How to Understand Things

Heh I get this a lot in my daily sysadmin/developer duties. When you need to turn on a knob somewhere you don't just wanna know that you have to turn it on, you want to know why you need to turn it on and follow the chain up until you get to facts you already know. But it's not always possible to get that far, there's too many layers of abstraction, the source code is not available or you just don't have the time.

It’s possible. It just takes time; very few people are in a position to devote that time.

This gives an advantage I haven’t seen discussed: when you put in the time, you make connections no one else thought of. It happens time and again, and it’s a clear pattern at this point.

It takes months of daily study, often tedious, with no clear benefit. But the benefits sometimes come. (I wrote “usually” rather than “sometimes,” but that’s not really true. The usual result is that you go to sleep more confused than you started. It’s not till much, much later that the connections even seem relevant.)

Great post. Personal anecdote:

I don't think I really understood anything in school, but I was decent at going through the motions of carrying out certain methods and recalling certain facts when I needed to.

I went on to study Maths at university, and for most of my first year, I had the same surface level "methods + facts" knowledge that got me through school. After some studying, I could recite definitions and theorems, I'd memorised some proofs, and I could occasionally manipulate a problem to get an answer. I think about half of the cohort was in a similar position. But it was clear that there were others in a completely different league.

When we were studying for our first year exams, I was struggling to remember the proof of a specific theorem (it felt quite long). A friend was trying to help me learn it, and he asked me what "picture" I had in my head for the theorem. I didn't have any pictures in my head for anything.

It turned out that a simple drawing could capture the entire statement of the theorem, and from that drawing, the proof was trivial to derive. It was long-ish to write out in words, sure, but the underlying concept was really simple. This blew my mind — I realised I didn't have a clue what we'd been studying the whole year.

The worrying thing is that I actually thought I understood that stuff. Before that incident, I didn't know what it feels like to actually understand something, and I didn't have an appreciation for the layers of depth even within that. I suspect lots of people go through the entire education system like this.

OP here -- that is a fantastic anecdote!

Do you have an example of a "drawing" of a theorem, in this context? (I've seen these for fairly trivial theorems but not for more complex ones, so I'm curious.)

I can draw a parallel to this in software development. Some product features that require development "from scratch", where you can get down to the original code and logic - this is where "taking time to think" really pays off.

But when you are basically composing a final product from components, libraries and features - this is where figuring something out may take really long time and a lot of effort. It today's world many libraries are open source, so you actually can get to the bottom of many issues. But the time and effort cost of that is almost never acceptable.

My conclusion is - if you are a "slow" thinker, prefer getting to the bottom and figuring stuff out - try and choose the "fundamental" type of work. Where you are "done is better than perfect" kinda person - you'll thrive in the upper layers of development stack where shipping stuff out is of utmost importance. Focus on your strengths.