What the Tortoise Said to Achilles (1895)
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Re: What the Tortoise Said to Achilles (1895)
#2The Wikipedia page is worth reading: https://en.wikipedia.org/wiki/What_the_Tortoise_Said_to_Achi...
Particularly this quote:
The Wittgensteinian philosopher Peter Winch discussed the paradox in The Idea of a Social Science and its Relation to Philosophy (1958), where he argued that the paradox showed that "the actual process of drawing an inference, which is after all at the heart of logic, is something which cannot be represented as a logical formula ... Learning to infer is not just a matter of being taught about explicit logical relations between propositions; it is learning to do something" (p. 57). Winch goes on to suggest that the moral of the dialogue is a particular case of a general lesson, to the effect that the proper application of rules governing a form of human activity cannot itself be summed up with a set of further rules, and so that "a form of human activity can never be summed up in a set of explicit precepts" (p. 53).
Re: What the Tortoise Said to Achilles (1895)
#3The Wikipedia page is worth reading: https://en.wikipedia.org/wiki/What_the_Tortoise_Said_to_Achi... Particularly this quote: The Wittgensteinian philosopher Peter Winch discussed the paradox in The Idea of a Social Science and its Relation to Philosophy (1958), where he argued that the paradox showed that "the actual process of drawing an inference, which is after all at the heart of logic, is something which cannot…
Being led down the rabbit hole is one of the reasons why I love Wikipedia.
I had searched 'What the Tortoise Said to Achilles' on Google, and ended up reading about the arrow paradox's rebuttals, which were really interesting.
But more to the point of the original article, it shows that there are definitely gray areas within morality, and it's impossible to use boolean logic to try to categorize humans.
Re: What the Tortoise Said to Achilles (1895)
#4Is the point here that the very structure of syllogism itself can be denied? That however inexorable a "If A and B, then C" argument is, someone else could always argue that it's not quite valid yet? It kind of reminds me of the point made (turgidly, but still) by Yudkowsky in The Simple Truth[1] - sometimes you just have to throw up your hands and declare the counter-arguments specious.
Re: What the Tortoise Said to Achilles (1895)
#5Reading this story in Hofstadner's GEB destroyed my ability to accept mathematical proofs as "proven". I just don't find them convincing; but more like using authorised forms o argument within an artificially stylised tradition (like English Literature). And I wonder if alien mathematics will reveal our mathematics as embarassingly parochial - and not the universal common ground usually assumed.
So, instead of proof, I have to fall back on intuition and working code, with their severe limitations.
However... studying mathematical proof has at times informed and grown my intuition, by revealing new ways to see a problem and new (bizarre and unintuitive) ways to decompose it.
I might have been better off never having seen this story.
Re: What the Tortoise Said to Achilles (1895)
#6Reading this story in Hofstadner's GEB destroyed my ability to accept mathematical proofs as "proven". I just don't find them convincing; but more like using authorised forms o argument within an artificially stylised tradition (like English Literature). And I wonder if alien mathematics will reveal our mathematics as embarassingly parochial - and not the universal common ground usually assumed. So, instead of proof,…
That was/is such a dilemma.
Re: What the Tortoise Said to Achilles (1895)
#7Reading this story in Hofstadner's GEB destroyed my ability to accept mathematical proofs as "proven". I just don't find them convincing; but more like using authorised forms o argument within an artificially stylised tradition (like English Literature). And I wonder if alien mathematics will reveal our mathematics as embarassingly parochial - and not the universal common ground usually assumed. So, instead of proof,…
> Reading this story in Hofstadner's GEB destroyed my ability to accept mathematical proofs as "proven".
That's too bad, because the anecdote doesn't challenge the basis for mathematical proofs or logical reasoning, in fact it requires it as a precondition for the anecdote to move forward. Remember that Gödel's incompleteness theorems don't argue that there are no true statements, only that some of them cannot be proven true.
> So, instead of proof, I have to fall back on intuition ...
You might be better off reviewing the structure of logic and mathematical proof. Start here:
http://en.wikipedia.org/wiki/Euclid's_theorem
My reasoning is that, if there's one proof sufficiently transparent to win acceptance from a skeptic of logic, then there might be two ... ad infinitum.
Re: What the Tortoise Said to Achilles (1895)
#8Reading this story in Hofstadner's GEB destroyed my ability to accept mathematical proofs as "proven". I just don't find them convincing; but more like using authorised forms o argument within an artificially stylised tradition (like English Literature). And I wonder if alien mathematics will reveal our mathematics as embarassingly parochial - and not the universal common ground usually assumed. So, instead of proof,…
Why would you be better off? I don't see any advantage in having the overly optimistic belief that there's some universal, correct set of axioms.
You can still do all the math you could do before -- and if Carroll or GEB gets you more interested in the fundamentals of math, you can do even more.
Yes, you have to accept some basis of mathematics, and you now understand that some true things will be unprovable in the basis you just accepted. But that doesn't stop you from proving things.
I think you might have just transferred your optimism about math to code instead. How do you know your programming language is doing what you asked it to? That you asked it to do the right thing at all? That the compiled code has the correct behavior? That your hardware works as advertised and is not failing at the moment? In both code and math, you have to accept some abstractions that you're not going to worry about, but the things you do with math are certainly more verifiable.
Re: What the Tortoise Said to Achilles (1895)
#9Reading this story in Hofstadner's GEB destroyed my ability to accept mathematical proofs as "proven". I just don't find them convincing; but more like using authorised forms o argument within an artificially stylised tradition (like English Literature). And I wonder if alien mathematics will reveal our mathematics as embarassingly parochial - and not the universal common ground usually assumed. So, instead of proof,…
Why would you be better off? I don't see any advantage in having the overly optimistic belief that there's some universal, correct set of axioms. You can still do all the math you could do before -- and if Carroll or GEB gets you more interested in the fundamentals of math, you can do even more. Yes, you have to accept some basis of mathematics, and you now understand that some true things will be unprovable in the b…
> I think you might have just transferred your optimism about math to code instead.
To add to your point, if someone begins to doubt the utility of mathematical logic and responds by refocusing his attention from logic to code, he's somehow overlooking the fact that code is built on a foundation of mathematical logic.
Re: What the Tortoise Said to Achilles (1895)
#10Reading this story in Hofstadner's GEB destroyed my ability to accept mathematical proofs as "proven". I just don't find them convincing; but more like using authorised forms o argument within an artificially stylised tradition (like English Literature). And I wonder if alien mathematics will reveal our mathematics as embarassingly parochial - and not the universal common ground usually assumed. So, instead of proof,…
Wait wait wait, proofs are still possible! The lesson of Godel's theorem is that you can have correctness or completeness, but not both within the same formal system. So you can have a system that yields only true statements, it just won't be able to encompass all true statements; or you can have a system that does encompass all true statements, but from which it is impossible to exclude some falsehoods.