Well, algebra and calculus have been around for a while.
And then, Calculus took about 3 years for me to begin to scratch the surface... and realise I will never need it as numeric methods took over completely with the advent of cheap compute.
And then, Calculus took about 3 years for me to begin to scratch the surface... and realise I will never need it as numeric methods took over completely with the advent of cheap compute.
But there's understanding and there's understanding. If all you have is a piece of paper then there's no way around analytical approach. People used to be unbelievably good at it, including the applied crowd, like engineers.
Having matlab at hand makes it possible to save years of analytical tinkering and simplifications and approximations.
It is useful to understand what happens exactly but we no longer need 3-4 years of calculus depths... I mean, this used to be THE subject in most engineering programs round the world. These days it's more like bootstrapping the intuition.
And they use poor and imprecise approximations that they can't debug, because they just plug in numbers and hope for answer.
What point are you trying to make?
Yep, what you want is a good and predictable approximation.
> What point are you trying to make?
In practice most Matlab solutions you are describing are awful, in terms of performance and in terms of accuracy due to ignorance about underlying theory.
If you want to calculate a numerical solution to a novel analytic problem you absolutely need a good understanding of analysis. Being able to plug things into Matlab is only useful if Matlab implements a good solver for that problem.
>It is useful to understand what happens exactly but we no longer need 3-4 years of calculus depths.
Indeed. We need 5+ years of analysis now.
What if you want to make a numerical method for something you can't look up the recipe for?
Numerical methods are quite well-studied by now. If you need a new method or a variation then you probably specialise in these things, and that's a different question.
> The point is that 3-5 years of analysis might be an overkill
Rarely have I met an engineer and said, wow that guy knows TOO much about math. What do you think they should substitute?
But the amount of time available is fundamentally limited and it has to be balanced.
E.g., in my personal circumstances probability theory, statistics, deeper proof understanding would all be useful. I had all of these at varying depths.
But it's only analysis that we were getting for years and years and years...
And I see why and how this is such an important subject historically. It's just not that big anymore relative to other things.