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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

#21

Post by resource0x »

__tmk__ wrote: post #2

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 pro…

And then erase all intermediate steps and just publish the end result - which no one will understand (including yourself a couple of years later), but that's OK /s

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

#22

Post by contravariant »

JohnKemeny wrote: post #6
itissid wrote: post #4

Earlier quoted context omitted.

Overheard at the watercooler: The only way a CS guy knows how to prove an algorithm works/does not work is to find counterexamples.

Seems hard to prove that an algorithm works by finding counterexamples.

I think most bugs are the result of someone's attempt to do just that. As is test-driven development I suppose.

Then again proving code works isn't everything either. There's a reason Knuth once stated "Beware of bugs in the above code; I have only proved it correct, not tried it."

Type checking is a nice intermediate, though not all languages allow all properties you care about to be encoded in types.

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

#23

Post by hgomersall »

resource0x wrote: post #21
__tmk__ wrote: post #2

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 pro…

And then erase all intermediate steps and just publish the end result - which no one will understand (including yourself a couple of years later), but that's OK /s

I've lamented this effect of the paper writing process for some time. I think a paper should present the steps taken by the authors to arrive at the new results, not just the new results in isolation. Those steps often provide so much useful insight.

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

#24

Post by Ar-Curunir »

resource0x wrote: post #21
__tmk__ wrote: post #2

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 pro…

And then erase all intermediate steps and just publish the end result - which no one will understand (including yourself a couple of years later), but that's OK /s

Increasingly this evolution is spread across multiple papers, and is not enclosed within a single paper.

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

#25

Post by Muirhead »

cykotic wrote: post #19
isaacfung wrote: post #14

Earlier quoted context omitted.

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 singularit…

Successful research mathematicians do consistently good work on many problems. While there may be one or two highlights that really boost a career, few people are one-hit wonders that stumble upon a solvable problem.

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

#26

Post by AnimalMuppet »

panda-giddiness wrote: post #8
JohnKemeny wrote: post #6

Earlier quoted context omitted.

Seems hard to prove that an algorithm works by finding counterexamples.

Just express it as an n-state Turing machine and see if it halts within BB(n) steps. /s

And, if n is large enough, then I can retire today.

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

#27

Post by Ar-Curunir »

resource0x wrote: post #21
__tmk__ wrote: post #2

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 pro…

And then erase all intermediate steps and just publish the end result - which no one will understand (including yourself a couple of years later), but that's OK /s

Increasingly this evolution is spread across multiple papers, and is not enclosed within a single paper. So you can partially follow the chain of thought.

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

#28

Post by cykotic »

Muirhead wrote: post #25
cykotic wrote: post #19

Earlier quoted context omitted.

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 singularit…

Successful research mathematicians do consistently good work on many problems. While there may be one or two highlights that really boost a career, few people are one-hit wonders that stumble upon a solvable problem.

Most are one hit wonders. Most research mathematicians end up having at most 1 or 2 Ph.D. students. Most publish minor extensions of known results. The greats have an insight and end up doing a lot in that one minor area. Very few people have more than a couple of truly remarkable theorems.

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

#30

Post by einpoklum »

hgomersall wrote: post #23
resource0x wrote: post #21

Earlier quoted context omitted.

And then erase all intermediate steps and just publish the end result - which no one will understand (including yourself a couple of years later), but that's OK /s

I've lamented this effect of the paper writing process for some time. I think a paper should present the steps taken by the authors to arrive at the new results, not just the new results in isolation. Those steps often provide so much useful insight.

I think so too, but then, (many/most) papers will need to be 2x or 3x longer - which is fine by me, but conferences and journals have strict length limits.
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