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).
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.
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.
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.
My "optimism" is more for my intuition; working code is experimental confirmation.
It's easier to be optimistic about code than proofs. Firstly, working code only needs to work in the specific cases you're using (that you test for); but a proof must work in every possible case. Thus, working code is simpler and easier to check, because it's aiming at less. My code almost always confirms my intuition.
Yes, it's also helpful to have the automatic, mechanical check of executable code; and as you say, this relies on compilers, OSes, silicon, hardware. (Though, anecdotally, I have noticed subtle problems that I eventually diagnosed and confirmed to be compiler and hardware bugs.) BTW, yes I have tried COQ (proof assistant; somewhat mechanical proof checking), but simple ideas become very complex to prove, and the problem of bugs in COQ itself etc is of greater concern, for the next reason:
Secondly, and relatedly, is that the standard is much lower for code. It just needs to work. Whereas a mathematical proof is supposed to be absolutely true. In other words, I don't ask as much from code. If there turns out to be a bug, it's just learning more about the problem; about the world. It's an engineering flaw. But if my proof is wrong, the game is lost.
An argument against my intuition is probably more telling. Though my faith in it has turned out to be justified many many times, I certainly can be wrong. My only real excuse is that, as a human being, I have nothing else to fall back on but my sense of reality and reason. That's my hardware; if it's wrong, I really am lost. So I might as well trust it. Fortunately, it's almost always right; probably because I try to see things from many angles and check them in many ways before my intutive sense is fully formed.
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.
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.
Though I still don't feel fully convinced by it; I don't fully see it. It's entirely possible my obstacle is not so much my skepticism as my stupidity :-)
I'm not sure I understand why. If the tortoise had insisted that one plus one is three, would it have destroyed your ability to accept arithmetic?
I'm not sure what Carroll intended by this piece, but what I take from it is that there's no sense arguing with irrational people. One can certainly claim to accept A, and accept if-A-then-B, but deny B; one can also claim that up is down and the sky is candy-striped.
