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?
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.
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.
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.
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!
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.
The Clay institute should let discoverers of independence proofs to take out a million dollar loan with no interest or repayment schedule.
What if they are all wrong also assumes reality has this property of an observer being able to characterize things as right and wrong.
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.
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).
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.
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.
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.
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.
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!
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".
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.
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 =)