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).
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.
Is 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.
Reading 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.
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.
Better off, because I could have learnt the conventional axioms as a skill, like the perceptual and motor skills comprising reading, writing, arithmetic. And only later question them.
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.
> 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.
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.
> 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:
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.
In my humble opinion, proofs by contradiction are not the best examples of mathematical reasoning to be presented to the uninitiated. It has been my experience that people untrained in mathematics find it difficult to (intuitively) accept them as valid.
The overall structure of the proof is not by contradiction, but one of the steps is. The Wikipedia article calls this out, right after the sentence you quoted.
Also, even though it's correct that Euclid didn't pose it as a proof by contradiction, it can certainly be posed that way, and I often use that form when presenting it to nonmathematicians.
Thanks for the link; I now feel a little more convinced by Euclid's Theorem than last time I looked at it.
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'd be interested in hearing why you're not fully convinced.
There seem to be 2 parts to the proof. If you have a list of primes:
You can generate another number from that list
You can always get a prime from that number to add to the list
I'm guessing it's the second part that isn't clicking with you, but perhaps I'm wrong.
As for 'stupidity', I wouldn't worry about it. The only people I've ever had call me a moron or question my intelligence in any way have always been people who were less intelligent than I am. And that's not because I'm a genius ;-)
I can follow the steps, but not see it. Like turn-by-turn directions, but no map. Perhaps also because I couldn't come up with it on my own - I don't see the family of which it is an instance (partly, this is the magic open-endedness of mathematics, it's not predictable).
But I'm seeing more: start with some primes. They needn't be consective or ordered, just some primes. Any old primes will do. eg 2 and 5 are OK (skipping 3).
Now multiply them all to get p. Obviously, p is divisible by all the primes we started with, because we just created it by multiplying them. eg 2 * 5 = 10
Note that p will generally be quite a bit bigger than the primes. Typically, you'll have the primes bunched up near the left of the number line, perhaps with some primes skipped between them, then a big gap to p, and continuing to infinity on the right.
Now we add one to p. This is just to the right of p on the number line. This p+1 is either prime or it isn't.
1. If it's prime, then there is a prime other than the ones we started with. eg 10 + 1 = 11
2. If it's not prime, it has divisors. This proof claims it must include divisors that are prime but are not among those we started with. <-- THIS IS THE BIT I DON'T GET
You can keep doing this, including that new prime (ie either p+1 itself or a prime divisor of it), showing there are infinitely many primes.
> 2. If it's not prime, it has divisors. This proof claims it must include divisors that are prime but are not among those we started with. <-- THIS IS THE BIT I DON'T GET
This step of the proof invokes the Fundamental Theorem of Arithmetic [0]: the statement that every natural number can be expressed as a unique product of primes (uniqueness isn't the important bit here, just the fact that such a factorization exists).
So you need to accept the Fundamental Theorem of Arithmetic as true before you can fully understand this proof.
EDIT I can see the divisor that must exist cannot be one of the given primes: taking just one of them, multiplied by the product of the rest, the next number it divides after p must be one extra addition of it, which will be greater than our number p+1. Therefore, it isn't a divisor. The same argument excludes all the other initial primes.
So this means: it has a divisor not in the initial primes (actually, I think it must have two). But why should it be prime?
I think a given divisor does not need to be prime; but it must not be divisible by an initial prime. I guess this means that either it itself is prime, or it has divisors which in turn are either prime or have divisors etc. None of these divisors are an initial prime, because then they would also be divisors of p+1, which we have established they are not.
So I guess that's the proof... but I don't feel sure of it. There are too many steps, and I'm not 100% sure of them, and can't see the whole. Perhaps I've not covered some possibility in some step - how could I be sure I've covered them all? Maybe as it becomes more familiar, I will come to see it.
To see the power of a given proof, try to imagine what would be required to refute it, falsify it. This is by no means the only avenue of attack, but it's instructive. Also, it resembles the approach used by scientists with respect to falsifiable scientific theories (which aren't the same thing as mathematical proofs).
> Reading this story in Hofstadner's GEB destroyed my ability to accept mathematical proofs as "proven".
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.
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).