3 ms·

I applaud this person for disproving conjectures and think disproving conjectures is important, but the reason why the conjectures he disproved didn't get much

I applaud this person for disproving conjectures and think disproving conjectures is important, but the reason why the conjectures he disproved didn't get much attention is that they're all relatively minor conjectures, and he probably would have gotten just as little attention if he had proved them. If someone disproved the Riemann hypothesis, it would be the biggest math story of the century -- certainly since the proof of Fermat's last theorem, I am sure. I also think that working out the consequences of a conjecture being true is _indistinguishable_ from attempting to disprove the conjecture. How could you possibly work out something like a proof (or disproof) by contradiction without doing that? Really working toward a proof in general is the same as working toward a disproof in almost every case I can think of. An exhaustiveness proof is also a search for a counterexample, etc.

bubblyworld 2y ago
An interesting aside about your last paragraph - there are some results in number theory (like Linnik's theorem) where the original proof was split into two cases. One where the existence of Siegel zeroes was assumed (a famous conjecture), and another where the converse was assumed instead! Sometimes a statement A is a non-trivial consequence of both a conjecture B and it's negation independently. So in a technical sense A doesn't take you any closer to a proof or disproof of the conjecture in question (and it can be difficult to tell whether your consequence is of this type or not).
- empath75 2y ago
  Well, sure, but you don't know that until you work out the consequences -- you're just exploring the space of possibilities. Eventually you'll either find a contradiction or you won't, but you'll _never_ find a contradiction if you don't look.
  - bubblyworld 2y ago
    You said that working out the consequences was indistinguishable from attempting to disprove the conjecture. I was just noting that there are sometimes subtle differences. I agree in general though, this wasn't intended as a criticism =)
- cvoss 2y ago
  Aside to your aside: When A can be proven from B, and A can also be proven from !B, it is really tempting to conclude that A is proven. But many people will argue this is incorrect in general, that is, they deny the excluded middle axiom. What if B is independent of your system of logic? We know then that A is not disprovable (since a proof of !A would prove !B). But that doesn't prove A. Unless A can be proven without reference to B, A might be independent too.
  - BalinKing 2y ago
    Not an expert, but I don't think that independence is particularly relevant—intuitionists deny LEM for more fundamental philosophical reasons AFAIK, even for statements that could theoretically be proven true or false within your system of logic.
    - cvoss 2y ago
      I don't understand there to be anything deeper about it, though happy to be corrected. In a consistent system, there are three kinds of statements: the provable ones, the disprovable ones, and the independent ones. This is straight from Godel Incompleteness I. The sin of LEM is exactly that it overlooks the reality of incompleteness. But, if course, you also can't fix it with an axiom encoding the trichotomy I just stated, because the independence of a statement is itself a statement that lives in a higher ambient system.
      - whatshisface 2y ago
        LEM doesn't overlook completeness within the context of an axiom system. Independent results remain, by definition, without correspondence to any proof. Believe it or not, constructive math actually does have a LEM, but a derived one. Any computable value can be shown to either equal or not equal another. Only for uncomputable predicates does the absence of LEM result in our inability to conclude x from not not x.
  - bubblyworld 2y ago
    Yeah, I believe it's invalid to conclude "B" from "A => B" and "not A => B" in constructive logic, for this reason. But most mathematicians reason classically without thinking twice about it!

bubblyworld 2y ago

An interesting aside about your last paragraph - there are some results in number theory (like Linnik's theorem) where the original proof was split into two cases. One where the existence of Siegel zeroes was assumed (a famous conjecture), and another where the converse was assumed instead! Sometimes a statement A is a non-trivial consequence of both a conjecture B and it's negation independently. So in a technical sense A doesn't take you any closer to a proof or disproof of the conjecture in question (and it can be difficult to tell whether your consequence is of this type or not).

empath75 2y ago
Well, sure, but you don't know that until you work out the consequences -- you're just exploring the space of possibilities. Eventually you'll either find a contradiction or you won't, but you'll _never_ find a contradiction if you don't look.
- bubblyworld 2y ago
  You said that working out the consequences was indistinguishable from attempting to disprove the conjecture. I was just noting that there are sometimes subtle differences. I agree in general though, this wasn't intended as a criticism =)
cvoss 2y ago
Aside to your aside: When A can be proven from B, and A can also be proven from !B, it is really tempting to conclude that A is proven. But many people will argue this is incorrect in general, that is, they deny the excluded middle axiom. What if B is independent of your system of logic? We know then that A is not disprovable (since a proof of !A would prove !B). But that doesn't prove A. Unless A can be proven without reference to B, A might be independent too.
- BalinKing 2y ago
  Not an expert, but I don't think that independence is particularly relevant—intuitionists deny LEM for more fundamental philosophical reasons AFAIK, even for statements that could theoretically be proven true or false within your system of logic.
  - cvoss 2y ago
    I don't understand there to be anything deeper about it, though happy to be corrected. In a consistent system, there are three kinds of statements: the provable ones, the disprovable ones, and the independent ones. This is straight from Godel Incompleteness I. The sin of LEM is exactly that it overlooks the reality of incompleteness. But, if course, you also can't fix it with an axiom encoding the trichotomy I just stated, because the independence of a statement is itself a statement that lives in a higher ambient system.
    - whatshisface 2y ago
      LEM doesn't overlook completeness within the context of an axiom system. Independent results remain, by definition, without correspondence to any proof. Believe it or not, constructive math actually does have a LEM, but a derived one. Any computable value can be shown to either equal or not equal another. Only for uncomputable predicates does the absence of LEM result in our inability to conclude x from not not x.
- bubblyworld 2y ago
  Yeah, I believe it's invalid to conclude "B" from "A => B" and "not A => B" in constructive logic, for this reason. But most mathematicians reason classically without thinking twice about it!