Every mathematician has only a few tricks (2020)

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.

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 mountains to find the most elegant, idiomatic way to express any program. Used the very best practices for testing, libraries, all that. He typed very fast.

* The former physicist who spent all his time reading obscure mailing lists on his favorite topics. His level of understanding of problems in his domains of interest was incredible.

I could go on and on! It's such a fun taxonomy to collect. All of these friends were marvelous at solving their particular flavor of problem.

As for myself, I like to think that my "trick" is to spend a long time poking at the structure of a problem. Eventually the solution I was looking for doesn't matter anymore, but the tools I developed along the way are pretty useful to everyone!

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.

A mathematician tricks.always. There is no point to untrick the trickster. Misuse of your time

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.

One of the answers links https://www.tricki.org/, which describes itself as a 'Wiki-style site with a large store of useful mathematical problem-solving techniques.' no longer maintained, but looks neat.

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.

For most of my computer science PhD the “trick” was just to get the inductive definition to work, and then how to tweak it for the next paper. Or, get enough structuret we can do an “abstract nonsense” proof[0].

[0]:https://ncatlab.org/nlab/show/category+theory#AbstractNonsen...

From a comment:

> These methods can be combined. First generalize the problem, making it more complicated. Then simplify along a different axis. – Stig Hemmer

Very relevant to software design too.

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