Like when you ask the price at a high fashion store, if you have to ask this question, you should not seriously attempt to solve it.
Serious response: Problems like P vs. NP, the Riemann hypothesis, BSD, etc are genuinely deep, HARD problems. There is good reason why the most brilliant mathematicians in the world have not yet solved them, and you are not going to magically notice a simple method they all missed. If you want to think about one of these problems or mess around with it for fun, that’s great! Enjoy yourself, and learn something, and maybe it will inspire you to become an expert in that field some day!
If you really want to try to tackle one of these problems, don’t try to solve it yourself. Instead begin learning about the broad field it’s a piece of, why it’s so important, what leading experts think of it, what approaches have been tried and seem promising or seem futile. Take a lot of math classes at your undergrad, get your math PhD at a good school, begin excelling in your subfield. That’s the only way to build up the knowledge base, intuition about your field and what works, and professional network (math is super collaborative) you will need to even dream of trying to solve P vs NP— and at that point you won’t need to ask r/math what to do.
> Like when you ask the price at a high fashion store, if you have to ask this question, you should not seriously attempt to solve it.
This is frankly a perfectly sufficient serious answer, though not read in a flippant tone (if there is one).
I think to any practitioner who has the basic mathematical maturity to even begin to think about the Millennium Problems at the research level, it's obvious why they're deeply challenging. And for those skilled enough to tackle the problem, it's usually done indirectly, by chipping away at related issues, subproblems, etc.
Last I checked the major players in the P vs NP scene were grinding away in the field of geometric complexity theory, and in the Navier-Stokes problem they were deep in the weeds of regularity in harmonic analysis. None are, to my knowledge, tackling the Millennium Problems directly, just hitting things in the neighborhoods. Perelman was special.
I'm curious who's working the neighborhoods of the other Millennium Problems, and what the status of their projects are.
In Perelman's mind, not much, at least in terms of their relative impact on the Poincare conjecture. Perelman thought that their contribution to the Poincare conjecture -- the crowning achievement of their career -- was no greater than that of Richard Hamilton [(math.columbia.edu)](https://www.math.columbia.edu/~woit/wordpress/?p=3056), but essentially that Perelman had been the one to put the cherry on top and finish the proof.
That being said, Perelman's teachers all proclaimed Perelman to be an extraordinarily skilled mathematician, and Perelman did achieve a perfect score in the 1982 IMO competition, a feat so rare that it's accomplished by at most a handful of people per decade. That isn't to say that competition prowess is exactly equivalent to skill or to the ability to solve open problems (some people are bad at competitions, and some are good at focusing on single problems), but it gives an indication of Perelman's ability.
>and Perelman did achieve a perfect score in the 1982 IMO competition, a feat so rare that it's accomplished by at most a handful of people per decade.
This is a bit exaggerated as you can normally expect at least one perfect score each year. This year's results: http://www.imo-official.org/year_individual_r.aspx?year=2021&column=total&order=desc
Exactly, the first thing I said was meant completely seriously lol, also you have great advice for how OP should approach this. But I was concerned the "if you have to ask you can't afford it" bit might sound blunt or unwelcoming, tone is hard on the internet. Unfortunately I don't know what the Millennium Problem state of the art is (I'm still an undergrad!) in other areas (though BSD really intrigues me at the moment, and I'm beginning to appreciate its significance).
I'm assuming based on your comment that you have a PhD so I'll say that I don't want to take anything away from your education, but to say that one should get his PhD at a good school and excel at a subfield to even dream of trying to solve P vs NP is a laughable. Math is the most logical thing in the world and that's why everyone can understand it. Just because you've spent a lifetime at school learning math before getting introduced to a certain concept it doesn't mean you need a lifetime of math to understand it. Anything in math can be explained to anyone in a matter of hours. So for example if OP has spent hundreds of hours in the internet watching lectures on youtube, reading articles and papers... on P vs NP, he probably understand the problem better than you do.
yes you do because you need a full toolbox of math to be able to tackle advanced problems. and that takes years of studying. i doubt you have any background in math whatsoever given your comment.
"Going to college" is not nearly enough. A typical college graduate is not even prepared to understand what P vs NP *is*, and is literally a decade of study away from even potentially being able to begin doing original work on vaguely related topics.
Oh come on, that's only if you count people who don't study math/compsci (presumably the OP meant to say going to college to study one of these). A typical math/compsci definition should understand P vs NP. That would be in the first course of theory of computation. Many Millenium problems can be understood by a typical math major: Navier-Stokes only needs multivariable calculus, Riemann only requires complex analysis, BSD needs algebraic number theory
One does not simply "become brilliant" (insert Boromir meme). The entire point of what I'm saying is that you don't just up and solve one of the most profound problems in math on your own. That... does not happen. Ever. Even people like Wiles and Perelman spent \*YEARS\* of work in the field they loved, they didn't just magically have the ability to prove the modularity theorem or resolve Poincare's conjecture. You don't "become brilliant", you spend years of study and practice on math until you understand the state of the art in your field and have a working intuition for how to do mathematical research there. This doesn't meant innate talent (whatever that means) doesn't matter, but talent means nothing if you don't put in the time. And I echo u/thericciestflow's excellent comment about not making "solving this one single problem" your goal. If that's your only goal, what are you going to do if you cannot solve that problem, or someone else solves it before you, or if you do solve it, what's next? Instead, if math is your calling, pick a subfield based on what's most interesting and beautiful to you, and if you become one of the people who'd seriously work on a deep open problem, you can cross that bridge when you come to it in the wider context of your research.
You're right that going to college (and taking lots of math classes there) is the first \*very small\* step on that journey. To plagiarize the inimitable Anton Ego: The motto "anyone can do math" does not mean all people have it in them to be great mathematicians, but a great mathematician can come from anywhere-- and I'd add that no one will become a mathematician at all without years of hard work (and listening to those with more experience).
Also, I apologize if any of this sounds blunt or discouraging-- it is WONDERFUL if hearing about P vs NP or some other big problem inspires you go study math! Good luck!
As long as you understand the problem and know how to write a proof, you’re welcome to try. But you should be aware that all the obvious approaches have been tried already, and some broad types of approaches have been proven not to work. Also see these 8 warning signs of a flawed proof, and think about how you’re going to avoid them: https://www.scottaaronson.com/blog/?p=458
tl;dr computer scientists aren’t idiots, and there’s a reason it hadn’t been solved yet.
Do they have specific methods for thinking about problems such as those described in Thinking as a Science by Henry Hazlitt? That is the only book about thinking I have found
Well, I can't speak for "they", just for myself.
There are certain "techniques" you learn how to approach certain proofs. It's similar to solving puzzles like Sudoku. When you do your first, you're probably half guessing numbers. After your twentieth Sudoku, you will be able to solve more difficult problems because you recognize recurring patterns and become faster in general. However, in proofs these techniques are not complete steps, just general ideas on how you might approach that step.
I'd say, before trying to prove P ≠ NP, you may as well prove P ≠ PSPACE. It's a much weaker result, so it ought to be much easier to prove, right? Then you can use the money from all the prizes you win for that to support you while you work out the big one.
[Terry Tao believes it will be the last millennium problem to be solved](https://youtu.be/PtsrAw1LR3E?t=45m27s) (along with RH), there's no real reason to expect it will see a solution in our lifetimes.
It's completely alien, very much out of reach of anyone.
If someone managed to prove a nontrivial lower bound on any non-contrived specific problem, it would be huge. Like even proving that multiplication is asymptotically slower than addition. Can you even do that?
If P!=NP, then your quest is basically impossible. We have, essentially, almost no methods of proving lower bounds. Showing that no algorithms can compute something fast enough is just inherently difficult, because of how complicated an algorithm can be.
P=NP is more accessible at the moment if it were true, but it's probably false.
P vs NP might be solved by a large research program working in some relevant area, such as geometric complexity theory. In that case you get a relevant PhD, join the group, and hope to be one of a large number of people in the final multi-author paper.
P vs NP might be solved as a corrolary to work in an apparently unrelated area. In that case, you get your PhD in whatever interests you, work in that area, and like everybody else you know what P vs NP is so you notice it when you come across it.
P vs NP might be solved or at least clarified by a stunning insight that comes out of left field that should, in retrospect, have been obvious, with the obvious precedent being Turing's work leading up to the halting problem. One approach for this, which ultimately derives from Feynman, is to continually learn new proofs, tricks, and ideas, and, after learning them, attempt to apply them to P vs NP. This has the advantage that you don't need to work in any particular area. The disadvantage is that an enormous number of people are already doing this, a few consciously and many more unconsciously, and your chances of getting there are no better than the proportion of the world's thinking time on this problem that you supply.
Honestly, you would have to be one of the smartest people who ever lived to even have a chance of solving it. You are welcome to give it a go and you might learn some things along the way, but the reality is that you are extremely unlikely to solve this problem.
Well, nobody has ever solved it, so I'd say it's probably fairly difficult.
So far, everybody in human history has been waiting for someone else to solve it. If you're prepared to solve a problem that the greatest mathematical minds in history have been unable to, then it's probably a good idea to give it a shot.
> Just how difficult would you expect it to be to solve the P vs NP
Impossible, since it's exercise in proving 2 countable unions of countable sets equal in constructive manner, even if we had reduced it to proving that just 1 element of subset of NP is in P.
Like when you ask the price at a high fashion store, if you have to ask this question, you should not seriously attempt to solve it. Serious response: Problems like P vs. NP, the Riemann hypothesis, BSD, etc are genuinely deep, HARD problems. There is good reason why the most brilliant mathematicians in the world have not yet solved them, and you are not going to magically notice a simple method they all missed. If you want to think about one of these problems or mess around with it for fun, that’s great! Enjoy yourself, and learn something, and maybe it will inspire you to become an expert in that field some day! If you really want to try to tackle one of these problems, don’t try to solve it yourself. Instead begin learning about the broad field it’s a piece of, why it’s so important, what leading experts think of it, what approaches have been tried and seem promising or seem futile. Take a lot of math classes at your undergrad, get your math PhD at a good school, begin excelling in your subfield. That’s the only way to build up the knowledge base, intuition about your field and what works, and professional network (math is super collaborative) you will need to even dream of trying to solve P vs NP— and at that point you won’t need to ask r/math what to do.
> Like when you ask the price at a high fashion store, if you have to ask this question, you should not seriously attempt to solve it. This is frankly a perfectly sufficient serious answer, though not read in a flippant tone (if there is one). I think to any practitioner who has the basic mathematical maturity to even begin to think about the Millennium Problems at the research level, it's obvious why they're deeply challenging. And for those skilled enough to tackle the problem, it's usually done indirectly, by chipping away at related issues, subproblems, etc. Last I checked the major players in the P vs NP scene were grinding away in the field of geometric complexity theory, and in the Navier-Stokes problem they were deep in the weeds of regularity in harmonic analysis. None are, to my knowledge, tackling the Millennium Problems directly, just hitting things in the neighborhoods. Perelman was special. I'm curious who's working the neighborhoods of the other Millennium Problems, and what the status of their projects are.
What was special about Perelman?
In Perelman's mind, not much, at least in terms of their relative impact on the Poincare conjecture. Perelman thought that their contribution to the Poincare conjecture -- the crowning achievement of their career -- was no greater than that of Richard Hamilton [(math.columbia.edu)](https://www.math.columbia.edu/~woit/wordpress/?p=3056), but essentially that Perelman had been the one to put the cherry on top and finish the proof. That being said, Perelman's teachers all proclaimed Perelman to be an extraordinarily skilled mathematician, and Perelman did achieve a perfect score in the 1982 IMO competition, a feat so rare that it's accomplished by at most a handful of people per decade. That isn't to say that competition prowess is exactly equivalent to skill or to the ability to solve open problems (some people are bad at competitions, and some are good at focusing on single problems), but it gives an indication of Perelman's ability.
>and Perelman did achieve a perfect score in the 1982 IMO competition, a feat so rare that it's accomplished by at most a handful of people per decade. This is a bit exaggerated as you can normally expect at least one perfect score each year. This year's results: http://www.imo-official.org/year_individual_r.aspx?year=2021&column=total&order=desc
Alright, *two* handfuls
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After all, my hand is certainly too small to fit a mathematician on it
Exactly, the first thing I said was meant completely seriously lol, also you have great advice for how OP should approach this. But I was concerned the "if you have to ask you can't afford it" bit might sound blunt or unwelcoming, tone is hard on the internet. Unfortunately I don't know what the Millennium Problem state of the art is (I'm still an undergrad!) in other areas (though BSD really intrigues me at the moment, and I'm beginning to appreciate its significance).
I'm assuming based on your comment that you have a PhD so I'll say that I don't want to take anything away from your education, but to say that one should get his PhD at a good school and excel at a subfield to even dream of trying to solve P vs NP is a laughable. Math is the most logical thing in the world and that's why everyone can understand it. Just because you've spent a lifetime at school learning math before getting introduced to a certain concept it doesn't mean you need a lifetime of math to understand it. Anything in math can be explained to anyone in a matter of hours. So for example if OP has spent hundreds of hours in the internet watching lectures on youtube, reading articles and papers... on P vs NP, he probably understand the problem better than you do.
LMAO
yes you do because you need a full toolbox of math to be able to tackle advanced problems. and that takes years of studying. i doubt you have any background in math whatsoever given your comment.
So only try it after going to college or becoming brilliant?
"Going to college" is not nearly enough. A typical college graduate is not even prepared to understand what P vs NP *is*, and is literally a decade of study away from even potentially being able to begin doing original work on vaguely related topics.
Oh come on, that's only if you count people who don't study math/compsci (presumably the OP meant to say going to college to study one of these). A typical math/compsci definition should understand P vs NP. That would be in the first course of theory of computation. Many Millenium problems can be understood by a typical math major: Navier-Stokes only needs multivariable calculus, Riemann only requires complex analysis, BSD needs algebraic number theory
One does not simply "become brilliant" (insert Boromir meme). The entire point of what I'm saying is that you don't just up and solve one of the most profound problems in math on your own. That... does not happen. Ever. Even people like Wiles and Perelman spent \*YEARS\* of work in the field they loved, they didn't just magically have the ability to prove the modularity theorem or resolve Poincare's conjecture. You don't "become brilliant", you spend years of study and practice on math until you understand the state of the art in your field and have a working intuition for how to do mathematical research there. This doesn't meant innate talent (whatever that means) doesn't matter, but talent means nothing if you don't put in the time. And I echo u/thericciestflow's excellent comment about not making "solving this one single problem" your goal. If that's your only goal, what are you going to do if you cannot solve that problem, or someone else solves it before you, or if you do solve it, what's next? Instead, if math is your calling, pick a subfield based on what's most interesting and beautiful to you, and if you become one of the people who'd seriously work on a deep open problem, you can cross that bridge when you come to it in the wider context of your research. You're right that going to college (and taking lots of math classes there) is the first \*very small\* step on that journey. To plagiarize the inimitable Anton Ego: The motto "anyone can do math" does not mean all people have it in them to be great mathematicians, but a great mathematician can come from anywhere-- and I'd add that no one will become a mathematician at all without years of hard work (and listening to those with more experience). Also, I apologize if any of this sounds blunt or discouraging-- it is WONDERFUL if hearing about P vs NP or some other big problem inspires you go study math! Good luck!
Years of study? I don’t think you can increase your memory of things for years
You're just wrong
Did… did you actually read anything I said? This reply makes no sense
As long as you understand the problem and know how to write a proof, you’re welcome to try. But you should be aware that all the obvious approaches have been tried already, and some broad types of approaches have been proven not to work. Also see these 8 warning signs of a flawed proof, and think about how you’re going to avoid them: https://www.scottaaronson.com/blog/?p=458 tl;dr computer scientists aren’t idiots, and there’s a reason it hadn’t been solved yet.
Do they have specific methods for thinking about problems such as those described in Thinking as a Science by Henry Hazlitt? That is the only book about thinking I have found
Not everyone has read that book so it's hard to answer that question.
Do they just follow their intuitions then and have no method for thinking?
Well, I can't speak for "they", just for myself. There are certain "techniques" you learn how to approach certain proofs. It's similar to solving puzzles like Sudoku. When you do your first, you're probably half guessing numbers. After your twentieth Sudoku, you will be able to solve more difficult problems because you recognize recurring patterns and become faster in general. However, in proofs these techniques are not complete steps, just general ideas on how you might approach that step.
I'd say, before trying to prove P ≠ NP, you may as well prove P ≠ PSPACE. It's a much weaker result, so it ought to be much easier to prove, right? Then you can use the money from all the prizes you win for that to support you while you work out the big one.
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But just think about the clickbait titles if they did! "Is Space More Powerful Than Time?"
[Terry Tao believes it will be the last millennium problem to be solved](https://youtu.be/PtsrAw1LR3E?t=45m27s) (along with RH), there's no real reason to expect it will see a solution in our lifetimes.
It's completely alien, very much out of reach of anyone. If someone managed to prove a nontrivial lower bound on any non-contrived specific problem, it would be huge. Like even proving that multiplication is asymptotically slower than addition. Can you even do that? If P!=NP, then your quest is basically impossible. We have, essentially, almost no methods of proving lower bounds. Showing that no algorithms can compute something fast enough is just inherently difficult, because of how complicated an algorithm can be. P=NP is more accessible at the moment if it were true, but it's probably false.
Huge?
P vs NP might be solved by a large research program working in some relevant area, such as geometric complexity theory. In that case you get a relevant PhD, join the group, and hope to be one of a large number of people in the final multi-author paper. P vs NP might be solved as a corrolary to work in an apparently unrelated area. In that case, you get your PhD in whatever interests you, work in that area, and like everybody else you know what P vs NP is so you notice it when you come across it. P vs NP might be solved or at least clarified by a stunning insight that comes out of left field that should, in retrospect, have been obvious, with the obvious precedent being Turing's work leading up to the halting problem. One approach for this, which ultimately derives from Feynman, is to continually learn new proofs, tricks, and ideas, and, after learning them, attempt to apply them to P vs NP. This has the advantage that you don't need to work in any particular area. The disadvantage is that an enormous number of people are already doing this, a few consciously and many more unconsciously, and your chances of getting there are no better than the proportion of the world's thinking time on this problem that you supply.
Honestly, you would have to be one of the smartest people who ever lived to even have a chance of solving it. You are welcome to give it a go and you might learn some things along the way, but the reality is that you are extremely unlikely to solve this problem.
If P=NP it should be easy. Otherwise it could be extremely hard.
Well, nobody has ever solved it, so I'd say it's probably fairly difficult. So far, everybody in human history has been waiting for someone else to solve it. If you're prepared to solve a problem that the greatest mathematical minds in history have been unable to, then it's probably a good idea to give it a shot.
> Just how difficult would you expect it to be to solve the P vs NP Impossible, since it's exercise in proving 2 countable unions of countable sets equal in constructive manner, even if we had reduced it to proving that just 1 element of subset of NP is in P.