> "physical world" is where the symbolic approach completely breaks down.

One needs to be able to solve problems that have all but the most essential details stripped out in order to develop a sense of how physical law actually works. Many times that is even "good enough" to get to a solution.

The best way to do that is through analytic methods, which give not only "an answer" but also tell you important features of the answer. These analytic solutions have "handles" you can use to ask "what-if" questions -- eg zero's in the denominator to indicate poles, behavior of the system as you take certain limits, geometric aspects such as symmetry, patterns in recurrence relations, etc, etc, etc..

As you see it clicks for many people, and doesn't mean much for others. Back then it was a very welcome click.

After all, it should be possible to find antiderivatives and solve ODEs without doing the dance with substitutions and change of variables, but still, we find it many magnitudes easier than just looking at it and divining the correct solution.

Usually, whoever does the derivation, or someone who wants to understand things properly, will do computations on multiple steps of the derivation from the start to the finish. A lot of these computations can be done by hand - you don't need a computer. A lot of computations should be done by hand - even if they could be done by a computer - because you only get a feel for the equations if you play with them with your hands. To quote Dirac, 'I consider that I understand an equation when I can predict the properties of its solutions, without actually solving it.' That comes from solving a lot of them by hand.

Yes, oftentimes, doing numerical or symbolic computation with a computer helps. But is the pain point of that having to type the equation into the computer. Hardly. It would be nice, but nothing ground breaking.

That explains everything :-)

You probably don't solve quadratic equations anywhere near as much as an engineer/physicist would, and so prefer less memorization and more "being able to derive when I need it". The latter is just too time consuming to do it every time.

I'm guessing you don't recall off the top of your head what the integral of 1/sqrt(a^2-x^2) or of 1/(a^2+x^2) is (unless you happen to teach Calc II), but most physicists likely do as they encounter it often and likely just memorized it.

Also, in Physics, a lot of ODE are mysteriously integrable if the variable is x instead of t. (One reason is that it's easy to measure the force/fields, but the "real" thing are the potential, so you are measuring the derivative of a hopefully nice object.)

Also a lot of the theoretical advanced stuff to prove analytical solutions and to estimate the error in the numerical integrations use the kind of stuff you learn solving the easy examples analytically.

And also historical reasons. We have less than 100 years of easy numerical integrations, and the math curriculum advance slowly. Anyway, I've seen a reduction in the coverage of the most weird stuff like the substitution ?=atan(x/2) (or something like that, I always forget the details). It's very useful for some integrals with too many sin and cos, but it's not very insightful, so it's good to offload it to Wolfram Alpha.

Unfortunately, engineers insist on designing devices which are neither perfect spheres nor perfectly flat planes (ridiculous!), and they might only have equations to describe how properties change in time or space. In this case, it can be easier to discretize the thing into a mesh, and use a matrix to describe how the physical phenomena at the points of that mesh relate to each other.

Consider the most important function of calculus - the integral. In layman's terms, it measures the area under a graph. Okay - that's a little bit useful, if you care about the physics of moving objects (A bus is accelerating at 2 m/s^2 for 5 seconds, how far does it travel..?)

Yet, if you know how to integrate, a mountain of not-immediately obvious physics problems - say, anything that has to do with electromagnetism (Maxwell's equations) immediately become tractable.

To directly explain the mechanics of what should otherwise be simple or everyday phenomena (like with three-particle orbits or the flow of water in your bathtub) often requires many equations and parameters. Since physical systems tend to be not-so-heterogeneous in structure, we can use experimental insight and exploit symmetries to reduce the number of equations and parameters up front. But this amounts to finding ways to simplify because we don't understand the precise system in play, only an analogous one.

For larger systems we have historically relied on equation solvers to do the reductions and find solutions. But there are systems for which the dimensionality cannot be easily reduced, like with language and vision. Then to be precisely predictive about these high-dimensional outcomes, we still need to compute beyond our ability to reduce the equations from the outset.

This all converges on deep learning - since in the end, it's much of the same linear algebra used by solver packages, just recursively applied to enormous high-dimensional datasets. Maybe calling it intelligence gives more agency to the process than we can stand to attribute. But in many ways it's just an extension of our usual ways of mathematical modeling before computers, but moving into so many dimensions and datapoints that it produces outcomes similar to the behavior we use our entire brain to recreate instinctually (like image recognition and language use).

What happens is you learn numerical approximation methods and then recreate the tables with human 'computers', like we used to do before electrical computers came into play. Or create mechanical analog computers like they did in the 1800s or greek times.

In the mechanics library that comes with the book (which I'm building on in my own work), functions can take either numerical or symbolic values. If you have a computation involving symbolic values, you can manipulate it just like you would with pencil and paper (except you can operate at a higher level of abstraction, never get writer's cramp, and never have to laboriously recopy line after line of symbols to make sure you got the right number of minus signs). If, as so often happens, you find yourself up against an intractable integral, you pass the whole thing to a numerical solver and get a number back right away. With enough calculations you can build up a qualitative understanding of the system's behavior. My understanding is that this is what mechanics people do all day, but not how mechanics is taught to newcomers.

[1]. https://arxiv.org/abs/1205.5935 [2]. https://en.wikipedia.org/wiki/Nonstandard_calculus

I know exactly what you mean. I can see stuff in physics and I intuitively know what is going to happen when I take a metallic hollow piston with air inside of it and place it partially in a magnetic field. It's observation of Solenoids and other stuff that allows me to do that, but I really couldn't do the same with math, and now I am trying to do that.