In my experience, it helps to approach from both sides, and that's also part of the point I'm trying to make. If you come only from the "concrete examples and a

In my experience, it helps to approach from both sides, and that's also part of the point I'm trying to make. If you come only from the "concrete examples and applications" side, you might not actually fully understand, despite thinking you do, and miss some finer subtleties. Those are often the "gotchas" that get pointed out in textbooks (but even then are easily forgotten). I am definitely a "hands on" math person. Almost every piece of math I learn has application ultimately, personally I would not really be motivated otherwise. I learn math because I want to use it. I then also find joy in the abstract beauty, but I would not uncover that beauty without something I could use it for in the first place in mind. For me specifically, a large application of the math I'm learning is signal processing (both digital and analog), and analog circuits (e.g. "EE stuff"). And in that, some of the learning of "meaningless abstractions" first, and later entangling it with actual application, definitely, positively helped in getting a good understanding of the underlying math, which I can in turn integrate with what I'm doing in the "real world".