My stance in general is not that we shouldn't teach students why things work; just that rigorous proofs aren't at all a good way to do it. Indeed, as far as I c

My stance in general is not that we shouldn't teach students why things work; just that rigorous proofs aren't at all a good way to do it. Indeed, as far as I could tell, most of the mathematicians who taught math classes I took had at best a rudimentary knowledge of how to use any of the theory they were teaching to do anything. I studied physics in undergrad and the general method was, yes, it involved detailed understanding of all the theory, far better than anyone learned it in the relevant math class (differential equations, multivariable calculus, and linear algebra, in particular --- a 1-semester course in QM has to teach everyone linear algebra because the math course on the subject is garbage; a 1-semester course in E&M has to teach multivariable calculus also because the math course was garbage, etc.) But yeah, absolutely you have to really learn e.g. group theory and algebraic number theory to do cryptography. I actually quit engineering and switched to physics precisely because I was so annoyed that an intro electrical engineering class used the Fourier transform as black magic without bothering with why it worked. In particular many of the exact theorems that are used in formal math are totally irrelevant and historical accidents. Calculus could have been developed in a thousand different ways; we happened to pick one, the particular names of the theorems or what order the results are constructed in is, IMO, totally irrelevant to anything outside pure mathematics (particular examples: which construction of the integers you use; which construction of the real numbers you use; which definition of the derivative; which definition of the integral, stuff like that. In the case of integrals, yes, different constructions have different theoretical properties in terms of which functions can be integrated --- but, in any application, the answers intrinsically can't be affected based on your choice of definition, so the actual definition can't matter.