My lecturer was once unable to solve a problem at the blackboard.
After trying for some time, he said he needed a short break to go back to his office to look at his notes. He brought the notes, and there he found a hint he had written for himself: "Use a trick"
Every mathematician has only a few tricks (2020)
31–40 of 87 posts
Re: Every mathematician has only a few tricks (2020)
#32My 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 r…
Nobody has enough memory or patience for third order derivatives so Newton's method (aka Newton Raphson) it is.
Re: Every mathematician has only a few tricks (2020)
#33For programmers I would say graphs or at least thinking in graphs is a common one.
Some would say SAT is also a handy trick but I’ve personally never used it.
Re: Every mathematician has only a few tricks (2020)
#34My 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 r…
My dad was also an Engineer. He was also a Fortran fan.
Once he asked me to explain OOP. After I explained the basics he said it was useless and never looked back.
Re: Every mathematician has only a few tricks (2020)
#35My 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 r…
My dad was also an Engineer. He was also a Fortran fan. Once he asked me to explain OOP. After I explained the basics he said it was useless and never looked back.
Chad Dad
Re: Every mathematician has only a few tricks (2020)
#36My impression from math lectures as a CS student was, that you have to know the tricks, otherwise you just sit there dumbfounded, not knowing how you can make progress. Obviously, there will also be some kind of mathematical understanding of things at play, but a lot of the math stuff is pattern recognition and applying tricks. If you don't know them, it becomes very frustrating. I've not ever seen a way of teaching math though, that focuses on teaching learners these tricks or techniques. They always fell out of things on the side or the teaching person assumed you know it, or was surprised, that the learning people were not able to see it, or able to apply some trick. Or they just didn't care, as long as a small portion of the learners somehow were able to solve it, because then it obviously cannot possibly have been their way of teaching, but other students must have been lazy...
Re: Every mathematician has only a few tricks (2020)
#37Earlier quoted context omitted.
Relatedly, I've found Newton-Raphson is a great example of an algorithm where Knuth's "I have only proven it correct, not tried it" rears its head very prominently. The obvious implementations can work flawlessly on toy examples and then fail miserably on real-world examples.
There's a reason why numeric analysis is still actively studied by research mathematicians. If we could just throw something as simple as newton's method at any nonlinear problem, we'd only need people to learn this once in school and everyone could solve everything.
Yeah for that you need Euler’s method… I mean of course Runge-Kutta. … By that I’m of course referring to rk4. … I mean, you have a point.
Joking aside I think having a few basic numerical methods like say Newton, rk4, brent root, monte carlo simulation etc in your general toolbox of techniques you know how to do can make you unreasonably effective in a wide range of situations. Just yesterday I had to solve a problem in a relatively small space so I first used a brute force method to check all the feasible solutions and having got the full list of actual solutions out, figured out the analytical solution. It meant I could be very confident that my analytical solution was correct when I had it.
Re: Every mathematician has only a few tricks (2020)
#38My 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 r…
My dad was also an Engineer. He was also a Fortran fan. Once he asked me to explain OOP. After I explained the basics he said it was useless and never looked back.
I guess as a typical engineer (not CS person type of engineer or software engineer) it is easy to think that. One might be working with machines or buildings and so on, all which require _calculation_ of processes. Those are typical cases for "just write a correct function", possibly one that takes many things into account. For such scenario OOP is truly useless and only over-complicates the matter. However, when we get to simulations, where maybe there is no known formula, or the precise calculation would be too expensive, then OOP can make sense. Doesn't have to, but could.
Re: Every mathematician has only a few tricks (2020)
#39Earlier quoted context omitted.
Relatedly, I've found Newton-Raphson is a great example of an algorithm where Knuth's "I have only proven it correct, not tried it" rears its head very prominently. The obvious implementations can work flawlessly on toy examples and then fail miserably on real-world examples.
There's a reason why numeric analysis is still actively studied by research mathematicians. If we could just throw something as simple as newton's method at any nonlinear problem, we'd only need people to learn this once in school and everyone could solve everything.
But I'm not sure I'd recommend going into the field. There's something demoralizing about doing research on something which already has dozens of valid and successful methods, of which you are trying to create a slightly more optimized version.
Re: Every mathematician has only a few tricks (2020)
#40Earlier quoted context omitted.
There's a reason why numeric analysis is still actively studied by research mathematicians. If we could just throw something as simple as newton's method at any nonlinear problem, we'd only need people to learn this once in school and everyone could solve everything.
But I'm not sure I'd recommend going into the field. There's something demoralizing about doing research on something which already has dozens of valid and successful methods, of which you are trying to create a slightly more optimized version.
It's not for a field for everyone.
But that "slightly more optimized version" may mean "one that does not quietly produce disastrously incorrect results for some input values".