About as useful as Feynman's algorithm for solving problems: 1. Write down the problem. 2. Think very hard. 3. Write down the solution.
You’d be surprised how many fail at step 1, skip step 2 and try straight step 3.
How to be successful as a research mathematician? Follow your gut
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Re: How to be successful as a research mathematician? Follow your gut
#12That'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 pro…
Polya‘s „How to solve it“ might also be worth a look. Summary: https://www.math.utah.edu/~alfeld/math/polya.html
I loved this book and respect the author but to be honest, for anyone that doesn't already know, it is geared towards highschool level mathematics. Still a great book though.
Re: How to be successful as a research mathematician? Follow your gut
#13That'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 pro…
Overheard at the watercooler: The only way a CS guy knows how to prove an algorithm works/does not work is to find counterexamples.
I would've picked proof by induction for CS peeps!
Re: How to be successful as a research mathematician? Follow your gut
#14Being 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.
Can you name a few successful research mathematicians who solved an important open problem by just being "lucky"?
Re: How to be successful as a research mathematician? Follow your gut
#15About as useful as Feynman's algorithm for solving problems: 1. Write down the problem. 2. Think very hard. 3. Write down the solution.
You’d be surprised how many fail at step 1, skip step 2 and try straight step 3.
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.
Re: How to be successful as a research mathematician? Follow your gut
#16About as useful as Feynman's algorithm for solving problems: 1. Write down the problem. 2. Think very hard. 3. Write down the solution.
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Re: How to be successful as a research mathematician? Follow your gut
#17Does this kind of advice apply to all research or specifically mathematics?
Re: How to be successful as a research mathematician? Follow your gut
#18Earlier quoted context omitted.
Polya‘s „How to solve it“ might also be worth a look. Summary: https://www.math.utah.edu/~alfeld/math/polya.html
I loved this book and respect the author but to be honest, for anyone that doesn't already know, it is geared towards highschool level mathematics. Still a great book though.
it uses high school math for examples which is great because most readers will already be familiar with the problems
you can apply the techniques anywhere though - I think the book is much more philosophy than math
Re: How to be successful as a research mathematician? Follow your gut
#19Being 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.
Can you name a few successful research mathematicians who solved an important open problem by just being "lucky"?
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.
Re: How to be successful as a research mathematician? Follow your gut
#20Earlier quoted context omitted.
You’d be surprised how many fail at step 1, skip step 2 and try straight step 3.
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.
I think it may also be missing a step 0, which is "Identify that there exists a problem". I find that this is often an entirely distinct step from Step 1, which is it's own hill to climb.
I'm in CS research (with most of my work being applied to biological problems), so I'm a few steps removed from pure math, but I can completely related to this. It's why I find writing grants so difficult -- because they generally require you to identify and describe the problem and propose some potential solutions. But that's like 90% of the work for the whole thing!