitur ambulando.
You see the distances were constantly diminishing; and so--"
"But if they increased instead?" the Tortoise interrupted. "How then?"
"Then I shouldn't be here," Achilles modestly replied; "and you
would have got several times 'round the world, by this time!"
"You flatter me; flatten, I mean" said the Tortoise; "for you
are a heavy weight, and NO mistake! Well now, would you like to hear
of a race-course, that most people fancy they can get to the end of
in two or three steps, while it really consists of an infinite
number of distances, each one longer than the previous one?"
"Very much indeed!" said the Grecian warrior, as he drew from his
helmet (few Grecian warriors possessed pockets in those days) an
enormous notebook and a pencil. "Proceed! And speak SLOWLY, please!
Shorthand isn't invented yet!"
"That beautiful First Proposition of Euclid!" the Tortoise murmured
dreamily.
"You admire Euclid?"
"Passionately! So far, at least, as one can admire a treatise that
won't be published for some centuries to come!"
"Well, now, let's take a little bit of the argument in that First
Proposition -- just two steps, and the conclusion drawn from them.
Kindly enter them in your notebook. And in order to refer to them
conveniently, let's call them A B and Z:
(A) Things that are equal to the same are equal to each other.
(B) The two sides of this Triangle are things that are equal
to the same.
(Z) The two sides of this Triangle are equal to each other.
Readers of Euclid will grant, I suppose, that Z follows logically
from A and B, so that any one who accepts A and B as true, must
accept Z as true?"
"Undoubtedly! The youngest child in High School -- as soon as
High Schools are invented, which will not be till some two
thousand years later -- will grant THAT."
"And if some reader had NOT yet accepted A and B as true, he might
still accept the sequence as a valid one, I suppose?"
"No doubt such a reader might exist. He might say 'I accept as true
the Hypothetical Proposition that, if A and B be true, Z must
be true; but I don't accept A and B as true.'
Such a reader would do wisely in abandoning Euclid, and taking to
football."
"And might there not also be some reader who would say 'I accept
A and B as true, but I don't accept the Hypothetical.' ?"
"Certainly there might. He, also, had better take to football."
"And neither of these readers," the Tortoise continued, "is AS YET
under any logical necessity to accept Z as true?"
"Quite so," Achilles assented.
"Well, now, I want you to consider me as a reader of the second kind,
and to force me, logically, to accept Z as true."
"A tortoise playing football would be--" Achilles was beginning.
"--an anomaly, of course," the Tortoise hastily interrupted.
"Don't wander from the point. Let's have Z first, and football
afterwards!"
"I'm to force you to accept Z, am I?" Achilles said musingly.
"And your present position is that you accept A and B, but
you don't accept the Hypothetical--"
"Let's call it C," said the Tortoise.
"--but you don't accept
(C) If A and B are true, Z must be true."
"That is my present position," said the Tortoise.
"Then I must ask you to accept C."
"I'll do so," said the Tortoise, "as soon as you've entered it
in that notebook of yours. What else have you got in it?"
"Only a few memoranda," said Achilles, nervously fluttering the
leaves: "a few memoranda of ... of the battles in which I have
distinguished myself!"
"Plenty of blank leaves, I see!" the Tortoise cheerily remarked.
"We shall need them all!" (Achilles shuddered.) "Now write as I
dictate:
(A) Things that are equal to the same are equal to each other.
(B) The two sides of this Triangle are things equal to the same.
(C) If A and B are true, Z must be true.
(Z) The two sides of this Triangle are equal to each other."
"You should call it D, not Z." said Achilles. "It comes next to
the other three. If you accept A B and C, you must accept Z."
"And why _must_ I?"
"Because it follows logically from them. If A B and C 'are' true,
Z must be true. You don't dispute that, I imagine?"
"If A B and C are true, Z must be true," the Tortoise thoughtfully
repeated. "That's another Hypothetical, isn't it? And, if I failed
to see its truth, I might accept A B and C, and still not accept Z,
mightn't I?"
"You might," the candid hero admitted; "though such obtuseness would
certainly be phenomenal. Still, the event is possible. So I must
ask you to grant one more Hypothetical."
"Very good. I'm quite willing to grant it, as soon as you've written
it down. We will call it:
(D) If A B and C are true, Z must be true.
have you entered that in your notebook?"
"I have!" Achilles joyfully exclaimed, as he ran the pencil into its
seath." And at last we've got to the end of this ideal race-course!
Now that you accept A B C and D, of course you accept Z."
"Do I?" said the Tortoise innocently. "Let's make that quite clear.
I accept A B C and D. Suppose I still refused to accept Z?"
"Then Logic would FORCE you to do it!" Achilles triumphantly replied.
"Logic would tell you; 'You can't help yourself. Now that you've
accepted A B C and D, you _must_ accept Z!' So you've no choice,
you see?"
"Whatever Logic is good enough to tell me is worth writing down,"
said the Tortoise. "So enter it in your book, please. We will call it:
(E) If A B C and D are true, Z must be true.
Until I've granted that, of course I needn't grant Z.
So it's quite a necessary step, you see?"
"I see," said Achilles; and there was a touch of sadness in his tone.
Here narrator, having pressing business at the Bank, was obliged to
leave the happy pair, and did not again pass the spot until some
months afterwards. When he did so, Achilles was still seated on the
back of the much-enduring Tortoise, and was writing in his notebook,
which appeared to be nearly full. The Tortoise was saying, "Have
you got that last step written down? Unless I've lost count, that
makes a thousand and one. There are several millions more to come.
And would you mind, as a personal favour, considering what a lot of
instruction this colloquy of ours will provide for the Logicians of
the Nineteenth Century -- would you mind adopting a pun that my
cousin the Mock-Turtle will then make, and allowing yourself to be
renamed Taught-Us?"
"As you please!" replied the weary warrior, in the hollow tones of
dispair, as he buried his face in his palms. "Provided that you, for
your part, will adopt a pun the Mock-Turtle never made, and allow
yourself to be re-named A Kill-Ease!"
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Transcribed into hypertext by Andrew Chrucky, July 10, 1997.