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Interesting take. You have casually dismissed the Goldbach Conjecture (perhaps the deepest centuries old problem in number theory) as 'trite, trivial and uninte

Interesting take. You have casually dismissed the Goldbach Conjecture (perhaps the deepest centuries old problem in number theory) as 'trite, trivial and uninteresting'... Suggesting that you are only minimally familiar with the issue... then toss about an inapplicable phrase 'self-referential recursive logic' as if you are deeply immersed in such matters, perhaps even _much_ smarter than the thousands of mathematicians (including the likes of Euler) that have applied themselves to this problem. An odd contradiction! I do believe this opinion places you very high on the 'confidence' axis, but not especially far along the 'competence' axis.

adastra22 4y ago
I was talking about Gödel, not Goldbach: > The proof of Gödel's result's involves very carefully formalizing what statements and proofs mean so that they can be encoded as statements about arithmetic. He then shows there is a statement with encoding G that says "The statement with encoding G cannot be proved" – if it is true, then it cannot be proved. Sorry I meant to quote this bit at the beginning of my comment. Parent comment which I was replying to talks about both.
- dudeguy3301 4y ago
  wait, so you are casually dismissing Gödel's work as a logical "gotcha"? don't premise opinions about complex proofs on other peoples woefully impoverished and misrepresented explanations of said proofs that is disrespectful and very very very short sighted. also, go read up (...on Gödel, Cantor, Turing, Tarski, etc...)
  - gsinclair 4y ago
    I understand precisely what adastra22 meant from the moment I read it. Unfortunately two people now have rushed to unkind characteristics of what that (perfectly sensible) meaning was.
    - dudeguy3301 4y ago
      sure, what was written was sensible but has no bearing on the proof. speculation isn't critical thinking, as adastra22 says, be careful to what you are responding to. my point is the lack of rigor and understanding because the proof isnt being discussed, hand waving about an unread and very important mathematical work is irresponsible. thats MY point
  - adastra22 4y ago
    Please try to give the people you talk to the benefit of the doubt, and read carefully what you are responding to. My training is as an applied physicist. We physicists have an interesting relationship with math. Obviously math is essential to the work that we do, but the physical world decides whether the math is right, not the other way around. Our mathematical models technically permit things like negative mass, time flowing backwards, or magnetic monopoles. But that doesn't mean tachyons, time machines, or fundamental magnetic particles exist--they don't, so far as we know. So I'm trained to actively disregard non-physical, not relevant mathematical implications. I'm sorry if this offends a pure mathematicians sensibilities, but pragmatically it is very useful. Or take a different field: in computational semantics, a branch of formal linguistics, there are many models for inferring a formal logical statement from an example written sentence or spoken utterance, and then determining the validity (truth) of the statement. These models get caught up on stuff like "This sentence is false." What's the truth value for that sentence? If it is true then it must be false, and if it is false then it must be true. Error, validity of this statement can't be determined! But hey, it turns out that in practice this basically never happens unless the speaker is really confused, misspeaks, or deliberately evasive. Real sentences don't have this self-referential, circular logic structure because that's not how people think or communicate. Now "this sentence is false" goes back to the greeks, IIRC, and Gödel's theorem is slightly different. Gödel's main work is in the formalization of proofs and proof systems, and I don't want to take away from that in any way. But the incompleteness theorem always seems to be explained through these sorts of self-referential examples and I have yet to ever see it reduced to a practical problem with real-world implications. Hence my question. Does Gödel's incompleteness theorem actually constrain a real world application of proof systems, where we tend to be interested in non-cyclical logical arguments?
    - dudeguy3301 4y ago
      still doesn't sound like you have ever read the proof, nor about it. so aren't you engaged in this discussion strictly by speculation? logical fallacy? just because you have some point to make has no bearing on the work you are adjacent to and not actually discussing. making your own points and arguments about logical gotchas is great, has nothing to do with critically understanding a work you haven't read. asking for giving the 'benefit of the doubt' to speculation informed only by a comment section is ridiculous. do your homework before asking for generous responses to your internet quibbling
- ColinWright 4y ago
  I'm late to the party, so I'm not sure you'll see this. Hope you do. Different people will each have their own contexts and their own concepts of "useful" or "interesting". I'm making assumptions about yours ... apologies if I misrepresent you. The proof of Gödel's result uses the paradoxical statement "This statement is false", but that's being used to prove this very general result about all systems. So the hunt is then on to find "natural" statements that are True but Unprovable. But the "unprovable" bit should more completely be stated as "unprovable in a specific axiomatic proof system". If we want to prove that statement S is "True but Unprovable" then we must actually prove that it's true. So if we've proved it's true, what does it mean to say it's unprovable? We just proved it! What's going on? So let's take a specific example. Peano Arithmetic (PA)[0] is an axiomatic proof system intended to capture Natural Numbers and their behaviour. The "Goodstein Sequence" G(m) of a number m is a sequence of natural numbers ... you can find the definition here[1]. It's not hard, but it's longer than I want to reproduce here. Goodstein's Theorem (GT) says that for every integer m greater than 0, G(m) is eventually zero. It has been proven that GT cannot be proved in PA, but it can be proved in stronger systems, such as second-order arithmetic. So the statement of GT is not self-referential, along the lines of "This Statement Is False" sort of thing. It's an actual statement about the behaviour of integers, so it's not a self-referential trick. Your question now is: What's the point? How is this useful or relevant? Much of modern (pure) mathematics is chasing things because the mathematicians find them interesting. The vast, vast majority will never, of themselves, be useful by (what I expect are) your standards. But it was once thought that factoring integers was of no practical use, and only pursued or investigated by cranks. Imaginary Numbers were thought to be bizarre, useless, and dangerous. Non-Euclidean Geometry was thought to be utter nonsense, and held up as part of the "proof" that the fifth postulate was unnecessary and was deducible from the other four. All three of these now form critical components in modern technology. Even more, to the average person on the street, anything to do with algebra is completely pointless. For you, Gödel's theorem is completely pointless and useless and probably of no interest at all, but it helps us understand the limitations of formal systems. The techniques that have been developed in the time since it was proved have helped us understand more about what computer verification systems might or might not be able to accomplish. Of itself, Gödel's theorem might not be of direct, immediate, and practical use, but the work it has inspired has tangentially been useful, and may yet be moreso. But not for everyone. After all, some people don't care about the Mona Lisa, or Beethoven's Fifth Symphony, or Michaelangelo's David, or the fact that people have walked on the Moon, so why should people care about results in Pure Mathematics? That's the thing about Pure Mathematics. Sometimes it ends up being useful in ways we never expected. [0] https://en.wikipedia.org/wiki/Peano_axioms https://en.wikipedia.org/wiki/Peano_axioms [1] https://en.wikipedia.org/wiki/Goodstein's_theorem#Goodstein_sequences https://en.wikipedia.org/wiki/Goodstein's_theorem#Goodstein_...
wikfwikf 4y ago
The Goldbach conjecture is of very little interest to mathematicians, unlike either Fermat's Last Theorem or the Riemann Hypothesis. Statistically it is clearly true, in the sense that there are way more prime numbers than you would need for it to be true. Finding a counterexample would be interesting in the sense that it would be very surprising (but not mathematically interesting). It is fated to be proved as a trivial corollary to some more important mathematics; a corollary that no one would have bothered with if not for the historical importance. The unsolved Twin Prime Conjecture, of roughly the same age, is expected to lead to much more interesting mathematics if it is proved.

adastra22 4y ago

I was talking about Gödel, not Goldbach: > The proof of Gödel's result's involves very carefully formalizing what statements and proofs mean so that they can be encoded as statements about arithmetic. He then shows there is a statement with encoding G that says "The statement with encoding G cannot be proved" – if it is true, then it cannot be proved. Sorry I meant to quote this bit at the beginning of my comment. Parent comment which I was replying to talks about both.

dudeguy3301 4y ago
wait, so you are casually dismissing Gödel's work as a logical "gotcha"? don't premise opinions about complex proofs on other peoples woefully impoverished and misrepresented explanations of said proofs that is disrespectful and very very very short sighted. also, go read up (...on Gödel, Cantor, Turing, Tarski, etc...)
- gsinclair 4y ago
  I understand precisely what adastra22 meant from the moment I read it. Unfortunately two people now have rushed to unkind characteristics of what that (perfectly sensible) meaning was.
  - dudeguy3301 4y ago
    sure, what was written was sensible but has no bearing on the proof. speculation isn't critical thinking, as adastra22 says, be careful to what you are responding to. my point is the lack of rigor and understanding because the proof isnt being discussed, hand waving about an unread and very important mathematical work is irresponsible. thats MY point
- adastra22 4y ago
  Please try to give the people you talk to the benefit of the doubt, and read carefully what you are responding to. My training is as an applied physicist. We physicists have an interesting relationship with math. Obviously math is essential to the work that we do, but the physical world decides whether the math is right, not the other way around. Our mathematical models technically permit things like negative mass, time flowing backwards, or magnetic monopoles. But that doesn't mean tachyons, time machines, or fundamental magnetic particles exist--they don't, so far as we know. So I'm trained to actively disregard non-physical, not relevant mathematical implications. I'm sorry if this offends a pure mathematicians sensibilities, but pragmatically it is very useful. Or take a different field: in computational semantics, a branch of formal linguistics, there are many models for inferring a formal logical statement from an example written sentence or spoken utterance, and then determining the validity (truth) of the statement. These models get caught up on stuff like "This sentence is false." What's the truth value for that sentence? If it is true then it must be false, and if it is false then it must be true. Error, validity of this statement can't be determined! But hey, it turns out that in practice this basically never happens unless the speaker is really confused, misspeaks, or deliberately evasive. Real sentences don't have this self-referential, circular logic structure because that's not how people think or communicate. Now "this sentence is false" goes back to the greeks, IIRC, and Gödel's theorem is slightly different. Gödel's main work is in the formalization of proofs and proof systems, and I don't want to take away from that in any way. But the incompleteness theorem always seems to be explained through these sorts of self-referential examples and I have yet to ever see it reduced to a practical problem with real-world implications. Hence my question. Does Gödel's incompleteness theorem actually constrain a real world application of proof systems, where we tend to be interested in non-cyclical logical arguments?
  - dudeguy3301 4y ago
    still doesn't sound like you have ever read the proof, nor about it. so aren't you engaged in this discussion strictly by speculation? logical fallacy? just because you have some point to make has no bearing on the work you are adjacent to and not actually discussing. making your own points and arguments about logical gotchas is great, has nothing to do with critically understanding a work you haven't read. asking for giving the 'benefit of the doubt' to speculation informed only by a comment section is ridiculous. do your homework before asking for generous responses to your internet quibbling
ColinWright 4y ago
I'm late to the party, so I'm not sure you'll see this. Hope you do. Different people will each have their own contexts and their own concepts of "useful" or "interesting". I'm making assumptions about yours ... apologies if I misrepresent you. The proof of Gödel's result uses the paradoxical statement "This statement is false", but that's being used to prove this very general result about all systems. So the hunt is then on to find "natural" statements that are True but Unprovable. But the "unprovable" bit should more completely be stated as "unprovable in a specific axiomatic proof system". If we want to prove that statement S is "True but Unprovable" then we must actually prove that it's true. So if we've proved it's true, what does it mean to say it's unprovable? We just proved it! What's going on? So let's take a specific example. Peano Arithmetic (PA)[0] is an axiomatic proof system intended to capture Natural Numbers and their behaviour. The "Goodstein Sequence" G(m) of a number m is a sequence of natural numbers ... you can find the definition here[1]. It's not hard, but it's longer than I want to reproduce here. Goodstein's Theorem (GT) says that for every integer m greater than 0, G(m) is eventually zero. It has been proven that GT cannot be proved in PA, but it can be proved in stronger systems, such as second-order arithmetic. So the statement of GT is not self-referential, along the lines of "This Statement Is False" sort of thing. It's an actual statement about the behaviour of integers, so it's not a self-referential trick. Your question now is: What's the point? How is this useful or relevant? Much of modern (pure) mathematics is chasing things because the mathematicians find them interesting. The vast, vast majority will never, of themselves, be useful by (what I expect are) your standards. But it was once thought that factoring integers was of no practical use, and only pursued or investigated by cranks. Imaginary Numbers were thought to be bizarre, useless, and dangerous. Non-Euclidean Geometry was thought to be utter nonsense, and held up as part of the "proof" that the fifth postulate was unnecessary and was deducible from the other four. All three of these now form critical components in modern technology. Even more, to the average person on the street, anything to do with algebra is completely pointless. For you, Gödel's theorem is completely pointless and useless and probably of no interest at all, but it helps us understand the limitations of formal systems. The techniques that have been developed in the time since it was proved have helped us understand more about what computer verification systems might or might not be able to accomplish. Of itself, Gödel's theorem might not be of direct, immediate, and practical use, but the work it has inspired has tangentially been useful, and may yet be moreso. But not for everyone. After all, some people don't care about the Mona Lisa, or Beethoven's Fifth Symphony, or Michaelangelo's David, or the fact that people have walked on the Moon, so why should people care about results in Pure Mathematics? That's the thing about Pure Mathematics. Sometimes it ends up being useful in ways we never expected. [0] https://en.wikipedia.org/wiki/Peano_axioms https://en.wikipedia.org/wiki/Peano_axioms [1] https://en.wikipedia.org/wiki/Goodstein's_theorem#Goodstein_sequences https://en.wikipedia.org/wiki/Goodstein's_theorem#Goodstein_...

wikfwikf 4y ago

The Goldbach conjecture is of very little interest to mathematicians, unlike either Fermat's Last Theorem or the Riemann Hypothesis. Statistically it is clearly true, in the sense that there are way more prime numbers than you would need for it to be true. Finding a counterexample would be interesting in the sense that it would be very surprising (but not mathematically interesting). It is fated to be proved as a trivial corollary to some more important mathematics; a corollary that no one would have bothered with if not for the historical importance. The unsolved Twin Prime Conjecture, of roughly the same age, is expected to lead to much more interesting mathematics if it is proved.