EDIT: FFS why does this subject always get people frothing at the mouth before they even read the main point stated, only to go on and accidentally agree with it eventually? Pls read first before getting mad at stuff that I explicitly argued against.
EDIT 2: OK apparently there's still miscommunication, and I think the 1st edit somehow made it worse. When I say "useful" I put it in scare quotes on purpose and as I clarify in the 1st, 4th and 5th paragraps, it is NOT about value but about practical/technological utility.
I originally posted this on R*ddit to an audience of math nerds (so be warned that it is written with reddit STEMlords in mind) because there was a relevant convo going on and it would be fun to also have it here.
Sure, there is a lot of modern math that is practically useful, but the majority of pure math really isn't "useful' in any way, shape or form for now, and probably won't be any time soon, possibly forever. Like, even areas which are apparently "useful", like computer science, is full of things that have absolutely 0 practical utility and are solely of academic interest. Whether P does or doesn't equal NP doesn't really matter to anyone doing practical work. People wouldn't get upset about their discipline getting slighted or whatever if this stupid idea that scientific research should have "practical application" (which generally means "someone can sell it for money") hadn't proliferated, starting from schools.
Even when someone finds an "application" through some kind of far fetched (or not so far fetched) reasoning, it's some application to, like, highly theoretical physics that may or may not actually have something to do with the real world, and even if it does, it is only relevant in extremely niche experimental circumstances to the extent that it can't ever conceivably lead to technological progress. And even IF it does, sometimes it's just progress relevant only to more research about more stuff without application.
So even then you have to resort to saying something like "the result is not useful but maybe one of the methods used to prove it can be used for something else", and then that something else turns out to also not be useful but again "maybe one of the methods used to find that something else is useful for another something else and that other something else is useful for another other something else and then that other other something else has a practical application that is only relevant to research, but then maybe that relates to some other other other...", etc and it gets kind of silly. That or someone says something abstract like "it's useless now but it may be useful some time!". Maybe. Or maybe not.
In the end of the day the same arguments could be used to justify anything being useful via some contrived butterfly effect style conjecture. This of course is usually done because otherwise people can't get grant money otherwise, governments demand that research will produce results they can use to blow up people or sell stuff. Also the result of a bad educational system that emphasizes this kind of "usefulness", which therefore renders it unable to convince students that something is worth learning unless it is "useful". Of course "why should I learn this if it's not useful to me" is a very valid concern of students, but the problem is somewhere else. First, schools DON'T really teach any of the stuff that is useful and interesting to most people. If they did, then math would get a lot less attacks on that front. Schools teach with 30% of the students in mind, the ones who will really apply the things they learned. The other 70% can just go to prison or whatever as far as the educational system is concerned. Second, schools are very boring and antagonistic towards kids and since kids are miserable learning stuff, they need extra justification to learn them. Third, the schools themselves teach kids to think like that so it's no surprise that they do. Fourth, school math mostly sucks and is super boring for most people.
So yes, most modern pure math is indeed "useless". That is not the issue. The issue is, why does this matter? Why is it bad? Should it be bad? I don't think so. It's a false idea that gets perpetuated at many levels starting from school. But then there is the issue of mathematics being very exclusionary and distant from most people, which makes it harder for them to care, which brings us to the issue of outreach but whatever, that's a different matter.
P=NP is monumentally important to practical work. If it's true, all problems are easy to solve.
Leaving aside from that terrible abysmal awful example to address your general point: there's a difference between basic research and engineering. We need to find out basic facts about what sort of world we're in in order to do engineering later. We obviously can't know the "practical application" of things that we don't know yet; we need to find them out first. Did Rutherford think about the "practical application" of his model of the atom? Or did the street eventually find its own use for it?
that doesn’t necessarily follow right? like what if all NP problems are solvable in n^googolplex steps or somethinh
Even n^googolplex is still sub-exponential time, and in practice ridiculously shitty poly time algorithms can often be reduced in magnitude, whereas if you have an exponential time algorithm you need to find something completely different.
it sounds like you’re saying they can’t necessarily be reduced though
Yes, not necessarily, but they've pretty much always been up until now. There's like, three or four practical algorithms in n^10.
All of them are approximations of NP-class problems, suboptimal, or literally invented to be intractable.
That's usually because the longer ones aren't practical. Or because they can't find them. But I did a google search and there's some algorithms which are theoretically useful for... something, and they're in like n(10100).
The algorithms I found for n10100 are just approximations of non-polytime algorithms.
And by that I mean that they don't solve a practical problem, not that they aren't practical to use.
There was one which was about a neat little word problem which had to do with hanging a picture or something and it was like n^500000 or something. The rest I wasn't sure what they were supposed to be.
The one for hanging a picture was published to a journal about silly computer science solutions, it's a problem that was invented to have a ridiculously stupid polytime solution
The rest were either estimations, or they had to do with combinatorics.
Well many of those kinds of problems are "silly".
Well yeah but the problem was invented just to make the solution harder, NP-hard problems are encountered often.
Then is it unreasonable to bet that if NP problems are actually P problems, their best algorithms might be n^(something huge)?
It's not, because fundamentally they are trying to approximate non-polynomial algorithms. We saw this play out for other algorithms before a polytime solution was found.
And even if they could, then one could approximate those high exponent algorithms and have a huge speedup.
It is not, at least not necessarily. Whether or not such a solution conceivably exists doesn't really matter at all to a programmer. Let's say it turns out that P=NP. Cool, then said fast solution exists. Does that mean the programmer can find?Not necessarily. But they can try. Let's say P does not equal NP. Cool. Then a faster solution may or may not exist. But the programmer doesn't generally know if the solution they found is the fastest one possible, so they probably will still try to find a better one. Nothing really changes for the programmer whether or not P equals NP. The programmer will keep looking for a faster solution to the extent they are willing or able to, unless they know the solution they found is the fastest possible, which is not something P versus NP can tell you alone. Oh, also forgot to mention that even if P=NP, a large number of problems won't have solutions which can feasibly be solved in polynomial time anyways due to other restrictions.
It is an example of something that SOUNDS like it has important applications but doesn't really in itself. This is similar to Navier-Stokes. The act of trying to solve NS will probably give immensely valuable insight into turbulence etc, however in terms of practical applications a strict solution of NS is not particularly important, because the systems involved are massively chaotic and real fluids don't truly obey NS anyways. In Mathematics existence and smoothness of solutions in many kinds of differential equations is a big problem, but anyone who does anything practical just ends up approximating them anyways.
The other argument you make is the same kind of hand wavy thing that people say and never convinces anyone. Some things you can tell are gonna have practical consequences, including Rutherford's model. Others maybe not but they do turn out to have some. But then there's all the other stuff.
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Not extensively comp sci in particular except for a few classes (idk if that counts), but applied math in general.
This opinion is not unique to me. Like, many people who actually solely do research on the field will say as much. If P!=NP, well, that's what everyone kind of expects already and nothing much changes. If P=NP but no one finds a polynomial time solution of an NP hard problem, that's big news, but it still doesn't change anything practically. If someone DOES find that, well, it might be useful, or it might not be, depending on a few other things, but just proving P=NP won't give you that.
EDIT: I just saw your edit, hold on a while.
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That is not necessarily true, as for many things in math. You don't have to find the thing you want to prove exists to prove that it exists. You can just prove that it can't not exist for example. You may even manage to find some kind of independence proof which states that it can not be decided based on your axioms.
Also "hard" and "easy" mean different things for comp sci and programmers.
There exist problems in P for which no polynomial algorithm is known like "does a given graph belong to a fixed minor-closed family (well, it also depends on method of representation of this family)", although for some such families like planar graphs efficient algorithms are known.
Btw before I said "a fast solution may or may not exist". I meant "a faster solution may or may not exist", in the sense that a solution faster than what the programmer did already may or may not exist. Sorry about that.