There is no way to pass through these obstacles (without spending the multiple lifetimes it took to forge them from first principles) except to memorize them and gradually extend understanding backwards from that memorization into the broader context of dependencies that converge into its formalization.
But having this predicament explained to me up front, ideally somewhere around the age one learns about something as basic as fractions, would have been enormously helpful.
That is not the only reason, but also a big part of it, yes. Some of the things (but not all) that I memorize I admittedly don't actually understand. But usually, the sudden realization comes at a later point, when I understood more of something else.
Literally happened last week to me. I had been memorizing a "stupid" theorem for a while[1], not realizing why it's useful, until something I read was about discontinuities in the n-th derivation of a function, and what that means for the terms of the function's Taylor series, and it all lit up in me, tying it all together.
It's a good feeling!
[1] https://www.dsprelated.com/freebooks/sasp/Spectral_Roll_Off....
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".
Not-memorizing forces you to practice more and develop deeper intuition which helps you reconstruct info. Memorization is a side-effect or learning, not a cause.
Our brains are connection machines, not encyclopedias.
But I got there by, at first, "blindly" memorizing FS, FT, DFT, DTFT, and Laplace and z transforms. This allowed me to not only apply them to problems while, at the time, not being fully comfortable with them yet, but also, and this is really the biggest part of the point I'm trying to make, to "look at them" while I was either solving problems or just furthering my understanding of the field when, say, lying in bed before falling asleep.
To an extent that would not have been possible if I had to "refer back" to their definitions on paper.
It was a tremendous tool (that is what memorization is, a tool) to just be able to recite, say, the DFT, or the DTFT, even while my intuition was still gradually building up. And to internally look at the differences between DFT and DTFT. The same is true for the countless other little factoids in my arsenal of flashcards. I literally built a lot of math understanding, both abstract and of actual engineering problems I was working on, when sitting in my car, on my 45 minutes commute to work or back home.
> Our brains are connection machines, not encyclopedias.
And memorization is a tool for making new connections.
I elaborated as a reply to a sibling comment, but repeat after me: You need to memorize things in math. In my experience, there is no way around it, especially if you want full understanding. It is a tool.
I realized this when I saw one of the higher math geniuses in university, one who really understood most things better than most of us, learn equations and lemmas from flashcards one day.
That memorization may become superfluous after you worked with something for a while, but until then, it is not just tremendously useful, but downright necessary, to make an equation, or set of rules, or whatever, just "pop up" in your head while you're thinking something through.
Relatedly:
> I needed to deeply understand it at the most universally root level and create a painstaking mental model of it
That is my modus operandi in math. I want to fully understand it, down to every little detail. But in my experience, memorization is one step for that model to really build itself up, and stick.
Reading through the pages of a math book and going "oh, yeah, I totally understood that, neat" is useless if you later encounter a problem and go "huh, so, what was the exact equation of the Fourier transform again"?
And not just because you now have to look it up to apply it, but also because an equation for example is not just a jumble of symbols that you write down and fill in. It has structure, it has meaning. If you can recite it in your sleep, then you also immediately see properties of it when they are relevant.
As Andrew Wiles said: "Math is not a spectator sport."
It's hard to fully bring across what memorization does for math, but since I started just using a flashcard app (Anki) several years ago, I literally sometimes lie in bed at night, eyes closed and no notepad, and work through math in my head, trying to further understand some aspects. And because I can "look" at what I memorized in my head, it works really well.