[HN Gopher] How to be successful as a research mathematician? Fo...
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How to be successful as a research mathematician? Follow your gut
Author : webmaven
Score : 70 points
Date : 2023-08-15 01:06 UTC (21 hours ago)
(HTM) web link (www.nature.com)
(TXT) w3m dump (www.nature.com)
| xiasongh wrote:
| Does this kind of advice apply to all research or specifically
| mathematics?
| [deleted]
| JohnKemeny wrote:
| About as useful as Feynman's algorithm for solving problems:
|
| 1. Write down the problem. 2. Think very hard. 3. Write down the
| solution.
| [deleted]
| siva7 wrote:
| You'd be surprised how many fail at step 1, skip step 2 and try
| straight step 3.
| AnotherGoodName wrote:
| 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.
| gms7777 wrote:
| 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!
| amelius wrote:
| Or brute force the hell out of everything.
| loa_in_ wrote:
| 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.
| isaacfung wrote:
| Can you name a few successful research mathematicians who
| solved an important open problem by just being "lucky"?
| cykotic wrote:
| 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:
|
| _Godel '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.
| Muirhead wrote:
| 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.
| cykotic wrote:
| 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.
| __tmk__ wrote:
| 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...
| [deleted]
| itissid wrote:
| Overheard at the watercooler: The only way a CS guy knows how
| to prove an algorithm works/does not work is to find
| counterexamples.
| JohnKemeny wrote:
| Seems hard to prove that an algorithm works by finding
| counterexamples.
| contravariant wrote:
| 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.
| panda-giddiness wrote:
| Just express it as an n-state Turing machine and see if it
| halts within BB(n) steps. /s
| AnimalMuppet wrote:
| And, if n is large enough, then I can retire today.
| aoki wrote:
| My real analysis instructor drily pointed out "You really
| like proof by contradiction huh" and my defense was that my
| main life skill is spotting why plausible-seeming things
| don't work correctly
| morkalork wrote:
| I would've picked proof by induction for CS peeps!
| SkyBelow wrote:
| From the number of math channels I follow, the common advice I
| see for handling a complex problem is to start with a simpler
| version of it and solve it. Try simplifying it in a few
| different ways, find a few different ways to solve each
| simplification, and then see which ways of solving the problem
| might work to solve the more complex problem. With experience
| one gets better at finding the right simplification with less
| effort/time invested.
| mathgenius wrote:
| 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 wrote:
| 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.
| steppi wrote:
| Tao's advice is characteristically excellent but I think
| Eugenia Cheng's advice (not just the article title but as a
| whole) is actionable, just not so much for someone who is ready
| for specific advice like Tao's. Her message seems to be
| targeted towards those who are just getting started and who may
| not otherwise have been interested in studying math.
| resource0x wrote:
| 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
| Ar-Curunir wrote:
| 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.
| hgomersall wrote:
| 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.
| einpoklum wrote:
| 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.
| Ar-Curunir wrote:
| Increasingly this evolution is spread across multiple papers,
| and is not enclosed within a single paper.
| _Microft wrote:
| Polya's ,,How to solve it" might also be worth a look.
|
| Summary: https://www.math.utah.edu/~alfeld/math/polya.html
| morkalork wrote:
| 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.
| tomjakubowski wrote:
| 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
| PartiallyTyped wrote:
| This is a great way to go about implementing features that deal
| with complex situations in NP-Complete problems, where you know
| a subset is trivially solvable, you build patterns / approaches
| for said problem, and for the remaining cases, just use
| approximate solutions.
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