I think what's more fundamental than just answering P vs NP, is what exactly makes boolean functions hard to (pseudo)inverse? And it's the latter where answering (unless the answer comes from an oracle) the former will help.
And this question is crucial, because everything we can ask computers (or restated, math with real-world applications) can be rephrased in terms of finding a solution to a SAT instance. It can be a very large instance but if there is a practical way of approaching the problem, it shouldn't matter.
I think you're assuming that future computer scientists and mathematicians will continue to run (many different) algorithms, and I don't. I think there might be a single algorithm, a SAT solver, which will subsume all others.
Do you not realize that SAT might just be O(2^n) at best? Even if by a gigantic miracle SAT is in P, unless there is another incredibly great miracle and it is something like O(n^3), which would almost certainly be the greatest discovery of CS, it would not replace almost any other algorithms? Simply because many of the most important ones are O(n^2) or even less.
>I think there might be a single algorithm, a SAT solver, which will subsume all others.
This is legitimately insane. You might as well hope for free energy, FTL Travel, or magic.
> Do you not realize that SAT might just be O(2^n) at best?
Yes, but that's the worst case on all possible instances. Even with this worst case performance, there might be an algorithm that works on par with other algorithms that specialize only on certain instances. E.g. there can be an O(2^n) algorithm for general SAT which performs in O(n^(3/2)) on XORSAT. We simply don't understand the problem well enough to tell.
Also the set of worst-case instances can be vanishingly small in practice.
You have a completely utopian idea about what might be computationally feasible. I don't even know what you are arguing. Certainly it is thinkable that SAT is somehow ridiculously simple to calculate, just in the same way that achieving simple and near free fusion energy is thinkable. It just isn't going to happen and speculating on it is pure science fiction.
> Certainly it is thinkable that SAT is somehow ridiculously simple to calculate, just in the same way that achieving simple and near free fusion energy is thinkable. It just isn't going to happen and speculating on it is pure science fiction.
These are, as I'm sure you know, utterly different. Nuclear fusion is real, 'simple' and 'near free' are engineering problems. If human civilization lasts long enough, and continues to progress technologically, we'll have that, for some value of 'simple' and 'near free'.
On the other hand, the conjecture that SAT is algorithmically feasible may be disproven, and in my opinion, will be. That would be "just isn't going to happen".
And this question is crucial, because everything we can ask computers (or restated, math with real-world applications) can be rephrased in terms of finding a solution to a SAT instance. It can be a very large instance but if there is a practical way of approaching the problem, it shouldn't matter.
I think you're assuming that future computer scientists and mathematicians will continue to run (many different) algorithms, and I don't. I think there might be a single algorithm, a SAT solver, which will subsume all others.