Another comment on this thread talked about the difference between a mathematical mindset and an engineering mindset, and I think that's exactly right. As a re

Another comment on this thread talked about the difference between a mathematical mindset and an engineering mindset, and I think that's exactly right. As a researcher and controls engineer, I certainly appreciate the necessity of rigorously proving the behavior of new algorithms. On occasion I develop those proofs myself. I'm not very good at it, and I appreciate those who are. But. Compared to the amount of time I spent in grad school honing my ability to generate proofs (courses in both real and numerical analysis, on top of the very applied math-flavored controls courses), the amount of time I spend either reading or producing stability proofs is minuscule. On the other hand, I spend an enormous amount of time solving constrained optimization problems. Both require calculus. But guess which topic I never got formally taught in grad school? The point being that while I appreciate proofs and mathematical rigor and clever algebraic manipulations, including changes of variables, in practice not only do I virtually never use those things, but the mindset I need when approaching an engineering problem is really quite distinct from the mindset I had to adopt when working my way through real analysis. I'll go one more step and say that in a lot of cases the two mindsets can actually be mutually exclusive. My advisor was brought up in a nonlinear controls lab where rigor was everything. Today, controls is being consumed by deep reinforcement learning, which is driven primarily by experiment. We get an idea for a new approach that might work, we try it out and see whether it works. We then go back and try to understand mathematically why it works. My advisor is unable to work that way; he must have things proven first, and only then will he even look at experimental results. He's been completely left behind by the field. He's unable to brainstorm about new things to try, because he can't accept that experimentalists can move faster than theoreticians. We've had vigorous arguments about this. I think that if enough experimentalists can show repeatable results, that's a good indication that mathematically there's something there worth investigating. He just thinks the experimentalists are all building a house on sand.