Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

What does it feel like to do maths? (2016)

plus.maths.org

Post reply on HN
Print view View original

1–10 of 20 posts

Re: What does it feel like to do maths? (2016)

#4

Post by gjulianm »

I always recommend the book "Love and Math", by Edward Frenkel. The book is fairly approachable I think, and it mixes his own biography with examples of interesting math concepts and connections.

Funnily enough, Frenkel works in the Langland program that Wiles talks about here, and explains a little bit more about it and why is it interesting.

Also, I really like this quote from the article

> Yes, some people are brighter than others but I really believe that most people can really get to quite a good level in mathematics if they're prepared to deal with these more psychological issues of how to handle the situation of being stuck.

I foud this true in experience. Sometimes people find maths hard at the beginning because it's a different way of thinking, much more abstract, and you don't usually have an intuition when you start. But in the end, most of it is practice. Also, some things only click when you go back to them, or need them for more advanced concepts.

Re: What does it feel like to do maths? (2016)

#6

Post by lordnacho »

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?

Re: What does it feel like to do maths? (2016)

#7

Post by atomicnature »

lordnacho wrote: post #6

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 blu…

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-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.

Re: What does it feel like to do maths? (2016)

#8

Post by Jensson »

lordnacho wrote: post #6

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 blu…

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.

Re: What does it feel like to do maths? (2016)

#9

Post by gjulianm »

lordnacho wrote: post #6

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 blu…

> 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.

Re: What does it feel like to do maths? (2016)

#10

Post by 082349872349872 »

gjulianm wrote: post #9
lordnacho wrote: post #6

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 blu…

> 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 on…

> 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

    Problems worthy
    of attack
    prove their worth
    by fighting back.
Post reply on HN