people lack nuance with the word βunderstandingβ β there are multiple resolutions / zoom levels and itβs normal that mathematicians donβt have 100% internalized understanding of every theorem loaded because nobody does
for instance i know that if i call sorted(scores), i get the scores back in order. one level down i understand how it builds sorted runs and merges them. further down iβm working through the invariants that prove why this produces a sorted result without losing or duplicating anything. i donβt need all of that loaded to build a leaderboard, i need to understand the guarantee iβm using and when it applies
same with a theorem β they need to understand what it lets them conclude and the conditions where it applies. they donβt need to understand every dependency to its core, they donβt need to rederive everything, they can rely on other people having checked it and go into the proof when they need to
engineering would not work if everybody forced full resolution for everything. the resolution you need depends on what youβre trying to do, and that can change when something breaks
the reason why we want our experts to understand stuff deeply is because we put them in situations where they need to move up and down the scales / zoom levels of magnification and debug stuff and diagnose which level the problem is at, or go 5 levels deep to find the cause. that doesnβt mean they need to keep all 5 levels fully loaded every time they use the system
just trying to reverse engineer how a mathematician works
like I wasn't aware they actually did that; take a well known theorem and then obsessively apply it to your own work, without really understanding the underlying proofs of that theorem
so, in a way, every mathematician is working on their own slice of the mathematical space, which is shaped by the theorems they are familiar with. and then human intuition kicks in and they start noticing patterns in that space, which allows them to prove something new
so the process of aggregating knowledge could be described as kind of a random search on top of a space you're familiar with. and the raw symbolic knowledge loop is, thus, just: we accumulate a book of theorems (memory), we search the proof for new theorems on top of it (compute), then we drop the proofs, put the new theorems in context, and go solve something else. then loop