3 ms·

The big thing I always wondered was what do you do when you are stuck with something that nobody else has an answer to? I did contest math as a kid, loved it,

The big thing I always wondered was what do you do when you are stuck with something that nobody else has an answer to? I did contest math as a kid, loved it, but of course there's always an answer. Often if I didn't get the answer, there would be some kind of "oh yeah why didn't I think of that" when I heard the solution. So you would always be bailed out eventually. But also sometimes I'd get this bolt from the blue and solve the thing. It would be some weird little trick that didn't seem to come from anywhere or lead anywhere other than this one solution. I'd also see these bolts in other people's solutions. That is both satisfying and insanely frustrating, because how do you know you'll keep thinking of these things? At some point, you will hit a rock that nobody you know, even really smart people who have done this their whole lives, can jiggle. What do you do? I also wonder about how it works working on "proper" mathematical problems. Are people who do that basically experts at the little problems? Or is it like coding, where you can design a fairly large system and rely on the parts to come together one way or another? Because when I read a math text, it seems really quite dense with things that require a lot of details to be correct. If it's a house of cards spanning a hundred page thesis, how brittle is it?

Jensson 4y ago
You solve it yourself, that is all there is to it. I have solved novel math problems, all you do is think about it until you have solved it, there are many strategies for doing that but in the end its all you thinking about different ways to solve it.
atomicnature 4y ago
I recently read a 900 pager on Newton's life, which includes a lot of details about his mathematical accomplishments (book link: https://www.amazon.in/Never-Rest-Biography-Cambridge-Paperback/dp/0521274354 https://www.amazon.in/Never-Rest-Biography-Cambridge-Paperba...). Key things that stand out: + Deep scholarship: Learn from your predecessors as much as possible; pick up their techniques, at the same time be sceptical, do not believe much, even when you're learning/mastering techniques. Ex: Newton mastered Descartes work early in his career, used that mathematics to open up Calculus, but at later stage became disillusioned with the analytic method, and converted his thought into pure geometric methods. + Imitate the masters before you: Newton was really one of the deepest students of the ancients. He'd ferret out obscure passages, and use them to solve his problems. Ex: He knew the intricacies of the analysis/synthesis method from Pappus, which most people barely know in the depth he knew of them. + Dedication to truth, desiring certainty and taking nothing less. Insanely high standards.
- 082349872349872 4y ago
  > Imitate the masters «Ved å studere mesterne, og ikke deres elever»
  - pbhjpbhj 4y ago
    Google translate says "By studying the masters, and not their students", apparently from Norwegian, apparently a quote from the mathematician Abel (who's name I know from Abelian groups but is probably better known to some here). en.wikipedia.org/wiki/Niels_Henrik_Abel
    - mejutoco 4y ago
      "Read the source, Luke" This one came to mind, which I think captures the same idea
gjulianm 4y ago
> At some point, you will hit a rock that nobody you know, even really smart people who have done this their whole lives, can jiggle. What do you do? Keep working. Or leave it, come back after some time (hours, days, weeks, months... even years). > Are people who do that basically experts at the little problems? Or is it like coding, where you can design a fairly large system and rely on the parts to come together one way or another? Because when I read a math text, it seems really quite dense with things that require a lot of details to be correct. If it's a house of cards spanning a hundred page thesis, how brittle is it? I haven't worked too much in pure math research, but from my understanding it's a process of adding details. You have something you want to prove, and a base you start from. You have an intuition of what you need to be true in order to prove the thing, so start developing those proofs too. In the way you'll find you need certain conditions, and you might find that certain concepts reappear so you find it helpful to define them. > Because when I read a math text, it seems really quite dense with things that require a lot of details to be correct. If it's a house of cards spanning a hundred page thesis, how brittle is it? While technically it is a house of cards (i.e., if a proof is wrong everything that depends on it fails), most of the time you won't find such fundamental errors. Maybe a proof has some missing details that can be amended, or the same statement can be proven another way, or the theorem needs just tiny changes or extra conditions to be fixed.
- 082349872349872 4y ago
  > even years the tide comes in for all problems, but for some more rapidly than others: cf https://en.wikipedia.org/wiki/Parallel_postulate#History https://en.wikipedia.org/wiki/Parallel_postulate#History Problems worthy of attack prove their worth by fighting back.
  - a9h74j 4y ago
    For some reason I associate that rhyme with Piet Hein, based on the game Soma.
    - 082349872349872 4y ago

Jensson 4y ago

You solve it yourself, that is all there is to it. I have solved novel math problems, all you do is think about it until you have solved it, there are many strategies for doing that but in the end its all you thinking about different ways to solve it.

atomicnature 4y ago

I recently read a 900 pager on Newton's life, which includes a lot of details about his mathematical accomplishments (book link: https://www.amazon.in/Never-Rest-Biography-Cambridge-Paperback/dp/0521274354 https://www.amazon.in/Never-Rest-Biography-Cambridge-Paperba...). Key things that stand out: + Deep scholarship: Learn from your predecessors as much as possible; pick up their techniques, at the same time be sceptical, do not believe much, even when you're learning/mastering techniques. Ex: Newton mastered Descartes work early in his career, used that mathematics to open up Calculus, but at later stage became disillusioned with the analytic method, and converted his thought into pure geometric methods. + Imitate the masters before you: Newton was really one of the deepest students of the ancients. He'd ferret out obscure passages, and use them to solve his problems. Ex: He knew the intricacies of the analysis/synthesis method from Pappus, which most people barely know in the depth he knew of them. + Dedication to truth, desiring certainty and taking nothing less. Insanely high standards.

082349872349872 4y ago
> Imitate the masters «Ved å studere mesterne, og ikke deres elever»
- pbhjpbhj 4y ago
  Google translate says "By studying the masters, and not their students", apparently from Norwegian, apparently a quote from the mathematician Abel (who's name I know from Abelian groups but is probably better known to some here). en.wikipedia.org/wiki/Niels_Henrik_Abel
  - mejutoco 4y ago
    "Read the source, Luke" This one came to mind, which I think captures the same idea

gjulianm 4y ago

> At some point, you will hit a rock that nobody you know, even really smart people who have done this their whole lives, can jiggle. What do you do? Keep working. Or leave it, come back after some time (hours, days, weeks, months... even years). > Are people who do that basically experts at the little problems? Or is it like coding, where you can design a fairly large system and rely on the parts to come together one way or another? Because when I read a math text, it seems really quite dense with things that require a lot of details to be correct. If it's a house of cards spanning a hundred page thesis, how brittle is it? I haven't worked too much in pure math research, but from my understanding it's a process of adding details. You have something you want to prove, and a base you start from. You have an intuition of what you need to be true in order to prove the thing, so start developing those proofs too. In the way you'll find you need certain conditions, and you might find that certain concepts reappear so you find it helpful to define them. > Because when I read a math text, it seems really quite dense with things that require a lot of details to be correct. If it's a house of cards spanning a hundred page thesis, how brittle is it? While technically it is a house of cards (i.e., if a proof is wrong everything that depends on it fails), most of the time you won't find such fundamental errors. Maybe a proof has some missing details that can be amended, or the same statement can be proven another way, or the theorem needs just tiny changes or extra conditions to be fixed.

082349872349872 4y ago
> even years the tide comes in for all problems, but for some more rapidly than others: cf https://en.wikipedia.org/wiki/Parallel_postulate#History https://en.wikipedia.org/wiki/Parallel_postulate#History Problems worthy of attack prove their worth by fighting back.
- a9h74j 4y ago
  For some reason I associate that rhyme with Piet Hein, based on the game Soma.
  - 082349872349872 4y ago