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Depends if you want to teach a mathematician or an engineer or a physicist. If you only need an engineer, numerical approximation is good enough. You teach abo

Depends if you want to teach a mathematician or an engineer or a physicist. If you only need an engineer, numerical approximation is good enough. You teach about slopes, and area under the curve. Automatic differentiation and numerical quadrature. Alternatively you can teach about function representation, by introducing Taylor and Fourier series, and then you have an easy closed formula to integrate and differentiate a function. If you want to teach a mathematician, integration by part is a great first introduction to the freedom of choice that you encounter when playing the find the proof game. There may be multiple paths to the solution. If you want to teach a physicist, you have to teach change of variables, and how to strike the infinitesimals to simplify them. Also teach them variation of constants and separation of variables. If you want a statistician or financier, just teach Monte-Carlo integration and Ito's formula. Education is pick and choose, depending on what you need it for.

iancmceachern 2y ago
The challenge is you often have all 3, and more, all om the same class.
- GistNoesis 2y ago
  Then it's more about gardening, than mathematics. Weed-out bad elements and behaviors, provide rich fertile grounds and nutrients, a direction to grow towards and let them grow by themselves. Promoting curiosity, building inner representation of abstract concepts and learning by doing with examples is usually a good recipe. But if you are more into agriculture, a combine harvester also works great.
burnt-resistor 2y ago
> If you only need an engineer, numerical approximation is good enough. No. Completely wrong. Engineers and scientists are given the very same rigorous fundamentals of mathematics. We (EE/CS) also had to take the very same weeder courses as mathematicians including abstract mathematics and concrete mathematics, in addition to modern physics for physicists and engineers, followed by the entire computer science and electrical engineering series. We did learn the Monte Carlo method for problems that may not be directly computable in reasonable time, but these are approximations that can be achieved with a qualified amount of error. In fact, I helped port a FORTRAN-based nuclear reactor simulator for UNIX to Windows that was, at its core, a quadruple integral Monte Carlo approximation considering configurations of voxels of matter and voids over time with their various properties. We started with the fundamental in high school on paper with area under the curve and learning mathematical transformations of integrals, and derivatives as rate of change. The Erable CAS found in the HP 48 helped provide transformations and reductions sometimes, but not every time for basic calculus, vector calculus, and ordinary diff eq. (We didn't have MAPLE, Mathematica, or GIAC which are far more powerful.) Finally, in college, we ended up with discrete FFTs and Laplace transforms in EE and the Schrödinger eave eq for hydrogen in physics. This isn't as far as physicists or mathematicians go in grad school such as groups, fields, and such. So, I think you need to reevaluate your assumptions about curricula that seems wholly inadequate.
- GistNoesis 2y ago
  Rather than talking about engineers, I am mostly speaking about the subject which is calculus, aka the study of continuous change, aka infinitesimal change, aka derivatives and integrals. At first order the engineer need to be able to provide some sort of solution to any problem and get something done. The numerical toolbox is kind of an easy reliable Swiss knife that will get you through anything related to derivatives and integrals. It Ten years down the line when your nuclear core is melting and you need to pull the right lever will you be confident enough to trust your rusty manual computations tricks and not miss some minus sign somewhere in the transcription of your derivative. The derivative and integral game, is kind of an interesting mathematical artefact about the study of functions from before the time we had computers. As far as I know it's not yet a solved game. It's more about heuristics and picking rules than using a specific formula. Rule based Integration https://rulebasedintegration.org/ https://rulebasedintegration.org/ is a kind of collection of all the tips and tricks to play this game, but even then it's not foolproof. Closed formula are nice to have but hard to get by in real problems. The thing which makes the difference between something that is nicely behaved or not, is whether or not the operations are closed ("closure" concept). When it's the case you can build some nice algebra with nice properties. The computations becomes deterministic and formulaic. When it's not you end-up playing a NP-complete game to find the solution. For example in the case of the study of infinitesimal rotations, the natural extension of calculus for physics, we have the Lie Algebra. Even there your numerical toolbox will be more useful than your pages of Feynman Diagrams.
lagrange77 2y ago
So you propose to teach them different methods of solving. I think, while your mapping from solution methods to professions makes sense, the act of solving should be taught as an afterthought. The goal should be to build a robust mental model. Personally i find playing around with computer visualisations helps a lot with that. The same is even more important for differential equations, imo. I wish i would have spent less time studying algorithmic DE solving techniques in my school days. The beauty and power of expression of these has nothing to do with the way you solve them, if you ask me.
- GistNoesis 2y ago
  The whole education can be seen as a sieving process routing people towards their natural abilities as needed by the economy. It starts very early with things like mental calculus, where you try to get kids to compute things like 99 times 101, 98 times 102, finding tricks and rules to make the computation "easier". We could instead teach them to follow mentally some multiplication algorithm, but instead we try to promote some exploration and looking for pattern. In chess, similarly you can train for puzzle to develop some vision without having to calculate, like multiplication table. The education system must also take care not to prevent it from adapting to further changes by being too rigid. If the cat can't get out of the box, the box won't ever grow bigger. But you can't have too many cats getting out, or you'll soon have an empty box and hangry cats. I don't know what the goals should be, far from me the idea of proposing to teach, I only show paths that can be taken.
  - lagrange77 2y ago
    I'm not sure i understand your answer, to be honest, especially the cat fable. > as needed by the economy Even if that is the purpose of education, a mathematician, engineer or physicist who has a deep understanding is worth more to the economy, than one who can remember and perform algorithms and computations, isn't he?
    - GistNoesis 2y ago
      From the individual intellectual point of view it's better to have a better education, but from the collective point of view having too many wild cards is not good for anyone. If there are too many physicists, they won't earn enough to justify the cost of the studies, and won't be happy in a menial job after spending years learning advanced mathematics. If anybody can do everything interchangeably, it also tends to create a monoculture which is a more fragile system. Often it's better not knowing how the sausage is made, and trusting that people who have specialized know what they are doing. Wild cats ("overeducated population") are dangerous.
RhysU 2y ago
> If you only need an engineer, numerical approximation is good enough. Fluid dynamics enters the chat. And whaddya mean "only" need an engineer?
- GistNoesis 2y ago
  Even in fluid dynamics the game is only as hard as you make it for yourself : https://en.wikipedia.org/wiki/Lattice_Boltzmann_methods https://en.wikipedia.org/wiki/Lattice_Boltzmann_methods You have well defined operations and conservation laws well behaved by construction. Don't let Navier & Stokes enter the chat, and you will be fine. The whole problem of integration is that mathematicians historically introduced new functions, and try to relate them together, in interesting way. Polynomials are good and well behaved. Algebras too.
  - deleted 2y ago
    [deleted]

iancmceachern 2y ago

The challenge is you often have all 3, and more, all om the same class.

GistNoesis 2y ago
Then it's more about gardening, than mathematics. Weed-out bad elements and behaviors, provide rich fertile grounds and nutrients, a direction to grow towards and let them grow by themselves. Promoting curiosity, building inner representation of abstract concepts and learning by doing with examples is usually a good recipe. But if you are more into agriculture, a combine harvester also works great.

burnt-resistor 2y ago

> If you only need an engineer, numerical approximation is good enough. No. Completely wrong. Engineers and scientists are given the very same rigorous fundamentals of mathematics. We (EE/CS) also had to take the very same weeder courses as mathematicians including abstract mathematics and concrete mathematics, in addition to modern physics for physicists and engineers, followed by the entire computer science and electrical engineering series. We did learn the Monte Carlo method for problems that may not be directly computable in reasonable time, but these are approximations that can be achieved with a qualified amount of error. In fact, I helped port a FORTRAN-based nuclear reactor simulator for UNIX to Windows that was, at its core, a quadruple integral Monte Carlo approximation considering configurations of voxels of matter and voids over time with their various properties. We started with the fundamental in high school on paper with area under the curve and learning mathematical transformations of integrals, and derivatives as rate of change. The Erable CAS found in the HP 48 helped provide transformations and reductions sometimes, but not every time for basic calculus, vector calculus, and ordinary diff eq. (We didn't have MAPLE, Mathematica, or GIAC which are far more powerful.) Finally, in college, we ended up with discrete FFTs and Laplace transforms in EE and the Schrödinger eave eq for hydrogen in physics. This isn't as far as physicists or mathematicians go in grad school such as groups, fields, and such. So, I think you need to reevaluate your assumptions about curricula that seems wholly inadequate.

GistNoesis 2y ago
Rather than talking about engineers, I am mostly speaking about the subject which is calculus, aka the study of continuous change, aka infinitesimal change, aka derivatives and integrals. At first order the engineer need to be able to provide some sort of solution to any problem and get something done. The numerical toolbox is kind of an easy reliable Swiss knife that will get you through anything related to derivatives and integrals. It Ten years down the line when your nuclear core is melting and you need to pull the right lever will you be confident enough to trust your rusty manual computations tricks and not miss some minus sign somewhere in the transcription of your derivative. The derivative and integral game, is kind of an interesting mathematical artefact about the study of functions from before the time we had computers. As far as I know it's not yet a solved game. It's more about heuristics and picking rules than using a specific formula. Rule based Integration https://rulebasedintegration.org/ https://rulebasedintegration.org/ is a kind of collection of all the tips and tricks to play this game, but even then it's not foolproof. Closed formula are nice to have but hard to get by in real problems. The thing which makes the difference between something that is nicely behaved or not, is whether or not the operations are closed ("closure" concept). When it's the case you can build some nice algebra with nice properties. The computations becomes deterministic and formulaic. When it's not you end-up playing a NP-complete game to find the solution. For example in the case of the study of infinitesimal rotations, the natural extension of calculus for physics, we have the Lie Algebra. Even there your numerical toolbox will be more useful than your pages of Feynman Diagrams.

lagrange77 2y ago

So you propose to teach them different methods of solving. I think, while your mapping from solution methods to professions makes sense, the act of solving should be taught as an afterthought. The goal should be to build a robust mental model. Personally i find playing around with computer visualisations helps a lot with that. The same is even more important for differential equations, imo. I wish i would have spent less time studying algorithmic DE solving techniques in my school days. The beauty and power of expression of these has nothing to do with the way you solve them, if you ask me.

GistNoesis 2y ago
The whole education can be seen as a sieving process routing people towards their natural abilities as needed by the economy. It starts very early with things like mental calculus, where you try to get kids to compute things like 99 times 101, 98 times 102, finding tricks and rules to make the computation "easier". We could instead teach them to follow mentally some multiplication algorithm, but instead we try to promote some exploration and looking for pattern. In chess, similarly you can train for puzzle to develop some vision without having to calculate, like multiplication table. The education system must also take care not to prevent it from adapting to further changes by being too rigid. If the cat can't get out of the box, the box won't ever grow bigger. But you can't have too many cats getting out, or you'll soon have an empty box and hangry cats. I don't know what the goals should be, far from me the idea of proposing to teach, I only show paths that can be taken.
- lagrange77 2y ago
  I'm not sure i understand your answer, to be honest, especially the cat fable. > as needed by the economy Even if that is the purpose of education, a mathematician, engineer or physicist who has a deep understanding is worth more to the economy, than one who can remember and perform algorithms and computations, isn't he?
  - GistNoesis 2y ago
    From the individual intellectual point of view it's better to have a better education, but from the collective point of view having too many wild cards is not good for anyone. If there are too many physicists, they won't earn enough to justify the cost of the studies, and won't be happy in a menial job after spending years learning advanced mathematics. If anybody can do everything interchangeably, it also tends to create a monoculture which is a more fragile system. Often it's better not knowing how the sausage is made, and trusting that people who have specialized know what they are doing. Wild cats ("overeducated population") are dangerous.

RhysU 2y ago

> If you only need an engineer, numerical approximation is good enough. Fluid dynamics enters the chat. And whaddya mean "only" need an engineer?

GistNoesis 2y ago
Even in fluid dynamics the game is only as hard as you make it for yourself : https://en.wikipedia.org/wiki/Lattice_Boltzmann_methods https://en.wikipedia.org/wiki/Lattice_Boltzmann_methods You have well defined operations and conservation laws well behaved by construction. Don't let Navier & Stokes enter the chat, and you will be fine. The whole problem of integration is that mathematicians historically introduced new functions, and try to relate them together, in interesting way. Polynomials are good and well behaved. Algebras too.
- deleted 2y ago
  [deleted]