[HN Gopher] What if they are all wrong? (2020)
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What if they are all wrong? (2020)
Author : macleginn
Score : 65 points
Date : 2024-11-05 13:13 UTC (9 hours ago)
(HTM) web link (igorpak.wordpress.com)
(TXT) w3m dump (igorpak.wordpress.com)
| empath75 wrote:
| 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 wrote:
| 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 wrote:
| 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 wrote:
| 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 wrote:
| 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 wrote:
| 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 wrote:
| 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 wrote:
| 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 wrote:
| 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!
| schoen wrote:
| You can conclude "not not B", which is also called "B is
| irrefutable". Irrefutability is like a weaker version of
| "true" in constructive logic that means "could be assumed
| without introducing a new contradiction".
|
| https://en.wikipedia.org/wiki/Double-negation_translation
| Xcelerate wrote:
| As someone not in the field of mathematics, I actually didn't
| realize there was such an emphasis on attempting to prove famous
| conjectures correct. I thought the famous conjectures were famous
| precisely because knowing the veracity of the conjecture (or
| independence with respect to some formal system) would be a
| monumental event in any case.
|
| I also assumed many mathematicians utilized the strategy of
| attempting to disprove something as a way to reveal the proof.
| Sort of like how the best chess players tend to spend most of
| their thinking time mentally challenging their planned move as
| opposed to novice chess players who tend to look for reasons
| confirming why their move will be a good one. Is this not the
| case?
| feoren wrote:
| This article is one professor's mix of opinion and elementary
| introduction to the concept of conjecture, and IMO is quite
| rambling and inconsistent. Why are you generalizing this one
| random person's thoughts into a conclusion about mathematics as
| a discipline? Don't let one poor thinker (or perhaps merely
| poor _writer_ ) spoil it for you.
|
| The author seems to have completely missed the actual reason we
| find conjectures interesting: we're asking ourselves, "do we
| have the techniques to prove this kind of problem?" That's the
| fundamental reason disproving a conjecture is less celebrated
| than proving it, because disproving is often done by finding a
| single counterexample, which doesn't teach us much about our
| proof techniques -- unless, of course, the process of finding
| that counterexample was particularly new and interesting!
|
| The most interesting question to ponder with a conjecture is
| not "is this true?", but rather "why is this hard to prove?".
| The most boring answer to the latter is "oh, because it's
| false", while the most exciting answer is "because we were
| missing this fundamental idea that can spawn an entire new
| branch of mathematics." It's all about the journey, not the
| destination.
| kevinventullo wrote:
| It would be a shameful travesty if the Clay Institute refused to
| award the prize for a correct proof that P vs. NP was independent
| of ZFC. In my mind, this would be a more interesting result than
| either a proof or a disproof!
| zusammen wrote:
| Not only would it be more interesting, but it would probably
| answer the practical question of whether efficient NP-solvers
| exist--in the negative. An independence proof boils down to,
| "it depends on these highly abstract axioms pertaining to
| massive infinite sets." So an independence proof would mean
| some model of mathematics exists where a polynomial-time NP-
| solver exists, but the lack of one within ordinary mathematics
| (which is the model-independent day-to-day stuff) could make it
| impractical in ways that are hard to imagine, since the
| mathematics leading to such a proof doesn't seem to exist.
| whatshisface wrote:
| The Clay institute should let discoverers of independence
| proofs to take out a million dollar loan with no interest or
| repayment schedule.
| wesselbindt wrote:
| I think this sentiment is coming from the same place as that of
| people who oppose more "pure" scientific pursuits in general. You
| can ask "what good does it do me that some guy was able to go to
| the moon?" or "who cares about theorizing about black holes,
| they're really far away", but ultimately pursuing these goals is
| what got us such crucial everyday stuff like GPS. Sure, you may
| not care about the goal (even though I think they're worthy
| goals), but you sure as heck benefit from the journey that got us
| there. Similarly, chasing mathematical conjectures, even the ones
| that turn out false, sets us off on journeys on which we
| discover/invent ever more sophisticated and powerful mathematical
| tools. And these tools have an amazing tendency to give us
| insight into broader problems, and help us prove other theorems,
| even if the conjecture we were originally chasing turns out to be
| false. It is in this way that chasing conjectures gets us closer
| to the truth, regardless of their own truthiness.
| gurubm wrote:
| Any theorem was a conjecture before its formal proof was accepted
| by the community. Thus the importance of conjectures is a proxy
| to the importance of theorems: they are proto-theorems.
|
| The fact that the human mind is capable of searching for new
| mathematical theorems specifically here instead of there, is
| quite interesting. It's as if a skilled mathematician has
| knowledge that is bigger than math itself.
|
| Penrose used this as argument for the special nature of
| consciousness, wrongly - it probably makes more sense to remember
| how the way the human mind is not exact and produces errors all
| the time plays a huge role in a creative process. And luckily we
| can amend the human mind by social processes that help
| eliminating errors again.
|
| This way, a conjecture can be thought of as a claim to truth in a
| competitive environment and that would explain why proving the
| truth is regarded so much higher.
| hshshshshsh wrote:
| What if they are all wrong also assumes reality has this property
| of an observer being able to characterize things as right and
| wrong.
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