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What the Tortoise Said to Achilles (1895)

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Re: What the Tortoise Said to Achilles (1895)

#11

Post by hyp0 »

rspeer wrote: post #8
hyp0 wrote: post #5

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,…

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…

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.

Re: What the Tortoise Said to Achilles (1895)

#12

Post by hyp0 »

lutusp wrote: post #7
hyp0 wrote: post #5

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,…

> 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 pro…

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 :-)

Re: What the Tortoise Said to Achilles (1895)

#13

Post by PhasmaFelis »

hyp0 wrote: post #5

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,…

> 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.

Re: What the Tortoise Said to Achilles (1895)

#14

Post by tjgq »

lutusp wrote: post #7
hyp0 wrote: post #5

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,…

> 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 pro…

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.

Re: What the Tortoise Said to Achilles (1895)

#15

Post by lutusp »

hyp0 wrote: post #12
lutusp wrote: post #7

Earlier quoted context omitted.

> 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 pro…

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 :-)

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).

Re: What the Tortoise Said to Achilles (1895)

#16

Post by phaemon »

tjgq wrote: post #14
lutusp wrote: post #7

Earlier quoted context omitted.

> 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 pro…

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.

Euclid's proof isn't a proof by contradiction. As the wikipedia page says:

>"Euclid is often erroneously reported to have proved this result by contradiction"

It simply says that if you are constructing a list of primes, you can always add one more to the list, therefore there are infinitely many.

Re: What the Tortoise Said to Achilles (1895)

#17

Post by phaemon »

hyp0 wrote: post #12
lutusp wrote: post #7

Earlier quoted context omitted.

> 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 pro…

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 ;-)

Re: What the Tortoise Said to Achilles (1895)

#18

Post by hyp0 »

phaemon wrote: post #17
hyp0 wrote: post #12

Earlier quoted context omitted.

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…

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.

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.

So, yes, it's the second part.

Re: What the Tortoise Said to Achilles (1895)

#19

Post by hyp0 »

hyp0 wrote: post #18
phaemon wrote: post #17

Earlier quoted context omitted.

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…

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 (sk…

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.

Re: What the Tortoise Said to Achilles (1895)

#20

Post by tjgq »

phaemon wrote: post #16
tjgq wrote: post #14

Earlier quoted context omitted.

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.

Euclid's proof isn't a proof by contradiction. As the wikipedia page says: >"Euclid is often erroneously reported to have proved this result by contradiction" It simply says that if you are constructing a list of primes, you can always add one more to the list, therefore there are infinitely many.

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.
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