[qubex]

I studied applied mathematics, but it took me ages to shake away (false) intuition engendered in me about “Calculus” and “infinitesimals” at high school. Sure, it worked, but learning “differentiation from first principles” with the limit taken when δ︎x→︎0 just by cancelling out did an unmeasurable amount of damage to my ability to absorb the formal Weierstrss formulation in terms of limits.

[romwell]

Congratulations. Now that you're a grown-up, you can re-do the damage by learning https://en.wikipedia.org/wiki/Non-standard_analysis

On a more serious note, you can understand most, if not all, of Calculus by saying that dx=0.0001, and that A ~= B if they don't differ by more than, say, 0.01 (say, that's the instrument error).

Then you get your limits, FTC, and so on, and verify the results with a four-function calculator.

Example: f=x^2, f' = ?

(f(x+dx) - f(x))/dx = (x^2 + 2x*dx + dx^2 - x^2)/dx = 2x + dx = 2x + 0.0001 ~= 2x

The mental effort you have to make here is that things on the LHS of ~= are "actual" values, and on the RHS are "measured" values, and that ~= is not an equivalence relation.

On a yet more serious note, learning about differential forms will help justify some of that high-school notation.

On a philosophical note, Weierstrass is not the end-all of Calculus. Neither Newton nor Leibniz did it that way. By adding rigor, some argue that the essence has been obscured (hence the non-standard analysis above).

[qubex]

Yeah, I’m aware of the numerics... but I’m also aware that if you do the same process backwards (naive numerical integration), for example for simulating planetary motions, you get into ridiculous situations where the collective momentum of a closed system rises exponentially after a ”close flyby”. This is precisely the kind of situation where ”false intuition” created by these shallow teachings cause the most harm.

[romwell]

I don't think it's harmful - without stumbling onto an example like that, it's hard to justify why we need solid foundations and exact solutions.

On the other hand, very brute numerics work for an awful lot of scenarios - that's why epsilon-delta came centuries after Calculus was invented.

For instance, "first-order optics" and "third-order optics" rise from chopping off the Taylor series after the 1st and 3rd term, resp. And it works! In many places, 1st order approximations are just good enough. A lot of scenarios are inherently stable.

So I don't think the intuition you build up is wrong - it just has a scope. There's nearly always a place for counter-examples where "things work the way you think they should" wouldn't apply, however you think about things :)

On a philosophical note, the continuity is a human construct - down there, things seem to be discrete, just with a very small step size. Continuity models these pretty well, until it doesn't - but that doesn't mean the intuition you build up is wrong. Just limited in scope.

[qubex]

I see what you are saying, but to each his own: I’m one of those kids (there's one in every mechanics class) that grow irate at the sin(ϴ︎)≈︎ϴ︎ approximation in the pendulum solution and then waste weeks using the full series expansion, only to find out the results differ only in the third or fourth decimal place. The thing is, I emerged from the experience thinking to myself “that was a useful learning moment”.