Really knowing things

• April 5, 2023

During undergrad, I got into competitive programming and noticed a strong tendency among contestants progressing quickly to spend almost all of their time actually solving problems. Contestants who progressed more slowly tended to spend a lot of time and energy on trying to memorize algorithms and then apply them to contest problems, rather than learning relevant algorithms “in the field.”

One of the top competitive programmers in the world, Um_nik, has a much-discussed tongue-in-cheek post titled Things I don’t know. At the end of the post, he says “if you know at least 3 of these things and you are not red — you are doing it wrong. Stop learning useless algorithms, go and solve some problems, learn how to use binary search.” Something about actually solving problems, thinking deeply about them, and internalizing their lessons helps one really know things.

On the other hand, it is very hard to theorize about how one might solve a problem. In the limit, people who spend almost all of their time learning new ideas but never applying them must do exactly that. It doesn’t work well. In some simplified but very real sense, they know lots of names, but don’t know what the objects the names point to really are.

Richard Feynman’s recollection of a conversation he had as a kid gets at something foundational about knowledge:

One Sunday all the kids were walking in little parties with their fathers in the woods. Then the next Monday we were playing in a field, and a kid said to me, “What’s that bird? Do you know the name of that bird?” I said, “I haven’t the slightest idea.” He said, “Well it’s a brown-throated thrush. Your father doesn’t teach you anything.” But my father had already taught me about the names of birds. Once we walked and he says “that’s a brown-throated thrush, in German it’s… in Chinese it’s… in Japanese it’s… and so on. But when you know all the names in every language of that bird, you know nothing, but absolutely nothing, about the bird.” And then we would go on and talk about the pecking and the feathers. So I had learned already that names don’t constitute knowledge. (source)

Names themselves are powerful and indeed even underrated. When something acquires a name, its essence is compressed, allowing for high-level discussions. It is for this reason that once something is discovered, it is given a name.

But the directionality is the point! Focusing too much on memorizing a name before grasping for oneself the thing the name describes is most often a mistake. The process of learning should follow the process of original discovery as closely as possible, and should deviate only when there is a better approach to discovery.

It is easy to confuse memorization of attributes with understanding. To arrive at fundamental, meaningful understanding one should focus on getting to a point where such attributes just make sense by questioning them, playing with them, and connecting them with everything one already knows about the world. This process itself seems to have a name: discovery learning.

Explanations

For people who focus on understanding in this way, the ability to explain things simply comes almost for free. Roughly, my model is that this ability emerges from extraordinary knowledge transfer – imagining all concepts that you know something about as nodes in a graph, the more edges to which you can connect a new concept, the more simply you’ll understand it, and in turn, be able to explain it.

If you practice discovery learning, you’ll find integrating new nodes easier since your whole knowledge graph is highly connected. On the other hand, if you don’t spend time trying to connect with other nodes and instead opt to memorize, you’ll end up with components of the graph that are much more isolated.

An implication of these claims is that we should generally expect someone who is at the very top of their field to discuss its ideas in ways that are less complex because their understanding is deep enough to do so. In a 1960 interview with the BBC, Orson Welles, creator of Citizen Kane, describes this phenomenon exactly when explaining the advantages of working with his cameraman:

… and of course again I had a great advantage not only in the real genius of my cameraman but in the fact that he, like all great men, I think, who are masters of a craft, told me right at the outset that there was nothing about camera work that I couldn’t learn in half a day, that any intelligent person couldn’t learn in half a day. And he was right… and I was lucky enough to be told that by the absolute best living cameraman. Like a doctor – you know very good doctors often tell you, “you know, we don’t really know anything much about medicine.” I’ve noticed an awful lot of good doctors do talk that way, but only the very good ones. (source)

This is not an attack on camerawork or medicine; Welles’ point is incredibly broad. A mid cameraman finds the sheer magnitude of angles, scenes, props, and lighting decisions impossibly complex, and necessarily conveys camerawork through such a lens. The very best, on the other hand, finds it all very natural, and indeed is able to convey camerawork as such to intelligent inquirers.

Extremely successful people have an extraordinary hold on the fundamentals, and that is precisely what makes them great. Most content in the world is unnecessarily complicated because its explainers have never experienced its discovery for themselves and thus are not capable of conveying it in the way it would best be discovered.¹

There is a tendency among such explainers to be proud of this fact, as if the horrifying web of arduous understanding they’ve achieved is a triumph. In reality, it is almost always a sign that some piece of their understanding could be made more complete, and the very best cameramen and doctors would tell you the same.

Discovery as a spectrum

As always, there are tradeoffs. Perhaps literally discovering the entirety of human knowledge from first principles would indeed give you the best possible understanding, but you’ll probably want some hints to steer your search towards better paths of discovery. At the same time, you want to avoid looking at the answers in the back of the book too much.²

If you do end up stuck, having tried something yourself before seeking out an explanation lets you transfer the discovery process of the explanation back into your own. If you never try, there is little hope of using an explanation to update on what your next step should’ve been. When learning something in a brand new domain, you can do something similar: start by figuring out which of its ideas are just “translations” of ones you already know.

I find that the optimal way for me to understand something new is to play around with it, to try to learn-by-doing, and ideally to have the person I know who understands the thing best close-by to answer any questions or keep me from getting stuck. I am lucky in that over time the percentile of “person I know who understands the thing best” has grown tremendously, and I plan for this to continue.³

There is an analogy to technical debt here, where when you need to “get” something quickly, you’ll focus on building up a less-than-ideal understanding that at least suffices. But over time, if you don’t tackle this debt, you might look back and realize that things are quite fragile.

The will to understand

If indeed discovery learning is so strong an idea, then why isn’t it more popular? It’s hard! It takes much more will to make attempts that often fail at integrating new knowledge from first principles than to stuff new facts into the brain in a minimally viable way for them to “serve their purpose.”

But there are some people for whom the perception that they do not fully understand something is so annoying that their will to change this is enormous. While writing this, I came across Nabeel Qureshi’s excellent essay on How To Understand Things, and he concurs:

The smartest person I’ve ever known had a habit that, as a teenager, I found striking. After he’d prove a theorem, or solve a problem, he’d go back and continue thinking about the problem and try to figure out different proofs of the same thing. Sometimes he’d spend hours on a problem he’d already solved.

Intelligent people simply aren’t willing to accept answers that they don’t understand – no matter how many other people try to convince them of it, or how many other people believe it, if they aren’t able to convince them selves of it, they won’t accept it. (source)

I am not sure how much of this trait comes from nature or nurture, but in my own experience it seems possible to deliberately cultivate it. To the extent that one can, I am confident that it is a trait worth cultivating.

“Playing football is very simple, but playing simple football is the hardest thing there is.” - Johan Cruyff

Thanks to Alex Labbane, Daniel Mota, Logan Baker, and Nabeel Qureshi for giving feedback on drafts and contributing ideas.