Here's a proof of P neq NP: https://zenodo.org/records/17913205
Authors write subtitle: "Conditional for Abstract Computation, Unconditional for Physical Reality"
I agree. Computational limits become physical law, not algorithmic puzzles. Cryptography is unconditionally secure. NP-hard problems require approximation, not solution. AI must be heuristic, not exhaustive. Understanding what physics forbids, not just what we haven't achieved -> focuses effort productively.
The fact that thousands of people have failed to prove that P=NP indication that it is probably not true. It has even been proven that it cannot be proven by some methods.
Couldn’t you equally say “The fact that thousands of people have failed to prove that P!=NP indication that it is probably not true”?
My completely unscientific hunch is someone will eventually prove that P=?=NP is independent of ZF(C). Maybe the universe just really wants to mess with complexity theorists
My philosophy of math muscles tingle at both sentences at about the same rate.
P=NP and P=!NP are both proven nor disproven. (There is redundant information in this sentence.)
History shows us that the historical / ‘effort’ argument is not applicable to mathematics. All proofs were unproven once until proven successfully for the first time. Harder problems need bigger shoulders to stand on. Sometimes this is due to new tools, sometimes it is a magically gifted individual focusing on the problem, usually some mix of both. All we know is that all before have failed. It’s one of the beauties in math.
Maybe I should have written: "Many have tried to find algorithms in P to solve NP problems and failed to find them." Even now, many people are working on algorithms to find solutions for NP problems. I understand that it has been proven that it is not possible to proof P=NP? using 'algorithms'. That might mean that even when a proof is found that P=NP that there still will be no P algorithm to solve NP problems.
Someone might eventually provide a non-constructive proof that P=NP - a proof that such an algorithm must exist but which fails to actually produce one.
Or even a galactic algorithm-an algorithm for solving an NP-complete problem that is technically in P, but completely useless for anything in practice, e.g. O(n^10000000)
This is a fairly new question; from the early 20th century, iirc.
There were many questions with no answers for literal centuries and thousands trying, and failing, to crack them. A solution was ultimately found despite that.
A new "math" might be needed, but an answer (affirming or not) will be found.
It is fairly new, but very relevant for daily life, like many others are not. Thousands of people have tried to write smart algorithms to solve NP problems and many have thought they found an algorithm in P only to be disproven later.
Whether the Riemann hypotesis is true or not, is not going to have any practical effect, accept for a small group of mathematisians who are working on it. Most people do not know what a Field medal is nor care about it.
> A new "math" might be needed, but an answer (affirming or not) will be found.
What if there exists a proof that P!=NP, but the shortest possible proof of that proposition is a googolplex symbols that long? Then P!=NP would be true, and provable and knowable in theory, yet eternally unprovable and unknowable in practice
That's exactly the kind of situation I had in mind when I wrote that.
Goodstein’s theory would take more symbols than there are atoms in the observable universe to write down in "classic" maths. To "fix" this, mathematicians had to use a "new" way of thinking about infinity known as transfinite induction.
I think if we're smart enough to detect(?) a proof, we'll find a way to express it in a finite manner.
Am I naive to think we've reached the point where anyone would be able to get a revolutionary thought out there quite easily? If I were such a brilliant nutjob, I'd post it on some math or computer science forum if I just wanted to be recognised. Even if just a few people see it, such an audience would likely be entrenched with the right communities to signal boost it.
Nah, cranks post inscrutable incorrect proofs (and/or bizarre unified theories) to math forums regularly. They often lack the vocabulary to even format it in a way the community can read and correct.
I recall there was a mathematician that was cataloging all the 'squaring the circle' methods people kept mailing him (it's been proven to be impossible).
If their idea were legitimately revolutionary and they had the vocabulary to express it, they could simply publish.
I agree. Computational limits become physical law, not algorithmic puzzles. Cryptography is unconditionally secure. NP-hard problems require approximation, not solution. AI must be heuristic, not exhaustive. Understanding what physics forbids, not just what we haven't achieved -> focuses effort productively.
My completely unscientific hunch is someone will eventually prove that P=?=NP is independent of ZF(C). Maybe the universe just really wants to mess with complexity theorists
P=NP and P=!NP are both proven nor disproven. (There is redundant information in this sentence.)
History shows us that the historical / ‘effort’ argument is not applicable to mathematics. All proofs were unproven once until proven successfully for the first time. Harder problems need bigger shoulders to stand on. Sometimes this is due to new tools, sometimes it is a magically gifted individual focusing on the problem, usually some mix of both. All we know is that all before have failed. It’s one of the beauties in math.
Or even a galactic algorithm-an algorithm for solving an NP-complete problem that is technically in P, but completely useless for anything in practice, e.g. O(n^10000000)
There were many questions with no answers for literal centuries and thousands trying, and failing, to crack them. A solution was ultimately found despite that.
A new "math" might be needed, but an answer (affirming or not) will be found.
Whether the Riemann hypotesis is true or not, is not going to have any practical effect, accept for a small group of mathematisians who are working on it. Most people do not know what a Field medal is nor care about it.
What if there exists a proof that P!=NP, but the shortest possible proof of that proposition is a googolplex symbols that long? Then P!=NP would be true, and provable and knowable in theory, yet eternally unprovable and unknowable in practice
Goodstein’s theory would take more symbols than there are atoms in the observable universe to write down in "classic" maths. To "fix" this, mathematicians had to use a "new" way of thinking about infinity known as transfinite induction.
I think if we're smart enough to detect(?) a proof, we'll find a way to express it in a finite manner.
I recall there was a mathematician that was cataloging all the 'squaring the circle' methods people kept mailing him (it's been proven to be impossible).
If their idea were legitimately revolutionary and they had the vocabulary to express it, they could simply publish.
From the original twit:
> I had a dream where P=NP.
Did this poster, in their dream, solve P=NP or they just heard it had already been solved?
Then after waking up from this dream they asked some slop slinger if P=NP?!?
From the follow up article:
> I guess by now you have a better understanding of why I thought I was crazy when I woke up thinking P=NP.
What do the details matter? Last week I had a dream that my childhood rat was the president of space. That's what dreams do.
> fun story: I still remember those “random oracles” that we used to proof cryptographic primitives in college
So someone who previously used 'random oracles' to prove 'cryptographic primitives' had to ask a slop slinger if P=NP?!?