8 ms·

How to be successful as a research mathematician? Follow your gut

99 points by webmaven 3y ago

amelius 3y ago
Or brute force the hell out of everything.
lambdaphagy 3y ago
I appreciated Eugenia Cheng's youtube lectures on category theory and I know interviews aren't the best format for sustained exposition, but the remark about mathematical rigor being a European colonialist thing seemed pretty forced. If anything, it underestimates the strangeness of the history of math in an effort to cram it into a simple frame: "Europeans bad". There was something distinctive about work in foundations of math in the late 19th / early 20th centuries, work which came out of European cultures (or more accurately Latin Christendom). Plausibly, this work was centuries in the making, the culmination of a trend towards increasingly analytic and systematizing modes of thought that begins with the recovery of Greek philosophy in the middle ages. But, ok, even if you don't buy that, how do you get from the Congo Free State to Frege, Hilbert and Goedel? It's a radical claim that would be utterly fascinating if it were true. But Chang just kind of tosses it off at the end and asks us to accept it as an unremarkable truism.
- nxobject 3y ago
  I've talked about this in a replay to another comment, but I don't think she's making any value judgment in the article, or (at least in the article) actually articulating any claim about colonialism and mathematics. In fact, I don't even think she's saying "Europeans bad": We might marvel at, say, Amish people raising a barn or Inuit people building a kayak without any European formal maths training. We might declare that this is maths — and I think it is. But it’s problematic if we think we’re doing them some great honour by bestowing the label of maths on what they do.
  - lambdaphagy 3y ago
    I think that if I were being interviewed and my interviewer said "you also discuss how colonialist ideology has influenced modern mathematics?", but I didn't think that colonialist ideology has influenced modern mathematics, I would say: "What? I don't think colonialist ideology has influenced modern mathematics at all." Rather than insinuating that Europe must have stolen math from other parts of the world and pawned it off as its own, or been really mean about rigor, or whatever. There were several totalizing ideologies in the 20th century that attempted to make all of human affairs about themselves: everything from literature to science to birdwatching had some sort of party line: Did this literature say the right things? Was this science done by the right kind of scientist? Is this bird the right kind of bird to be watching? Judging by their results, I wouldn't look forward to a do-over of that line of thinking.
    - nxobject 3y ago
      I think that's our disagreement: I think you can say, "maybe there's some relationship between what we have seen as colonialism and academic mathematics", without even thinking that it was about "stealing math", or thinking about colonialism determined everything about mathematics in some totalizing way. I'm sorry if I come off as a little obtuse, so I might as well say what I'm really thinking. My thinking leans towards Eugenia's, and it's a lot more conservative: the question itself is worth thinking about, but it's a lot to think through, and the link may as well be tenuous. But, in my thinking, if NATO once funded pure mathematics (as Grothendieck once protested against), if DARPA also currently funds pure mathematics, then it is worth asking what historical narrative explains it, and whether it applies to the slightly more distant past, too. Above all I don't believe that the academic enterprise has ever been a neutral and disinterested endeavor, either – it's a lot of public money, and wherever there's public money politics comes before or after. I don't think that's a totalizing way of understanding academia.
      - lambdaphagy 3y ago
        Cheng talks a lot about colonialism inn _Is Maths Real_ but here is the closest thing to an argument for the claim that I could locate: > “Now, at the same time as all this, it is true that mathematics as a field has grown carefully and rigorously according to a carefully constructed framework of logic and rigour. It is, arguably, the framework that is keeping some maths in and keeping some maths out. But... there are questions about the values built into the framework of mathematics. That framework is geared towards the principles of ‘progress’ and ‘development’. Deep down I have an uncomfortable suspicion that those principles are indelibly linked to colonialism, imperialism and the urge to conquer others.” (p. 320-1) That's the argument in its entirety. What humanity was up to in the ten millennia of recorded history before formal mathematics is left unexplained. The most parsimonious explanation for the military's funding of mathematics research is that mathematics often has applications to military technology. But even here, we have to be careful: it's typically applied math that's the most, well... applicable, which is precisely the sort of thing that pure mathematicians are supposed to look down upon as "not really math". (The pure math work at DARPA is AFAICT more a consequence of DARPA's idiosyncratic governance structure that gives program managers wide latitude to swing for the fences, accepting that there will be some strikeouts.) But the fact that modern Western mathematics is indispensable for advanced engineering projects at all, in a way that Sanskrit prosody and Inuit kayak architecture are not, is pretty crucial to understanding the funding situation: it's funded because it's useful, not useful because it's funded. If we don't grant that math is actually unreasonably effective in explaining / predicting / manipulating the world then sure, what math gets funded looks like another courtly intrigue. But it seems that the unreasonable effectiveness is precisely what needs to be explained away on Cheng's view. I'm all ears for the contrary view, but we've already written more on the topic than Cheng did in the entire book, which is light on argument and regrettably heavy on insinuation.
marmakoide 3y ago
The whole tirade on rigorous proofs and colonial mindset puzzles me. From my understanding (I am European ...), rigorous proofs were increasingly seen as a need because as things like calculus progressed, any loose definition had a tendency to become a minefield. Think of the definition of derivatives, continuity. Also, at a time where things were getting mechanize because we could and it was a gain, it was natural to check if we could somehow mechanize math. After all, Pascal made a machine that could do sums, why not a machine for proofs ?
- thoi23u4234324 3y ago
  I don't think 'colonial' is the right word; I think it's more to do with theology and Europe's very twisted (and ongoing) history with it. A very clear example is S Ramanujan's way of doing mathematics, where proofs were secondary to 'calculate, calculate, calculate, then hypothesize, then maybe prove' approach. This was not a one off thing, nor was his strange approach unique. Indian traditional mathematics also follows SR's approach of compute & then generalize & then prove (which most working mathematicians actually follow to some degree anyway). India was ahead of much of the world in Math until the cusp of renaissance(finally culminating in the dev. of calculus a good hundred years before Newton, and possibly having traveled to and 'inspired' him with the Jesuit missions), and its outcome's are simply extraordinary. Of course, the tradition was not given any credit whatsoever due to the prevalent racial-thinking in old-Europe; this now continues because Indologists continue to hold onto their inherited toxic racial thinking and prevent anyone else from touching this subject because 'Hindus are bad/dirty/nazis/<add adjective> people' (you see this with Yoga, NSDR, Mindfullness,... to this day). Regardless of all the 'credit allocation' business, and somewhat more importantly, it's actually really sad that this tradition is no longer alive, and is instead replaced with a cheap mimicry of Europe. C K Raju has written extensively on this. It's also not like this way of doing math. is inherently worse. A lot of actual formal math, starting from the construction of real-numbers is actually very 'unreal' when you get down to it. The kinds of hiccups thrown up by real-numbers in the theory of computation needs to be seen to be believed. Chaitin even has a paper by the title 'How real are real numbers'... We actually get around many of these weird issues because the construction of formal mathematics is actually based on ensuring that discretization is stable (this doesn't work in TOC). Still one also can't deny the strides that modern mathematics has made, in large part built on formalization. All said, to destroy any existing tradition (even Western) is extremely detestable. OTOH given that it is a jealous-monoculture (all of academia has become one big church-like monoculture at this point), the creation of a new tradition needs active support. I wish people would spend time on the latter instead of pointless race-wars (or DEI non-sense). (edit: expand)
vouaobrasil 3y ago
I have a PhD in math, did some successful research, but left math for a variety of personal reasons. My advice: 1. Network, network, network. Get famous or established mathematicians interested in your work. 2. Be a minority. Due to the prevailing ideology of academia, it's much easier to be successful if you are a minority. (I am not saying it doesn't take hard work, but the competition is not as extreme due to intense equity measures.) I will laugh if anyone gets mad at this point. Equity rules make this point an incontrovertible fact. 3. Solve problems related to the work of established mathematicians. Interesting or novel problems that are not related to the work of others will be ignored. 4. Forget about being curious. Okay, this one has some dark humour about it...but seriously, if you just explore your curiosity, you might solve cool problems but you won't be "career successful". It's much better just to make useless investigations into ever-more esoteric problems that validate the "successful" work of others which is becoming increasingly technical. A note: this might sound a bit cynical, but I thought I'd give another viewpoint to show a bit more of the balanced reality of the math world.
- thoi23u4234324 3y ago
  > Be a minority. Won't help if you're Asian IMO.
  - nxobject 3y ago
    Well, it just means you're not a minority in the world of mathematics.
  - vouaobrasil 3y ago
    I meant a minority w.r.t. to the current distribution in math.
    - kevinventullo 3y ago
      I’ve found the word “underrepresented” works a little better in this context to avoid misunderstandings.
- nxobject 3y ago
hyperthesis 3y ago
The author teaches maths to art students. The paragraph on maths research: But now I realize that, in maths research, you don’t just follow logical steps. If you do, you’ll never get anywhere interesting. You have to use your gut instinct and feel your way through something first, and then back it up with logic afterwards. Opposite to TFA, in school, I gave up on maths because explanations were not provided; you had to take things on faith (e.g. algebra of limits, epsilon-delta proofs were not given), and the questions were carefully manufactured to be solvable by the given methods. i.e. puzzle-solving, not general powerful methods. Also, no proof of the uniqueness of prime factorization. TBF, it's hard to load proofs on to school students, and since Euclid was dropped, the focus is on technicians and results - not truth. But it can be done: spend the time on building foundations; or omit subjects where you can't. I'd like an axiomatic approach.
majikaja 3y ago
Alternatively, just do whatever obscure research you want outside of the university system. There won't be anyone to judge your 'success' or lack of it. When was the last time these mathematicians received feedback from the outside world instead of just deciding within their group who to declare as #1? They literally get paid by tax payers to do what is often the equivalent of glorified crosswords full time then congratulate themselves that they're good at it.
__tmk__ 3y ago
That's not very actionable advice. I've found this article [0] from Terence Tao very insightful: [A]ctual solutions to a major problem tend to be arrived at by a process more like the following (often involving several mathematicians over a period of years or decades, with many of the intermediate steps described here being significant publishable papers in their own right): 1. Isolate a toy model case x of major problem X. 2. Solve model case x using method A. 3. Try using method A to solve the full problem X. 4. This does not succeed, but method A can be extended to handle a few more model cases of X, such as x’ and x”. 5. Eventually, it is realised that method A relies crucially on a property P being true; this property is known for x, x’, and x”, thus explaining the current progress so far. 6. Conjecture that P is true for all instances of problem X. 7. Discover a family of counterexamples y, y’, y”, … to this conjecture. This shows that either method A has to be adapted to avoid reliance on P, or that a new method is needed. 8. Take the simplest counterexample y in this family, and try to prove X for this special case. Meanwhile, try to see whether method A can work in the absence of P. (... 15 more steps) [0]: https://terrytao.wordpress.com/career-advice/be-sceptical-of-your-own-work/ https://terrytao.wordpress.com/career-advice/be-sceptical-of...
- mathgenius 3y ago
  That's about how to solve problems, but to do research you need to find good problems, and that turns out to be more important, imo. How do you pick "major problem X" anyway? This strategy from Tao just leads to incremental results...
  - fn-mote 3y ago
    1. If Tao (a Fields medalist) is telling you a process to follow, a dismissal of "just leads to incremental results" requires more evidence than just a bald claim. IMO, he's giving away his working process for free. 2. If "all" you want is tenure at a research 1 institution in the us, there's a lot to be said for this process. Another ingredient is "pick an area where other people will be interested in the results." PS I don't know what to make of your username, but perhaps you have more to say about picking a problem... in which case I'm sure many of us would love to hear it.
JohnKemeny 3y ago
About as useful as Feynman's algorithm for solving problems: 1. Write down the problem. 2. Think very hard. 3. Write down the solution.
- paulpauper 3y ago
  somewhere between steps 1-3: be as smart as him
- hyperthesis 3y ago
  42
- marmakoide 3y ago
  It's useful. It just need a note about the cyclic nature of it and the place of ego * Forget about glory and peer recognition. One day you'll die and as time passes you'll be forgotten, including anything you made. Don't waste energy fighting it, just embrace it and carry on. Do it for that 1 mn moment of bliss when you found something nice. * Accept repeated failure. It's ok. * Spend time to learn why attempt N failed before going to attempt N+1, * Before attempt N+1, go out for a walk, cook something nice, do something with your kids, have a beer, watch a movie
- siva7 3y ago
  You’d be surprised how many fail at step 1, skip step 2 and try straight step 3.
  - AnotherGoodName 3y ago
    I'm pretty sure that's the point and it's not entirely a joke. Stating the problem to a succinct degree is the hardest part and normally reveals the other steps. If your maths problem is actually a specific instance of a known broader problem then just stating that will point the way forward. In fact a proof that your problem is an instance of a broader problem can generally be a worthy paper in its own right.
    - hyperthesis 3y ago
loa_in_ 3y ago
Being a successful research mathematician, although I am not one, sounds nowadays like more about figuring out the market for your skills among multitudes of opaque research projects so I imagine following your gut you might just get into a good post if you're lucky.
- strikelaserclaw 3y ago
  We are truly in a time where there are plenty of (bad, unsubstantial) ideas out there, i reckon it takes a good instinct to be a good mathematician (or any other thing that does not rely on public opinion).
- isaacfung 3y ago
  Can you name a few successful research mathematicians who solved an important open problem by just being "lucky"?
  - cykotic 3y ago
    This is a philosophical viewpoint but in some sense all of them are lucky. Louis de Branges claimed for years to have a proof of Riemann Hypothesis. No one really agreed with his conclusion. He is certainly a first rate mathematician and had he came up with a proof that others accepted he’d be lauded as a great mathematician. Abhyankar claimed that no one has ever really understood Hironaka’s resolution of singularities paper. Hironaka is a Field’s Medalist and Abhyankar a great mathematician in his own right. He was never able to find a simpler proof for resolution of singularities. If he had gotten lucky then he would have. The point is, that luck plays a role in terms of whether or not the right idea pops into your head. How many brilliant people labored over problems that simply have no solution and thus aren’t considered one of the greats? Newton wrote more about alchemy than math or physics. He’s not considered a great chemist. One thing the greats have in common is spending a great deal of time thinking about problems. That increases the probability of coming up with a brilliant insight. Of course, it is not all luck. You do have to have good intuition. Here’s a quote by Chaitin: Gödel's incompleteness theorem tells us that within mathematics there are statements that are unknowable, or undecidable. Omega tells us that there are in fact infinitely many such statements: whether any one of the infinitely many bits of Omega is a 0 or a 1 is something we cannot deduce from any mathematical theory. More precisely, any maths theory enables us to determine at most finitely many bits of Omega. In some sense we get a survivorship bias when talking about the greats. They happened to work on a problem that was solvable. I suggest there are many more equally brilliant people who didn’t get lucky and thus are unknown.
xiasongh 3y ago
Does this kind of advice apply to all research or specifically mathematics?
- deleted 3y ago
  [deleted]

99 points by webmaven 3y ago

Or brute force the hell out of everything.

I appreciated Eugenia Cheng's youtube lectures on category theory and I know interviews aren't the best format for sustained exposition, but the remark about mathematical rigor being a European colonialist thing seemed pretty forced. If anything, it underestimates the strangeness of the history of math in an effort to cram it into a simple frame: "Europeans bad". There was something distinctive about work in foundations of math in the late 19th / early 20th centuries, work which came out of European cultures (or more accurately Latin Christendom). Plausibly, this work was centuries in the making, the culmination of a trend towards increasingly analytic and systematizing modes of thought that begins with the recovery of Greek philosophy in the middle ages. But, ok, even if you don't buy that, how do you get from the Congo Free State to Frege, Hilbert and Goedel? It's a radical claim that would be utterly fascinating if it were true. But Chang just kind of tosses it off at the end and asks us to accept it as an unremarkable truism.

nxobject 3y ago
I've talked about this in a replay to another comment, but I don't think she's making any value judgment in the article, or (at least in the article) actually articulating any claim about colonialism and mathematics. In fact, I don't even think she's saying "Europeans bad": We might marvel at, say, Amish people raising a barn or Inuit people building a kayak without any European formal maths training. We might declare that this is maths — and I think it is. But it’s problematic if we think we’re doing them some great honour by bestowing the label of maths on what they do.
- lambdaphagy 3y ago
  I think that if I were being interviewed and my interviewer said "you also discuss how colonialist ideology has influenced modern mathematics?", but I didn't think that colonialist ideology has influenced modern mathematics, I would say: "What? I don't think colonialist ideology has influenced modern mathematics at all." Rather than insinuating that Europe must have stolen math from other parts of the world and pawned it off as its own, or been really mean about rigor, or whatever. There were several totalizing ideologies in the 20th century that attempted to make all of human affairs about themselves: everything from literature to science to birdwatching had some sort of party line: Did this literature say the right things? Was this science done by the right kind of scientist? Is this bird the right kind of bird to be watching? Judging by their results, I wouldn't look forward to a do-over of that line of thinking.
  - nxobject 3y ago
    I think that's our disagreement: I think you can say, "maybe there's some relationship between what we have seen as colonialism and academic mathematics", without even thinking that it was about "stealing math", or thinking about colonialism determined everything about mathematics in some totalizing way. I'm sorry if I come off as a little obtuse, so I might as well say what I'm really thinking. My thinking leans towards Eugenia's, and it's a lot more conservative: the question itself is worth thinking about, but it's a lot to think through, and the link may as well be tenuous. But, in my thinking, if NATO once funded pure mathematics (as Grothendieck once protested against), if DARPA also currently funds pure mathematics, then it is worth asking what historical narrative explains it, and whether it applies to the slightly more distant past, too. Above all I don't believe that the academic enterprise has ever been a neutral and disinterested endeavor, either – it's a lot of public money, and wherever there's public money politics comes before or after. I don't think that's a totalizing way of understanding academia.
    - lambdaphagy 3y ago
      Cheng talks a lot about colonialism inn _Is Maths Real_ but here is the closest thing to an argument for the claim that I could locate: > “Now, at the same time as all this, it is true that mathematics as a field has grown carefully and rigorously according to a carefully constructed framework of logic and rigour. It is, arguably, the framework that is keeping some maths in and keeping some maths out. But... there are questions about the values built into the framework of mathematics. That framework is geared towards the principles of ‘progress’ and ‘development’. Deep down I have an uncomfortable suspicion that those principles are indelibly linked to colonialism, imperialism and the urge to conquer others.” (p. 320-1) That's the argument in its entirety. What humanity was up to in the ten millennia of recorded history before formal mathematics is left unexplained. The most parsimonious explanation for the military's funding of mathematics research is that mathematics often has applications to military technology. But even here, we have to be careful: it's typically applied math that's the most, well... applicable, which is precisely the sort of thing that pure mathematicians are supposed to look down upon as "not really math". (The pure math work at DARPA is AFAICT more a consequence of DARPA's idiosyncratic governance structure that gives program managers wide latitude to swing for the fences, accepting that there will be some strikeouts.) But the fact that modern Western mathematics is indispensable for advanced engineering projects at all, in a way that Sanskrit prosody and Inuit kayak architecture are not, is pretty crucial to understanding the funding situation: it's funded because it's useful, not useful because it's funded. If we don't grant that math is actually unreasonably effective in explaining / predicting / manipulating the world then sure, what math gets funded looks like another courtly intrigue. But it seems that the unreasonable effectiveness is precisely what needs to be explained away on Cheng's view. I'm all ears for the contrary view, but we've already written more on the topic than Cheng did in the entire book, which is light on argument and regrettably heavy on insinuation.

marmakoide 3y ago

The whole tirade on rigorous proofs and colonial mindset puzzles me. From my understanding (I am European ...), rigorous proofs were increasingly seen as a need because as things like calculus progressed, any loose definition had a tendency to become a minefield. Think of the definition of derivatives, continuity. Also, at a time where things were getting mechanize because we could and it was a gain, it was natural to check if we could somehow mechanize math. After all, Pascal made a machine that could do sums, why not a machine for proofs ?

thoi23u4234324 3y ago
I don't think 'colonial' is the right word; I think it's more to do with theology and Europe's very twisted (and ongoing) history with it. A very clear example is S Ramanujan's way of doing mathematics, where proofs were secondary to 'calculate, calculate, calculate, then hypothesize, then maybe prove' approach. This was not a one off thing, nor was his strange approach unique. Indian traditional mathematics also follows SR's approach of compute & then generalize & then prove (which most working mathematicians actually follow to some degree anyway). India was ahead of much of the world in Math until the cusp of renaissance(finally culminating in the dev. of calculus a good hundred years before Newton, and possibly having traveled to and 'inspired' him with the Jesuit missions), and its outcome's are simply extraordinary. Of course, the tradition was not given any credit whatsoever due to the prevalent racial-thinking in old-Europe; this now continues because Indologists continue to hold onto their inherited toxic racial thinking and prevent anyone else from touching this subject because 'Hindus are bad/dirty/nazis/<add adjective> people' (you see this with Yoga, NSDR, Mindfullness,... to this day). Regardless of all the 'credit allocation' business, and somewhat more importantly, it's actually really sad that this tradition is no longer alive, and is instead replaced with a cheap mimicry of Europe. C K Raju has written extensively on this. It's also not like this way of doing math. is inherently worse. A lot of actual formal math, starting from the construction of real-numbers is actually very 'unreal' when you get down to it. The kinds of hiccups thrown up by real-numbers in the theory of computation needs to be seen to be believed. Chaitin even has a paper by the title 'How real are real numbers'... We actually get around many of these weird issues because the construction of formal mathematics is actually based on ensuring that discretization is stable (this doesn't work in TOC). Still one also can't deny the strides that modern mathematics has made, in large part built on formalization. All said, to destroy any existing tradition (even Western) is extremely detestable. OTOH given that it is a jealous-monoculture (all of academia has become one big church-like monoculture at this point), the creation of a new tradition needs active support. I wish people would spend time on the latter instead of pointless race-wars (or DEI non-sense). (edit: expand)

vouaobrasil 3y ago

I have a PhD in math, did some successful research, but left math for a variety of personal reasons. My advice: 1. Network, network, network. Get famous or established mathematicians interested in your work. 2. Be a minority. Due to the prevailing ideology of academia, it's much easier to be successful if you are a minority. (I am not saying it doesn't take hard work, but the competition is not as extreme due to intense equity measures.) I will laugh if anyone gets mad at this point. Equity rules make this point an incontrovertible fact. 3. Solve problems related to the work of established mathematicians. Interesting or novel problems that are not related to the work of others will be ignored. 4. Forget about being curious. Okay, this one has some dark humour about it...but seriously, if you just explore your curiosity, you might solve cool problems but you won't be "career successful". It's much better just to make useless investigations into ever-more esoteric problems that validate the "successful" work of others which is becoming increasingly technical. A note: this might sound a bit cynical, but I thought I'd give another viewpoint to show a bit more of the balanced reality of the math world.

thoi23u4234324 3y ago
> Be a minority. Won't help if you're Asian IMO.
- nxobject 3y ago
  Well, it just means you're not a minority in the world of mathematics.
- vouaobrasil 3y ago
  I meant a minority w.r.t. to the current distribution in math.
  - kevinventullo 3y ago
    I’ve found the word “underrepresented” works a little better in this context to avoid misunderstandings.
nxobject 3y ago

hyperthesis 3y ago

The author teaches maths to art students. The paragraph on maths research: But now I realize that, in maths research, you don’t just follow logical steps. If you do, you’ll never get anywhere interesting. You have to use your gut instinct and feel your way through something first, and then back it up with logic afterwards. Opposite to TFA, in school, I gave up on maths because explanations were not provided; you had to take things on faith (e.g. algebra of limits, epsilon-delta proofs were not given), and the questions were carefully manufactured to be solvable by the given methods. i.e. puzzle-solving, not general powerful methods. Also, no proof of the uniqueness of prime factorization. TBF, it's hard to load proofs on to school students, and since Euclid was dropped, the focus is on technicians and results - not truth. But it can be done: spend the time on building foundations; or omit subjects where you can't. I'd like an axiomatic approach.

majikaja 3y ago

Alternatively, just do whatever obscure research you want outside of the university system. There won't be anyone to judge your 'success' or lack of it. When was the last time these mathematicians received feedback from the outside world instead of just deciding within their group who to declare as #1? They literally get paid by tax payers to do what is often the equivalent of glorified crosswords full time then congratulate themselves that they're good at it.

__tmk__ 3y ago

That's not very actionable advice. I've found this article [0] from Terence Tao very insightful: [A]ctual solutions to a major problem tend to be arrived at by a process more like the following (often involving several mathematicians over a period of years or decades, with many of the intermediate steps described here being significant publishable papers in their own right): 1. Isolate a toy model case x of major problem X. 2. Solve model case x using method A. 3. Try using method A to solve the full problem X. 4. This does not succeed, but method A can be extended to handle a few more model cases of X, such as x’ and x”. 5. Eventually, it is realised that method A relies crucially on a property P being true; this property is known for x, x’, and x”, thus explaining the current progress so far. 6. Conjecture that P is true for all instances of problem X. 7. Discover a family of counterexamples y, y’, y”, … to this conjecture. This shows that either method A has to be adapted to avoid reliance on P, or that a new method is needed. 8. Take the simplest counterexample y in this family, and try to prove X for this special case. Meanwhile, try to see whether method A can work in the absence of P. (... 15 more steps) [0]: https://terrytao.wordpress.com/career-advice/be-sceptical-of-your-own-work/ https://terrytao.wordpress.com/career-advice/be-sceptical-of...

mathgenius 3y ago
That's about how to solve problems, but to do research you need to find good problems, and that turns out to be more important, imo. How do you pick "major problem X" anyway? This strategy from Tao just leads to incremental results...
- fn-mote 3y ago
  1. If Tao (a Fields medalist) is telling you a process to follow, a dismissal of "just leads to incremental results" requires more evidence than just a bald claim. IMO, he's giving away his working process for free. 2. If "all" you want is tenure at a research 1 institution in the us, there's a lot to be said for this process. Another ingredient is "pick an area where other people will be interested in the results." PS I don't know what to make of your username, but perhaps you have more to say about picking a problem... in which case I'm sure many of us would love to hear it.

JohnKemeny 3y ago

About as useful as Feynman's algorithm for solving problems: 1. Write down the problem. 2. Think very hard. 3. Write down the solution.

paulpauper 3y ago
somewhere between steps 1-3: be as smart as him
hyperthesis 3y ago
42
marmakoide 3y ago
It's useful. It just need a note about the cyclic nature of it and the place of ego * Forget about glory and peer recognition. One day you'll die and as time passes you'll be forgotten, including anything you made. Don't waste energy fighting it, just embrace it and carry on. Do it for that 1 mn moment of bliss when you found something nice. * Accept repeated failure. It's ok. * Spend time to learn why attempt N failed before going to attempt N+1, * Before attempt N+1, go out for a walk, cook something nice, do something with your kids, have a beer, watch a movie
siva7 3y ago
You’d be surprised how many fail at step 1, skip step 2 and try straight step 3.
- AnotherGoodName 3y ago
  I'm pretty sure that's the point and it's not entirely a joke. Stating the problem to a succinct degree is the hardest part and normally reveals the other steps. If your maths problem is actually a specific instance of a known broader problem then just stating that will point the way forward. In fact a proof that your problem is an instance of a broader problem can generally be a worthy paper in its own right.
  - hyperthesis 3y ago

loa_in_ 3y ago

Being a successful research mathematician, although I am not one, sounds nowadays like more about figuring out the market for your skills among multitudes of opaque research projects so I imagine following your gut you might just get into a good post if you're lucky.

strikelaserclaw 3y ago
We are truly in a time where there are plenty of (bad, unsubstantial) ideas out there, i reckon it takes a good instinct to be a good mathematician (or any other thing that does not rely on public opinion).
isaacfung 3y ago
Can you name a few successful research mathematicians who solved an important open problem by just being "lucky"?
- cykotic 3y ago
  This is a philosophical viewpoint but in some sense all of them are lucky. Louis de Branges claimed for years to have a proof of Riemann Hypothesis. No one really agreed with his conclusion. He is certainly a first rate mathematician and had he came up with a proof that others accepted he’d be lauded as a great mathematician. Abhyankar claimed that no one has ever really understood Hironaka’s resolution of singularities paper. Hironaka is a Field’s Medalist and Abhyankar a great mathematician in his own right. He was never able to find a simpler proof for resolution of singularities. If he had gotten lucky then he would have. The point is, that luck plays a role in terms of whether or not the right idea pops into your head. How many brilliant people labored over problems that simply have no solution and thus aren’t considered one of the greats? Newton wrote more about alchemy than math or physics. He’s not considered a great chemist. One thing the greats have in common is spending a great deal of time thinking about problems. That increases the probability of coming up with a brilliant insight. Of course, it is not all luck. You do have to have good intuition. Here’s a quote by Chaitin: Gödel's incompleteness theorem tells us that within mathematics there are statements that are unknowable, or undecidable. Omega tells us that there are in fact infinitely many such statements: whether any one of the infinitely many bits of Omega is a 0 or a 1 is something we cannot deduce from any mathematical theory. More precisely, any maths theory enables us to determine at most finitely many bits of Omega. In some sense we get a survivorship bias when talking about the greats. They happened to work on a problem that was solvable. I suggest there are many more equally brilliant people who didn’t get lucky and thus are unknown.

xiasongh 3y ago

Does this kind of advice apply to all research or specifically mathematics?

deleted 3y ago
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