What the Tortoise said to Achilles, by Lewis Carroll

What follows is a short story written by Lewis Carroll, and published in 1895. It explores some anomalies of Logic, suggesting that it’s incomplete, or empirical. This famous piece drew responses from celebrated logicians such as Bertrand Russel, and Peter Winch.

Tortoise and Achilles

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 had been constantly increasing?” 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 note-book 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 he 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 a 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 he 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 note-book 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 that are 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 and B and C, you must accept Z.”
“And why must I?”
“Because it follows logically from them. If A and B and C are true, Z must be true. You don’t dispute that, I imagine?”
“If A and B and C are true, Z must he true,” the Tortoise thoughtfully repeated. “That’s another Hypothetical, isn’t it? And, if I failed to see its truth, I might accept A and 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 and 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 sheath. “And at last we’ve got to the end of this ideal race-course! Now that you accept A and B and C and D, of course you accept Z.”
“Do I?” said the Tortoise innocently. “Let’s make that quite clear. I accept A and B and 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 and B and 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 and B and 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 note-book, 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 re-named Taught-Us?”
“As you please!” replied the weary warrior, in the hollow tones of despair, as he buried his face in his hands. “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!”

Lewis Carroll, “What the Tortoise Said to Achilles,” Mind 4, No. 14 (April 1895): 278-280.

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BE in Computer Engineering, BSc in Mathematics from the American University of Beirut. MSc in Mathematics from the University of Edinburgh. Currently a PhD Mathematics student at the University of Copenhagen, specializing in Number Theory.


  1. Avatar photo

    Any Mathematical/scientific field should be based on a number of Axioms/rules to avoid falling into infinite causal loops. The axioms in physics per example, are based on observations/experiments, the charge of an electron is one of the central units of nature, measured, and called e. In mathematics, the case is different, axioms are not based on observations, since mathematics itself is not. So the set of axioms/rules, are definitions, that can be seen as a first cause in the long causal chain, before which we can’t go back: It is not possible to prove why 1+1=2; if it was possible, we would have to prove the premises used in our argument, then prove the premises used in the argument for the first argument, and so, end-up in an infinite causal chain. So “1+1=2” is the first cause before which we are not supposed to step.

    Likewise, “Philosophy of Logic” is based on some axioms, called “Logical Statements”, considered the fundamental tools we use to prove/disprove a certain hypothesis.

    Modus Ponens is one of the logical statements, it goes as following:

    1) p => q,
    2) p,

    3) Therefore, q.

    An example of this statement, would be what the Tortoise said to Achilles:

    (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.

    The same way you can’t prove why 1+1 is 2, there is no way you can prove this argument-statement, as you will end-up in an infinite loop as we saw in the article; the only thing we can do, would taking Modus Ponens for granted.

    It’s funny how a Greek philosopher, who existed around 500 years BC, can keep us all busy, analyzing his primitive paradoxes. Even though they’re quite old and most of the times explainable, Zeno’s Paradoxes will keep on raising questions in mathematics and Philosophy, that most of the Logicians still discuss today.

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