What the Tortoise Said to Achilles

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What the Tortoise Said to Achilles ( English for What the tortoise said Achilles ) is a short story by Lewis Carroll , in the philosophical journal for the first time in 1895 Mind was released. In it, Carroll has the two protagonists from Zeno's paradox, Achilles and the turtle, discuss a logical paradox that is sometimes known as the inference paradox .

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The story follows on from the race between Achilles and the tortoise: Achilles caught up with the tortoise. She asks in astonishment how he could do this, since he had to go through an infinite number of sections. Achilles explains that the sections were getting shorter and shorter, so that he actually only had to cover a finite distance. The turtle then offers to tell him about another "race" that also consists of an infinite number of steps, but these are getting longer and longer. To do this, she chooses an excerpt from a proof by Euclid from its elements :

  • A : If two things are equal to a third, they are equal to each other. ( 1st axiom )
  • B : The two sides of this triangle are equal to the third.
  • Z : The two sides of this triangle are equal to each other.

The turtle declared that they fulfill the conditions A and B accepts and asks Achilles them with logical means to force Z to accept.

Achilles states that if A and B are true, then Z must be true too . The turtle asks him to record this statement as C :

  • C : If A and B are true, then Z is true.

Achilles thinks that everything would be clear, but the turtle while accepting C but not to be true, still Z . Achilles' objection that she must now also accept Z is only formulated as a further statement:

  • D : If A and B and C are true, then Z is true.

This process is repeated one more time, whereupon the narrator has to leave the two and only comes back some time later. At this point in time, the two of them had already inserted a thousand intermediate steps of the same kind, and Achilles realized that he would not be able to convince the turtle of Z.

style

The typical Carroll nonsense , which is also known from many of his other works, can be found in the story . What is striking is a large number of anachronisms that Achilles and the tortoise are also aware of. There are also many word games , including a quote from Alice in Wonderland .

interpretation

The paradox has been dealt with by many logicians and philosophers. In an overview article, Pascal Engel presents four approaches to resolution.

The most common approach is that a distinction must be made between requirements A and B and the final rules (here modus ponens ). The turtle does not do this and thus puts C on a par with the requirements. From the inference rule formulated with the derivative operator in the metalanguage

becomes a statement in object language with her

The paradox arises from the inadmissible mixing of the two levels, with correct treatment it does not occur. Carroll himself took this approach, as did Bertrand Russell, for example .

There are also other approaches: Ryle is of the opinion that the rules of inference cannot be reformulated into statements because they contain no theoretical knowledge about statements, but practical knowledge about how to draw conclusions from assumptions. So the turtle evidently does not accept the final rule that Achilles wants to apply. So for Achilles there is in fact no way to convince the turtle of Z.

Quine used the narrative to reject Carnap's attempt to derive logical truths from conventions: for that would require logic to be applied again, which leads to a circular argument .

Finally, there is the interpretation that the turtle accepts the rule of inference as the logical truth, but sees no reason for action in logic, i.e. for the actual recognition of the conclusion.

reception

In addition to a professional discussion by logicians and philosophers, the work also found its way into popular scientific literature : Hofstadter used the story in Gödel, Escher, Bach as an introduction to a chapter and a model for further dialogues between Achilles, the turtle and other characters.

Simon Blackburn wrote another sequel to mark the 100th anniversary of the first publication.

Web links

Wikisource: What the Tortoise Said to Achilles  - Sources and full texts (English)

Individual evidence

  1. ^ Lewis Carroll: What the Tortoise Said to Achilles. In: Min. Volume IV, No. 14, 1895. pp. 278-280. doi: 10.1093 / mind / IV.14.278
  2. ↑ Michael Clark: Paradoxes from A to Z. Routledge , 3rd edition 2012, ISBN 978-1-136-21804-0 . P. 103.
  3. ↑ Martin Gardner : The Universe in a Handkerchief. Springer, 2007. ISBN 978-0-387-28952-6 . P. 73.
  4. ↑ Pascal Engel: Dummett, Achilles and the tortoise. ( Preprint )
  5. ↑ Timothy Smiley: A Tale of Two Tortoises. In: Min. Volume 104, No. 416, October 1995. pp. 725-736. JSTOR 2254480
  6. ^ Bertrand Russell: The Principles of Mathematics. 1903. § 38. ( online )
  7. ^ Gilbert Ryle: Knowing How and Knowing That: The Presidential Address. In: Proceedings of the Aristotelian Society. Volume 46, 1945, pp. 1-16. JSTOR 4544405
  8. ^ WV Quine: Carnap and Logical Truth. In: Synthesis. Volume 12, No. 4, December 1960. pp. 350-374. JSTOR 20114356
  9. ^ Simon Blackburn: Practical Tortoise Raising. In: Min. Volume 104, No. 416, October 1995. pp. 695-711. doi: 10.1093 / mind / 104.416.695