Your phrase "or even what it means to understand that thing" strikes a chord with me. As someone studying math and computer science on my own, without anyone a

Your phrase "or even what it means to understand that thing" strikes a chord with me. As someone studying math and computer science on my own, without anyone around who shares my research interests, I have to figure out from first principles not only how to gain an understanding of things but define what understanding it even means. For example, I have recently been pursuing the idea of identifying data streams with bijections with the natural numbers, and higher-order operations on these bijections. The bijection is arbitrary, so the order assigned the sequence "doesn't matter" (the subject of study happens to be associative and commutative in its pure form). However I ran into an inconsistency with this when modeling products of streams, and came to an epiphany: yes, the order (the specific bijection) is arbitrary BUT it is fixed: that is, you can pick any order, but it has to be committed to a priori. (An intuitive analogy would be to consider the difference between a card game where the dealer shuffles the deck before the game and deals from the top, versus a card game where the dealer selects the cards he wants to give you as the game proceeds.) I felt like that was a moment when I understood what it means to understand this thing I'm studying which is so abstract. I saw it as a fork in the road between mathematics and mysticism: prior to that realization I had been studying mysticism, "assume the (nondeterministic) computer 'magically' chooses the best next item in the sequence to pursue" versus "assume the sequence is a priori in the best order to search (but that order is fixed)". The former is mysticism because it asserts something exists without pinning it down. The latter is mathematical because, although it describes something abstract, the properties of the thing it's describing do not shift during the course of working with it.