Math is also useful. If someone showed that p = np tomorrow in a formally verified proof I don't care if no one can understand it.
Suppose an oracle tells us the Riemann Hypothesis is correct. There are a vast number of results of the form:
If RH is correct then A.
It would be very useful to have an oracle tells us whether or not RH is correct.
For what? Which product becomes better if it is correct?
I have published mathematics so I do value knowledge, but for most of mathematics the value of the knowledge isn't the thing you try to prove it is all the things you learn as you try to prove it. p = np is one such thing.
So the whole interesting bit about it is the proof, not the fact.
For what? Which product becomes better if it is correct?
This sentiment is anti-thetical to the whole point of pure math and theoretical science. No product became better when Euler proved the fundamental theorem of algebra.
Tying the worthiness of theoretical knowledge to the whether or not it improves a product is asinine in my opinion.
[0] and am only adding that "generally" because I can think of examples where I'd disagree, e.g. a kid that wants to count all stars in the night sky before it has dinner would just starve and then not be able to count stars, either.
We all believe it. It's a magic oracle. Now what?
But one can already study the consequences of P=NP right now. You don't need to know that it's provably true in order to do that.
Knowing an actual proof would be useful, but an oracle revealing merely that it's true (or even provable) without telling you the proof does not let you do anything you couldn't do before.
A lot of people spent a lot of time and effort to prove or disprove the Jacobian Conjecture. AI solved it easily. It is increasingly becoming the case that humans are not as good at mathematics as computers. You are free to ignore computer generated proofs but I don’t think this position will win out in the long run.
No, people constantly prove statements of the form "if P=NP, then strange implication X". They do not consider it wasted effort at all, because of the contrapositive: if X is indeed very strange, they might be able to prove that it is false, and then they've settled P!=NP.
At some point an AI will prove a result that is so long and complicated that no human will understand it. This should not preclude people from using that result. In general, whenever the body of knowledge is increased it is a good thing. Even if it isn’t increased by humans.
In any case, I think it's better to read PP as somebody would find a practical, albeit incomprehensible, algorithm for solving NP complete problems.
Although I probably disagree with PP, because even a candidate algorithm that mysteriously works without proof would have practical value, so this case is not predicated on proving.
I think a better example of genuinely practical but rather uninteresting (YMMV) mathematical proofs are proofs of convergence of numerical methods, FEM for example. (I have been through it in school, it was a torture.)
It would not not necessarily be practical, even if it ran in polynomial time. It may have cost O(n^c), with a totally out of order exponent like c=A(5,5) or whatever.
> Oh it would change a lot. It would be an enormous psychological boost for everyone to find a practical algorithm.
OK, I tell you that P=NP, and that I am a magic oracle. So, you have you psychological boost for finding a practical algorithm for free. :-)
a) I am already convinced that P=NP
b) You have to convince many other people as well (that you're a magic oracle), because for the effect to work, lot of people would have to work on the problem (or at least spend tokens)
Nevertheless, a plausible magic oracle (such as Lean-verified proof, even if non-constructive and incomprehensible for humans) would convince many to take a 2nd look.
Somebody, eventually, somewhere would understand it, or at least aspire to understand it. And even if he doesn’t, what have they learned in the process? About themselves, about their environment? About failure? I would bet a lot. How useful then, can we say that it is, not because we can understand it, but because we can try? That is useful. This is about the journey. Sometimes the journey is the point.
This is like if math was fascist, this is what would happen. When you start controlling the flow of knowledge like this, it will be bad news all around. And who is to say whether or not something can be understood?Aside from the math nazis.
If we are going to dictate what gets published like this, why bother publishing anything? This feels like a gatekeeping…that’s exactly what it is. Ya’ll getting nervous?
Furthermore, careful analysis of the latter would as likely as not yield further understanding and, actually /would/ help finding such algorithms.
Finally, it has been observed time and time again that often (again, nothing comes up and i don’t want to ask AI) the certainty that something is possible and has been done is motivation and inspiration enough for people to independently solve a problem. Sometimes it is even enough for someone new to simply not know that something is “hard” to solve.
It even “motivates” llms, it seems (eg https://cdn.openai.com/pdf/04d1d1e4-bc75-476a-97cf-49055cd98...)
Of course this is all pure speculation concerning a hypothetical proof that most likely doesn’t exist, or indeed might be so complicated as to not be approachable even after hundreds of lifetimes of study.
Nevertheless your conclusion does not follow from the premise
You would need an algorithm that finds solutions, not just a proof they exist. So the value here would almost entirely come from how you proved p = np, since that proof will probably be the first step towards finding the polynomial solutions. But if humans don't understand it good luck finding any.
We only compute with two kinds of things:
- small data; or,
- extremely lower power and coefficient algorithms
We lack the power to, eg, use a quintic algorithm in anything but nearly trivial cases.
Is Amazon still delivering food to your cat?
Humans don't need to understand what AI generates. We still can get the rewards.
In reality, it wouldn't depend on the truth value of the statement, but on the AI understanding the proof. If it understands it then it might be able to use it to find reductions.
So the point remains, knowing that P=NP isn't what's important, it's the proof that matters.