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.
It's a little silly to get angry/surprised when you say inflammatory or just straight up wrong things at the beginning of your post and don't clarify until near the end. Like, no kidding people don't want to read the whole thing when the first paragraph makes it look like you have zero clue what you're talking about
We've gone over this. The first paragraph is not wrong. Assuming this is what you are referring to, the process of solving P vs NP may bring something important for practical work, or it may not. But the actual answer to the question, which is considered extremely important theoretically and for good reason, won't change much. IF it turns out that P=NP and IF the proof actually involves finding an algorithm to solve NP hard problems in polynomial time and IF that algorithm is of a practicable form and not something insane like telling you that you can solve NP hard problems but in n^666 time, then alright, yeah. But that's a lot of ifs, and it's not directly linked to whether or not P=NP is solved. I brought this up specifically because it is one of the famous problems that the solution sounds like it may be massively important practically, but in reality kind of isn't.
Also I didn't "only" clarify near the end, the point is more or less contained in the first paragraph. It's just that for some reason this subject instantly pushes the buttons of people.
While the P=NP thing is a bad take for reasons people with more understanding than me have already explained, the worst part IMO is
This is a ridiculously presumptuous and weird thing to say, particularly given how fast computing is expanding and making it more relevant than ever.
If I'm understanding your argument correctly, the better way to approach this is to say that all fields of study are useful. Non-STEM things like art and music are important, as are mathematics and physics that are too experimental to find an immediate use for. Like, modern art does not improve people's material conditions in any significant way, does that mean it should be tossed out? Should history? Should philosophy? For all his writing, Marx hasn't improved my life one bit so far, toss him out too.
I know this kind of slippery slope argument sounds silly, but that's the point. It's a really bizarre way to come around to your final point, which seems to be that it should be studied despite being useless.
The P=NP thing isn't my personal take. Many people working in the field more or less agree. P=NP is about whether all NP hard problems can be solved in polynomial time. The answer doesn't necessarily tell you how to do it if it is even possible (and most don't believe it is). If it's not true, then not much changes practically. If it is, then there's many many reasons why it may still not really make a difference practically speaking. It could make a difference in some kind of optimal case where it turns out that the proof actually gives an algorithm, that algorithm is something practicable, and it isn't some kind of horribly unwieldy thing that renders any kind of practical solution in exponential time more practical for a typical problem (which is also possible, "fast" means something different to comp sci people and to programmers, for comp sci people fast is asymptotic, but in the real world you don't always care about the asymptotic behavior of something, in the real world many problems in polynomial time are practically not solvable), and a best case scenario solution is getting more unlikely by the day. Mathematical comp sci people don't care if they get that P!=NP or if they get a non constructive proof of P=NP or if they give some kind of independence proof or if they actually do find a constructive proof of P=NP but it is horribly impractical. Just getting any kind of proof is amazing news because it's such a deep result and such an important question theoretically, even though it probably won't change much in practical terms.
Well computing is expanding a lot but most relevant math is still relatively simple, bar a few niche contexts.
See, I kept putting useful in scare quotes for a reason. I also said this in the first paragraph:
Usually when presented with something that seems far fetched or hard to apply, people say "why is that useful? What can someone do with this? What can you make? How can you sell it? Why do we need this? Why are we spending time on this if it isn't useful?". This is a consequence of capitalism. Practically useful and valuable are different things. I think all the miscommunication arises from people uncharitably or hastily interpreting what I mean by useless.
So what, you put useless in "scare quotes" and that makes it sarcastic or something? Why not just say what you mean very directly? I think most people here are much more inclined to listen and try to understand/reply to you than the average person on reddit, you can be very straightforward and people will take it in good faith. The weird rhetorical devices just make you harder to understand, I'm kinda questioning if I ever actually knew what your point was now.
I mean I did, in the first paragraph, then the third paragraph, and then the last paragraph (I specifically thought wow, maybe I should make sure to put that in the first paragraph, and then I did and people still got angry), it's just that it seems like when you say something like that some people are geared to immediately disagree because they think you're attacking science. idk.
Simply P=NP being proven even with no implementation means we have to abandon assymetrical encryption, because there's no way to know if your opponent found it.
Math in computing gets pretty hairy pretty fast, it's just abstracted away from you. Do you know what kind of math is necessary for example to prove that a pseudo-random number generator is of cryptographic quality? Or the kind of math needed to optimise path-tracing algorithms. It gets into what one would call abstract math really fast.
Only in the case that an actual algorithm is found and it is practicable enough.
Yes, but you can never know if the algorithm has been found or not. All you know is that it can be found.
And if it is found, it's probably going be n^3 to n^4 at worst, see the recent prime factoring attempt.
You already don't know that but that doesn't stop anyone.
I'm not sure how someone could know that at this point, perhaps you know something that I don't about this though. I'm not a computer scientist.
We already don't know that but we are 99.999% certain P!=NP, so there is not much point.
As a CompSci student, I know of no absurdly impractical algorithm in polytime that isn't an approximation of an NP-class problem. Worst I've ever heard of is n^19 or so but that could be approximated in n^3 or so.
I think most think the odds are a bit more balanced than that. Knuth believes P=NP is more likely iirc, although he's kinda weird sometimes (in a good way). But, like, if people were practically certain, well, that's all engineers usually care about. They only care about being 99.9999% certain their plane won't randomly blow up. Just how the world is I guess.
Well, we're not talking about approximations here, P=NP is not about approximations. That's kind of the issue with these kinds of problems, you can go a long way with approximations in the real world, but these kinds of math problems are not concerned about them typically. The Goldbach conjecture is an open problem, but it's already been tested for as many numbers as you would conceivably ever "need" it for. No one knows the solution to Navier-Stokes and it is a big open problem, but Navier-Stokes is already just an approximation to real fluids. Very interesting problem, and something good will probably come out of trying to solve it, but the answer is not exactly game changing for practical stuff.
Yeah, Knuth is certainly weird about it.
As for approximations, that's not what I meant. I meant the full on algorithm was n^19 - which is not necessarily horrible in all cases, whereas the approximation of that was n^3.
For many NP class problems P=NP also means that better approximations are basically guaranteed.
Well, where I was getting at was that afaik you could approximate it even if it runs on exponential time, but the conjecture is not about approximations.
Approximations of exponential time algorithms are much worse and sometimes even impossible to useful degrees of proximity compared to approximations of polynomial algorithms.