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But when we are talking about unsolved problems, we are in the land of vanishing small probabilities in any case. Could it not be that the very fact that someon

But when we are talking about unsolved problems, we are in the land of vanishing small probabilities in any case. Could it not be that the very fact that someone has been trained in conventional mathematics a hindrance to them resolving an unsolved question, and someone thinking in isolation has an advantage? Here's Grothendieck, for example: “In fact, most of these comrades who I gauged to be more brilliant than I have gone on to become distinguished mathematicians. Still, from the perspective of thirty or thirty-five years, I can state that their imprint upon the mathematics of our time has not been very profound. They've all done things, often beautiful things, in a context that was already set out before them, which they had no inclination to disturb. Without being aware of it, they've remained prisoners of those invisible and despotic circles which delimit the universe of a certain milieu in a given era. To have broken these bounds they would have had to rediscover in themselves that capability which was their birthright, as it was mine: the capacity to be alone.” I can certainly see this happening: someone away from the pressures of publishing, the drudgery of admin tasks, the frustration of applying for grants or department politics, or the fear of looking "bad" to their peers etc. devoting their time to mathematics out of pure joy, play and drive and coming up with novel definitions and assumptions that those in establishment mathematics, out of pure sociological and psychological reasons, have not even dared venture into.

throwawaymath 7y ago
I can see a case for a trained mathematician contributing a legitimately new perspective to the community by withdrawing and spending time looking at things in an unorthodox manner. That's closer to that Grothendieck was describing than what you're posing. In the early to mid 20th century, it was feasible for someone to pick up a book on number theory and apply relative genius to an open problem to quickly solve it. The bottom line is that prerequisites for doing so were often just basic calculus and high school algebra. An understanding of sequences and series went a long way. This isn't the case in the 21st century. We're firmly out of that territory. You can't offer a profound new perspective on a thing which you can't understand, and the barrier to understanding research mathematics continually rises. The forefront of modern mathematics is so far removed from even graduate level mathematics course material that it's not going to just be intuited through untrained brilliance. You have to actively learn it, which is (unfortunately) vanishingly unlikely outside of academia. If the tools available to you are basic real analysis and linear algebra, P/NP is beyond your reach - full stop. At this point we actually have proofs that a valid proof of P/NP (and similar problems) cannot be achieved through large swathes of elementary techniques. We don't live in a world like Good Will Hunting where an amateur can succeed by being a genius. That's useful in the long term but not enough on its own.
- piyushahuja 7y ago
  What prerequisites, according to you, does a graduate from a good CS program need to understand P/NP enough to begin taking a stab at?
  - throwawaymath 7y ago
    For starters: 1. All math and CS prerequisites to an intro computational complexity course. 2. An intro computational complexity course. 3. All math and CS prerequisites to an advanced, graduate-level computational complexity course. Let’s say by this point you have worked through two linear algebra courses, three calculus courses, one or two discrete mathematics courses, some graph theory, some combinatorics, some logic, and a couple of algorithms courses. 4. The advanced, graduate-level computational complexity course. Hopefully you work on probabilistic computation and, in particular, quantum computation. 5. Now read and understand the relevant papers advancing the field (not just this one problem) for the last two decades. That one paper from 1990 about reducing the permanent to the determinant? You should know about that. 6. Finally, to reach the temple you must walk the path littered with the bodies of would-be explorers before you. Read the papers of failed proofs, starting with the easily refutable ones. The most sophisticated failed proofs have errors so subtle that they are useful for the research community in their own right as an exercise in peer review. As a rule, a grad student near their PhD in this area should know of everything Aaronson mentioned here: https://www.scottaaronson.com/talks/pvsnp.ppt https://www.scottaaronson.com/talks/pvsnp.ppt. None of that should be unfamiliar or unknown. A postdoc and beyond should actively have ideas percolating on how to chip away at some small aspect of a lesser problem featured therein. Here’s the reality: at almost every stage of a researcher’s career, “taking a stab at” a famous outstanding problem is the wrong way forward. Usually a problem is still outstanding because it actually needs a new theory - this is the practical utility in solving theoretical problems in the first place. Therefore the best bet for solving this problem is actually to chip away at it over a long period of time. Think hacking through a rainforest to reach a goldmine, not managing to somehow parachute to the goldmine directly when it’s hidden beneath the trees.
    - piyushahuja 7y ago

throwawaymath 7y ago

I can see a case for a trained mathematician contributing a legitimately new perspective to the community by withdrawing and spending time looking at things in an unorthodox manner. That's closer to that Grothendieck was describing than what you're posing. In the early to mid 20th century, it was feasible for someone to pick up a book on number theory and apply relative genius to an open problem to quickly solve it. The bottom line is that prerequisites for doing so were often just basic calculus and high school algebra. An understanding of sequences and series went a long way. This isn't the case in the 21st century. We're firmly out of that territory. You can't offer a profound new perspective on a thing which you can't understand, and the barrier to understanding research mathematics continually rises. The forefront of modern mathematics is so far removed from even graduate level mathematics course material that it's not going to just be intuited through untrained brilliance. You have to actively learn it, which is (unfortunately) vanishingly unlikely outside of academia. If the tools available to you are basic real analysis and linear algebra, P/NP is beyond your reach - full stop. At this point we actually have proofs that a valid proof of P/NP (and similar problems) cannot be achieved through large swathes of elementary techniques. We don't live in a world like Good Will Hunting where an amateur can succeed by being a genius. That's useful in the long term but not enough on its own.

piyushahuja 7y ago
What prerequisites, according to you, does a graduate from a good CS program need to understand P/NP enough to begin taking a stab at?
- throwawaymath 7y ago
  For starters: 1. All math and CS prerequisites to an intro computational complexity course. 2. An intro computational complexity course. 3. All math and CS prerequisites to an advanced, graduate-level computational complexity course. Let’s say by this point you have worked through two linear algebra courses, three calculus courses, one or two discrete mathematics courses, some graph theory, some combinatorics, some logic, and a couple of algorithms courses. 4. The advanced, graduate-level computational complexity course. Hopefully you work on probabilistic computation and, in particular, quantum computation. 5. Now read and understand the relevant papers advancing the field (not just this one problem) for the last two decades. That one paper from 1990 about reducing the permanent to the determinant? You should know about that. 6. Finally, to reach the temple you must walk the path littered with the bodies of would-be explorers before you. Read the papers of failed proofs, starting with the easily refutable ones. The most sophisticated failed proofs have errors so subtle that they are useful for the research community in their own right as an exercise in peer review. As a rule, a grad student near their PhD in this area should know of everything Aaronson mentioned here: https://www.scottaaronson.com/talks/pvsnp.ppt https://www.scottaaronson.com/talks/pvsnp.ppt. None of that should be unfamiliar or unknown. A postdoc and beyond should actively have ideas percolating on how to chip away at some small aspect of a lesser problem featured therein. Here’s the reality: at almost every stage of a researcher’s career, “taking a stab at” a famous outstanding problem is the wrong way forward. Usually a problem is still outstanding because it actually needs a new theory - this is the practical utility in solving theoretical problems in the first place. Therefore the best bet for solving this problem is actually to chip away at it over a long period of time. Think hacking through a rainforest to reach a goldmine, not managing to somehow parachute to the goldmine directly when it’s hidden beneath the trees.
  - piyushahuja 7y ago