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Every mathematician has only a few tricks (2020)

mathoverflow.net

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Re: Every mathematician has only a few tricks (2020)

#11

Post by tibbar »

philipov wrote: post #9
tibbar wrote: post #2

I think this is true for engineers as well! I enjoy getting to know the "theme" of my favorite coworkers over the years. There was: * The fellow who always looked for the simplest hack possible. Give him the most annoying problem, he'd pause, go Wait a minute! and redefine it to have a very easy solution. He typed very slowly, but it didn't really matter. * The one who truly loved code itself. He would climb mountain…

For me, it's tracing code/pipelines to figure out how a result was produced, typically in the context of that result being wrong somehow. Go To Definition is the most useful function in any editor. I'm always surprised by how frequently colleagues don't think to do this and are left helpless.

This reminds me of my further theory that everyone needs one 'heavy' and one 'light' technique. The 'light' technique is something that often works well as a heuristic and can be an effective unit of iteration. The 'heavy' technique is something that you can fall back on in difficult cases, something that can reliably solve hard problems, even if it's slow.

Sometimes the heavy technique is: just ask someone else. ;)

Re: Every mathematician has only a few tricks (2020)

#12

Post by eigenman »

There is a joke in applied mathematics that we’re like Taco Bell. We all use the same six ingredients, mixing them in different ways.

For myself, I’ve found several techniques I use over and over again. Some of this is a “when you’re a hammer, everything looks like a nail.” But fundamentally there are only a handful of ideas. One professor of mine once said the only groundbreaking result in the past few decades was compressive sensing.

Re: Every mathematician has only a few tricks (2020)

#13

Post by ArcHound »

I remember annoying one of our professors at the university. Whenever he was discussing a problem and found me falling asleep, he called my name. I woke up, said Chinese remainder theorem and with like 90% success rate it solved the problem handily. Yes, it was an Algebra class. We were still surprised how well it worked.

Re: Every mathematician has only a few tricks (2020)

#14

Post by cyberax »

philipov wrote: post #9
tibbar wrote: post #2

I think this is true for engineers as well! I enjoy getting to know the "theme" of my favorite coworkers over the years. There was: * The fellow who always looked for the simplest hack possible. Give him the most annoying problem, he'd pause, go Wait a minute! and redefine it to have a very easy solution. He typed very slowly, but it didn't really matter. * The one who truly loved code itself. He would climb mountain…

For me, it's tracing code/pipelines to figure out how a result was produced, typically in the context of that result being wrong somehow. Go To Definition is the most useful function in any editor. I'm always surprised by how frequently colleagues don't think to do this and are left helpless.

Another example: debuggers.

It's amazing that a lot of new developers don't know how to use them at all! I'm not even talking about using the command line gdb, but just the basic "Set Breakpoint" feature and attaching to processes.

Re: Every mathematician has only a few tricks (2020)

#16

Post by huflungdung »

tibbar wrote: post #2

I think this is true for engineers as well! I enjoy getting to know the "theme" of my favorite coworkers over the years. There was: * The fellow who always looked for the simplest hack possible. Give him the most annoying problem, he'd pause, go Wait a minute! and redefine it to have a very easy solution. He typed very slowly, but it didn't really matter. * The one who truly loved code itself. He would climb mountain…

[dead]

Re: Every mathematician has only a few tricks (2020)

#17

Post by TeMPOraL »

tibbar wrote: post #11
philipov wrote: post #9

Earlier quoted context omitted.

For me, it's tracing code/pipelines to figure out how a result was produced, typically in the context of that result being wrong somehow. Go To Definition is the most useful function in any editor. I'm always surprised by how frequently colleagues don't think to do this and are left helpless.

This reminds me of my further theory that everyone needs one 'heavy' and one 'light' technique. The 'light' technique is something that often works well as a heuristic and can be an effective unit of iteration. The 'heavy' technique is something that you can fall back on in difficult cases, something that can reliably solve hard problems, even if it's slow. Sometimes the heavy technique is: just ask someone else. ;)

> Sometimes the heavy technique is: just ask someone else. ;)

For a lot of people I know, this is the light technique!

Re: Every mathematician has only a few tricks (2020)

#19

Post by seanhunter »

My dad (an engineer not a mathematician) would use Newton-Raphson[1] to solve basically any problem that wasn’t very obviously linear. When I was a kid, some of my first programming memories were my dad getting me and my brother to implement Newton-Raphson in basic on an HP85a, getting me to implement Newton-Raphson in rpn on an HP calculator, debugging my dad’s (genuinely revolting) basic program[2] which wouldn’t run but (who would have guessed?) used Newton-Raphson to compute something or other.

He basically learned the one numerical root-finder and how to evaluate basic second derivatives and he was set for life on all the problems a career in chemical and process engineering could throw at someone.

[1] https://sheffield.ac.uk/media/31988/download?attachment

[2] He learned to program in FORTRAN and lived by the maxim that a determined FORTRAN programmer can write FORTRAN in any language.

Re: Every mathematician has only a few tricks (2020)

#20

Post by seanhunter »

Animats wrote: post #18

Feynman wrote in his autobiography that much of his success came from having different mathematical tricks than most of his peers. So when they were stuck, he could sometimes make progress.

Interestingly, one of his famous tricks of computing complex integrals by parameterizing[1] and then differentiating under the integral is known as “Feynman’s trick” in his honour spite of it having been invented by Euler over 250 years before.

[1] https://zackyzz.github.io/feynman.html

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