Thommo wrote:
Olivier's post starts with the word "proof" when he means "theorem" (or lemma or corollary), by the way.

OlivierK wrote:

Thommo wrote:
Olivier's post starts with the word "proof" when he means "theorem" (or lemma or corollary), by the way.

Well, yes, but if we're being pedantic, then "proven theorem" would be more correct than either.

scott1328 wrote:Have you actually read the elements?

Most of the theorems appearing in the Elements were not discovered by Euclid himself, but were the work of earlier Greek mathematicians such as Pythagoras (and his school), Hippocrates of Chios, Theaetetus of Athens, and Eudoxus of Cnidos. However, Euclid is generally credited with arranging these theorems in a logical manner, so as to demonstrate (admittedly, not always with the rigour demanded by modern mathematics) that they necessarily follow from five simple axioms. Euclid is also credited with devising a number of particularly ingenious proofs of previously discovered theorems: e.g., Theorem 48 in Book 1.

Book 7 deals with elementary number theory: e.g., prime numbers, greatest common denominators, etc. Book 8 is concerned with geometric series. Book 9 contains various applications of results in the previous two books, and includes theorems on the infinitude of prime numbers, as well as the sum of a geometric series. Book 10 attempts to classify incommensurable (i.e., irrational) magnitudes using the so-called “method of exhaustion”, an ancient precursor to integration.

Calilasseia wrote:Doesn't a system have to be complete in order to be able to prove its own consistency? My point stands.

Thommo wrote:...
If anyone can tell me the correct "value" or "use" (outside of mathematics) of a key result in logic like the
Löwenheim–Skolem theorem I'd certainly love to see them do it. I used to use that thing all the time, but I'm buggered if I can see any connection from it to anything non mathematical.

Prior work has also shown that crows, like many other animals, can discriminate different amounts. To investigate whether the ability is hardwired, Nieder and his colleagues used implanted electrodes to record activity from adult crows’ endbrains—the bird equivalent of the human cerebral cortex, the highest processing center in the brain—while the animals engaged in a task in which they were rewarded each time they correctly identified sets of dots that matched in color. This revealed that, despite not being trained to identify discrete quantities, the animals possessed neurons that fired faster when they saw specific numbers of dots.

zoon wrote:

Thommo wrote:...
If anyone can tell me the correct "value" or "use" (outside of mathematics) of a key result in logic like the
Löwenheim–Skolem theorem I'd certainly love to see them do it. I used to use that thing all the time, but I'm buggered if I can see any connection from it to anything non mathematical.

As Scott1328 pointed out in #31 above, mathematics has survival value when it's used to model the external world; if there's any chance that the Löwenheim–Skolem theorem could be involved in the calculations of physicists or engineers, then I think it has what most of us (probably including jamest) would describe as value (I don't know whether mathematicians would also find it beautiful, giving an intrinsic sense of satisfaction - this might also add a separate element of human value?).

Thommo wrote:

zoon wrote:

Thommo wrote:...
If anyone can tell me the correct "value" or "use" (outside of mathematics) of a key result in logic like the
Löwenheim–Skolem theorem I'd certainly love to see them do it. I used to use that thing all the time, but I'm buggered if I can see any connection from it to anything non mathematical.

As Scott1328 pointed out in #31 above, mathematics has survival value when it's used to model the external world; if there's any chance that the Löwenheim–Skolem theorem could be involved in the calculations of physicists or engineers, then I think it has what most of us (probably including jamest) would describe as value (I don't know whether mathematicians would also find it beautiful, giving an intrinsic sense of satisfaction - this might also add a separate element of human value?).

Trust me, it doesn't have the potential to be used in the calculations of physicists or engineers.

Thommo wrote:It's hard to see how.

As I said, I am buggered if I can see how it has any non mathematical value.

Q: So, pure maths… come for the pretty patterns, stay for the revolutionary insights?

A: That about covers it.

Thommo wrote:I would, it's a good read.

He's probably a fair bit more optimistic about potential offshoot applications than I am though. Pure mathematics (and particularly the logical subfields) has tended to become not only more abstract, but to add more layers of abstraction as time has gone by.

jamest wrote:I'm not denying that some (if not most) are valuable. I'm just curious as to whether they're ALL valuable. [ ] metaphysical potential and hence value?

Greyman wrote:It is interesting to note that nobody was blowing each other up over the effort to prove Fermat's Last Theorem or... well, any mathematical proofs or conjectures, ever.

ughaibu wrote:How about that case of Hippasus of Metapontum? Or Cardano and Tartaglia?

LucidFlight wrote:Hopefully, jamest will be back soon to guide the discussion into more metaphysical territory, with an example of what we ought to be thinking about.

jamest wrote:

LucidFlight wrote:Hopefully, jamest will be back soon to guide the discussion into more metaphysical territory, with an example of what we ought to be thinking about.

I'm happy enough to have inspired intelligent discussion amongst yourselves, for now. Once you're all congregated at one spot, I'll come back.