[gwnn]

the axiom of choice sounds banal to me, why isn't it?

[helene_t]

gwnn, on 2011-April-18, 11:53, said:

the axiom of choice sounds banal to me, why isn't it?

Well the fact that you can't construct a choice function for the real numbers makes it a defensible assertion that maybe there isn't one.

But yeah, agree, it sounds banal to me as well. Zorn's lemma even more so.

[kenberg]

When I was a grad student the gag was this: Consider two statements, the axiom of choice and the well-ordering theorem. The first is obviously true and the second is obviously false. The fact is that they are equivalent.

Now to get a feel for the axiom of choice it goes like this. First consider a nice sensible function defined on the positive integers, say f(n)= the largest prime number that is less than n. No one knows how to compute f(n) when n is large, but no one doubts that this is a function. f(23243u0953234394939403049393939993) is some number, the fact that I don't know what it is is my problem.

Now consider a different situation. Call a set A of real numbers nice if it has at least one number in it and furthermore if x and y are both in A then x-y is a rational number. Give nay one such set A, since it has at least one number in it we can start a proof by saying "Let x be a number in A...", The Axiom of Choice allows us to do more. It allows us to say "Let X be the collection of all nice sets, and let f be a function whose domain is the collection of all nice sets and such that for any nice set A, f(x) is a number in A." That is, we have some huge collection of sets and f chooses one number from each of them.

This is a highly non-constructive process. It seems innocent enough but it has some consequences, for example the Banack-Tarski paradox which no doubt can be found on Wik.

Which brings up another joke from my grad school years. Someone suggested that if we ever find another planet with intelligent life there is at least a reasonable chance that they will do all of their math w/o the Axiom of Choice because they decided they would not put up with having to accept Banach-Tarski. Probably this joke will never make it onto the Letterman show.

[nige1]

Private Eye had a series on celebrities and their spoons

[mycroft]

mycroft, on 2011-April-18, 10:29, said:

However, assume the Extended Riemann Hypothesis for a minute...or P ≠ NP; or the Axiom of Choice or the Parallel Axiom (those two are cheating, however, as it's been proven that with or without those axioms, the math is consistent (but different)).

PassedOut, on 2011-April-18, 10:38, said:

One can construct consistent systems using all sorts of different sets of axioms. The systems of real value model the real world in a useful, verifiable way.

Sorry, that was unclear - I was conflating two cases.

The first two (Extended Riemann Hypothesis and P ≠ NP) are unproven, but "likely" correct, and useful (so much so that if someone proves either of them incorrect, the world as we know it (the internet and world commerce, at least) may collapse "overnight").

The latter two (Axiom of Choice and Euclid's Parallel Axiom) are "unprovable", because consistent and complete mathematical models can be generated with and without those axioms. I'm not sure that there's a real-world-value argument (yet) for or against the Axiom of Choice; there certainly is real-world value in non-Euclidean Geometries (I'm taking the usefulness of Euclidean Geometry as proven for this discussion) - the "real world" is non-Euclidean.

[PassedOut]

mycroft, on 2011-April-18, 15:32, said:

there certainly is real-world value in non-Euclidean Geometries

Of course. But I'm having trouble figuring out if you are working on making a bigger point with these responses to me.

[mycroft]

Yes, sorry to confuse. Nigel's original comment was:
'We all "believe" things we don't understand advocated by authorities we trust. Earlier somebody mentioned Fermat's last Theorem. You can believe it to be true without understanding the proof.'
and you mentioned "proof involving magic", and our resident math professor.

My point is that the status of the ERT and P/NP is "belief without proof" - my google link was, explicitly, for "Assuming the Extended Riemann Hypothesis" -(and we certainly put a *lot* of faith in that belief, because it's useful and convenient), and the postulated axioms (which need not be postulated) are belief involving magic (or "flip of the coin" or "usefulness in context".)

Other beliefs without proof that are useful and convenient are, well, useful and convenient. They don't have to be "right" to be so. They don't even have to be "rational".

Having said that. in the original case, if the bookburner is the same idiot that threatened to do it last year, I investigated his "college student handbook" then. Same thoughts apply now.

As far as "but, but, but, killing!" - I have a word for societies that set things up in such a way that a bad consequence is likely-to-inevitable if the right levers are pushed attempting to look all nice and innocent when someone pushes the lever. It's not nice, and I'm in that society.

[PassedOut]

mycroft, on 2011-April-19, 11:47, said:

Yes, sorry to confuse. Nigel's original comment was:
'We all "believe" things we don't understand advocated by authorities we trust. Earlier somebody mentioned Fermat's last Theorem. You can believe it to be true without understanding the proof.'
and you mentioned "proof involving magic", and our resident math professor.

My point is that the status of the ERT and P/NP is "belief without proof" - my google link was, explicitly, for "Assuming the Extended Riemann Hypothesis" -(and we certainly put a *lot* of faith in that belief, because it's useful and convenient), and the postulated axioms (which need not be postulated" are belief involving magic (or "flip of the coin" or "usefulness in context".)

Okay, thanks for elaborating.

When I say "magic" I don't mean "something I don't understand" nor "something that has not yet been proved." Rather, I mean "something I could never understand in terms of the natural world."

I'm confident that I could understand the proof of Fermat's last theorem, given enough time and effort. And even if that were beyond my capabilities, I know that mathematicians have been on it like flies on roadkill trying to break the proof and Ken would know if it had been done.

If you use magic to (for the sake of example) turn Nigel into a frog, I won't ever understand how you did it. That's the nature of magic. As I said earlier in this thread, my "belief system" is that of pretty much everyone, except that I don't believe in magic. And therefore I don't believe in religious miracles either.

The point about my "belief system" being pretty much like everyone else's is to emphasize that there is no conflict at all between atheism and accepting that concepts are not material things. And my not believing in magic is inconsequential because I never encounter anything magical.

[blackshoe]

PassedOut, on 2011-April-19, 12:57, said:

…I don't believe in magic. And therefore I don't believe in religious miracles either.

Quote

From The Tome of the Shek-Pvâr: It is a grievous error to confuse magic and religion. There is a vital difference between a spell and a miracle, between magic and religion. As the mage Génin wrote:

Quote

He who names miracle magic, insults churches and gods, and he who attributes unto a spell divinity, insults its caster. Confusing magic and religion is both offensive and blasphemous.

OTOH

Quote

From Authentic Thaumaturgy: The difference between "magic" and "miracle" is one of degree. The gods are capable of manipulating many orders of magnitude more mana than any mere mortal.

Take your pick. The latter was written by P.E.I. Bonewits, the only person to ever receive a degree in magic from a major university (UC Berkeley, who after they granted this degree, changed their rules to prevent anyone else from embarrassing them further. )

[Winstonm]

Quote

The difference between "magic" and "miracle" is one of degree.

The difference is between illusion and delusion. In illusion, a trick is played on your mind. In delusion, your mind plays a trick on you.

[blackshoe]

No, it's not.

Bonewits operates from the basis that magic really exists (though it is difficult to master). Granted he's probably wrong, but his conclusion does follow logically from his premises. In his theory, a man might do little things (a spell to produce light, for example), a god might do bigger things (rain down fire on a city). A man (or woman, let's not be sexist here) might ask a god to do something (through prayer, for example); the god might channel the "mana" required through the man's mind. Or not. (In Bonewits' theory, magic works by manipulation of the world via one's mind, using "mana" as an energy source.

Larry Niven postulated, in The Magic Goes Away, that mana used to be all around us, but that it was a non-renewable resource, and we've used it all up, and that is why magic no longer works. Maybe Niven is right.

[Winstonm]

Quote

Larry Niven postulated, in The Magic Goes Away, that mana used to be all around us, but that it was a non-renewable resource, and we've used it all up

I have an easier explanation. We used to believe all sorts of nonsesical bs. Not so much anymore.

[blackshoe]

Heh. Just as much "nonsensical bs" out there, it's just different from the old stuff.