Talk:What the Tortoise Said to Achilles

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Paullusmagnus: Although the original dialog does indicate the premises as you do, with A,B,C,...,Z, I think this is faulty, because it implies, on some level, that there couldn't be more than 26 stages to the argument. That is why I thought 1,2,3,...,Z would be better; in that case no confusion can arise. You might object to having Z be the name for the final conclusion, but couldn't you propose another name, rather than getting rid of the #s altogether? --Ryguasu 20:45, 20 Sep 2003 (UTC)

The natural replacement for "Z" in today's notation would be "ω" (lowercase Omega), although if Achilles were more clever, he could come up with an ωth axiom. (And indeed he does so, in one of Hofstadter's dialogues.) Of course, the use of "ω" by mathematicians today (originally chosen by Cantor, I believe, and probably at about the same time as Carroll wrote his dialogue) is inspired by the same idea as Carroll's use of "Z" -- it is the last letter. -- Toby Bartels 09:45, 30 Aug 2004 (UTC)

--- I don't get it. Basically, this is my interpretation:

A tortoise doesn't accept modus ponens arguments as being self-evident. And, in order to make the tortoise accept them, Achilles tries to use another modus ponens argument (knowing full well the tortoise doesn't accept them). And he fails, obviously. When it's all over, we come to the conclusion that Achilles can't prove the original argument to the tortoise.

So, Lewis Caroll basically proved that if someone doesn't accept your type of argument, using the same type of argument to prove the first argument won't work.

Three questions: Why, god forbid, does Achilles try proving a modus ponens argument with another one, even though the tortoise doesn't accept them? Why is this a paradox, and why would philosophers try to "solve" it? And isn't Achilles' failure self-evident, without the need for this story?

If I've interpreted this incorrectly, please tell me.

-Its not really about modus ponens, its about logical consequence

all premises from c onward are basically the previous premises are true. what it shows that you need an axiom for all proofs. or atleast thats my take on it. 24.237.198.91 05:32, 30 September 2006 (UTC)

Personally can't accept any of those statements as there are THREE sides to a triangle :-P UnseemlyWeasel 02:54, 9 October 2006 (UTC)

It seems that all of you bring up some fine points. I cannot grasp why this should be qualified as a genuine paradox either, and there doesn't seem to be a definitive answer that philosophers can agree on. But this mysterious short paper did draw the attention of countless influential philosphers and they took them seriously so there probably is some deep stuff going on here.

A few points I'd like to make after reading the comments written above:

1. First of all, you are right that this is not about modus ponens in particular. It is about inference rules in general.

2. I don't think it's the case that the premises from c onward say that the previous premises are true. They say that IF the previous premises are true THEN the conclusion is true. They are conditional statements and they say nothing about the truth values of those premises.

3. "If A and B, then Z" is not an argument and therfore it is neither valid nor invalid. It is a tautological statement.

Basically I think the most fundamental lesson we can draw from this is that we should not confuse inference (A, B therefore Z) with implication ((A.B)-->Z). I guess this is essentially what Bertrand Russell said in Principles of Mathematics. I have to write a paper on this dialogue and my professor suggested that I use the deduction theorem as a central idea in resolving this "paradox."

Contents 1 Sex 2 Football reference 3 Kurt Godel 4 Title Alluding

[edit] Sex

As of Nov 20, there is some material on the article about Hofstadter's change of mind regarding the tortoise's sex. Suggest deleting as not germane. Also, the article refers to tortoise as "he" (which Hofstadter would disapprove of), suggest changing to "it". elpincha 19:29, 20 November 2006 (UTC)

[edit] Football reference

Hey, Do we really need Carroll's in-joke about the hypothetical readers being better off if they were to play football instead of reading Euclid? It might read well in the original but here it looks very strange and is confusing. The text here should anyway not be a verbatim copy from Carroll's text and clearly the paradox doesn't suffer from having these references removed. Thoughts? Andeggs 11:33, 20 December 2006 (UTC)

Well, the paradox wouldn't suffer from having the entire context removed, we could state it entirely in the abstract. But what a sad thing that would be! If it confusing and doesn't read well, my preference would be to rewrite it to read better. But since I'm not volunteering to do that now, I won't stop you from removing it, if you must. 192.75.48.150 16:14, 20 December 2006 (UTC)

Yes sorry to be a party-pooper but I did have to read it several times to understand what the football reference was all about. Andeggs 12:44, 21 December 2006 (UTC)

[edit] Kurt Godel

Since Hofstadter has been cited here I think it makes sense to mention Godel's famous "Incompleteness" Proof that is given in his landmark paper "On Formally Undecidable Proposisions of Principia Mathematica". If no one objects, I'll add a few lines to the discussion section.

We do have an article on Gödel's incompleteness theorems, and I don't see the point of adding them here. How would it be relevant? 192.75.48.150 15:00, 17 January 2007 (UTC)

[edit] Title Alluding

Not only the title alludes to the paradox stated. Within the first few lines of the text, the narrator says that Achilles has caught up to the turtle, and then they jabber about how the turtle thought the paradox couldn't be solved. Might want to include this in there, rather than just "the title alludes". 130.76.96.19 18:51, 13 March 2007 (UTC)