Do all mathematical proofs have value?

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Re: Do all mathematical proofs have value?

#41  Postby OlivierK » Apr 01, 2018 4:32 am

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. ;)
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Re: Do all mathematical proofs have value?

#42  Postby Calilasseia » Apr 01, 2018 4:32 am

Doesn't a system have to be complete in order to be able to prove its own consistency? My point stands.
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Re: Do all mathematical proofs have value?

#43  Postby Thommo » Apr 01, 2018 4:38 am

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. ;)


That's a tautology, an unproven theorem is not a theorem, it's a conjecture.

It doesn't matter though, I'm not saying you were wrong, or really even unclear, it was just a quibble.
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Re: Do all mathematical proofs have value?

#44  Postby Calilasseia » Apr 01, 2018 4:45 am

scott1328 wrote:Have you actually read the elements?


This is the version I refer to.

From the introduction:

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.


A little further on, we have this:

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.


Part of the problem, of course, is that the five axioms of Euclidean geometry are actually spread across two sections of Book 1, the first four axioms being present in the Definitions section, whilst the parallel lines axiom appears a little later in the Postulates section. Book 7, on number theory, is somewhat less disjointed in this regard, with all the relevant axioms in use being present in the initial Definitions section. As will be seen by anyone downloading and reading that text, which is based upon the Greek considered definitive in modern scholarship.
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Re: Do all mathematical proofs have value?

#45  Postby Thommo » Apr 01, 2018 4:48 am

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


If you want to contend the issue, then most of your points under discussion are wrong.

Most modern mathematical logic absolutely does not "establish that any set of axioms generated isn't internally inconsistent [and] only then ... move on to the process of establishing that if a given set of axioms holds for any system, then various theorems derive as a consequence for those same systems", for the simple reason it provably can't in a huge range of cases.

It's also a point that doesn't stand that "actually, the correct result arising from Gödel's Incompleteness Theorem is..." because what you were correcting was in fact correct and there is more than one incompleteness theorem, with more than one correct consequence.

Your statement of the result of the first incompleteness theorem is correct, but unfortunately it has no real bearing on your other claims.

Scott is also correct about Euclid's elements, whilst it can be referred to as formal and axiomatic, these words are used in a historical context, it has paltry similarity to modern formalised symbolic logic and its axiomatisation and construction is lacking in various regards as well. Most of the proofs contained within are ruler and straight edge constructions that students might do in high school, not formal logic.

ETA: An example of an incomplete system that proves its own consistency:
https://math.stackexchange.com/question ... ic-systems
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Re: Do all mathematical proofs have value?

#46  Postby LucidFlight » Apr 01, 2018 5:22 am

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.
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Re: Do all mathematical proofs have value?

#47  Postby zoon » Apr 01, 2018 9:10 am

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?).

The capacity to enjoy abstract mathematics for its own sake is quite possibly hardwired into us (some more than others) through evolution, because of its usefulness in modelling. A study published last month and described in a magazine "The Scientist" here showed that the brains of untrained crows are hardwired to respond differently to different numbers of objects. From the article in "The Scientist":
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.

As pointed out in the article, the crows may in fact have learned to do this in spite of not having had specific training, the experiment would need to be repeated in juveniles, but even the capacity to learn has to be hardwired. Some capacity for abstract maths appears to have evolved at least twice through natural selection, as bird and primate lineages separated more than 300 million years ago and use different brain structures for complex thinking. The original article, in Cell Biology, I think issue of 15th March 2018, is here in full.
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Re: Do all mathematical proofs have value?

#48  Postby Thommo » Apr 01, 2018 9:20 am

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.
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Re: Do all mathematical proofs have value?

#49  Postby zoon » Apr 01, 2018 9:34 am

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.

Might it have any use in cryptography?
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Re: Do all mathematical proofs have value?

#50  Postby Thommo » Apr 01, 2018 9:46 am

It's hard to see how.

As I said, I am buggered if I can see how it has any non mathematical value.
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Re: Do all mathematical proofs have value?

#51  Postby zoon » Apr 01, 2018 11:19 am

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.

Thanks, you intrigue me, would you go along with this piece on "Why do we pay pure mathematicians?"? Quoting from it:
Q: So, pure maths… come for the pretty patterns, stay for the revolutionary insights?

A: That about covers it.
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Re: Do all mathematical proofs have value?

#52  Postby Thommo » Apr 01, 2018 11:26 am

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.
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Re: Do all mathematical proofs have value?

#53  Postby zoon » Apr 01, 2018 11:55 am

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.

Perhaps, for the pure mathematicians themselves, it may even be better to avoid thinking about practical applications, as that could detract from the quality of the maths, the building of the foundational structures? Whereas, for the taxpayer, it's well to be aware that it has the potential for useful results in wholly unexpected places, in the way that George Boole wasn't thinking of computers when he developed Boolean algebra. Similarly, biochemists (for example) may be alarmed if too much of their funding comes from companies which want useful results, as this can compromise the basic research.
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Re: Do all mathematical proofs have value?

#54  Postby Thommo » Apr 01, 2018 12:34 pm

Yes, that's a reasonable way of looking at it.

Personally I'm plenty happy with its inherent value, without any expectation of external value or reward. I wouldn't suggest we defund English literature PhD's for example, and I'm even less optimistic of the chances of those resulting in revolutionary new technologies.
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Re: Do all mathematical proofs have value?

#55  Postby scott1328 » Apr 01, 2018 2:02 pm

If one actually wants to do rigorous thought exercises that has no application to the real world,one only has to focus attention on trans finite arithmetic and the higher order infinities.

Of course it used to be said that advanced number theory was a branch of mathematics with. No application to the real world until modern cryptographic systems were developed.
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Re: Do all mathematical proofs have value?

#56  Postby ughaibu » Apr 06, 2018 6:13 am

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?
Can you give an example of a valuable proof and explain how it is valuable, please.
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Re: Do all mathematical proofs have value?

#57  Postby ughaibu » Apr 06, 2018 6:29 am

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.
How about that case of Hippasus of Metapontum? Or Cardano and Tartaglia?
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Re: Do all mathematical proofs have value?

#58  Postby Sendraks » Apr 06, 2018 9:41 am

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


Go on.
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Re: Do all mathematical proofs have value?

#59  Postby jamest » Apr 06, 2018 11:01 pm

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.
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Re: Do all mathematical proofs have value?

#60  Postby OlivierK » Apr 07, 2018 6:31 am

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.

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