My mental model of learning | German Capuano

My current employer has a seemingly inoffensive question in its job<br>applications. "What’s your most memorable meal?" It's an opportunity<br>for applicants to tell us about their personality. These days, most<br>applicants lose points for obvious AI-generated answers. The following<br>is one of the common variants.

"My most memorable meal was at<br>; someone had made a<br>. Nothing<br>about it was ——but<br>."

I tested the prompt and noticed how difficult it is to get something<br>that sounds remotely human. Even after detailed prompts and multiple<br>corrections, I barely got something passable. This issue lingered in<br>my mind for a long time and eventually shaped my mental model of<br>learning. It also gave me a habit. Now when I enter a new field, I try<br>to capture my early ideas before expertise edits them out of me. That<br>habit has helped me stay creative and produce output even when<br>entering deep rabbit holes. Here I’ll do my best to explain why.

Knowledge collapses the search space

Here’s another anecdote. My PhD thesis advisor once told me that I was<br>spending too much time reading papers, and that I was going to learn<br>the bad ways to solve the problem. He also told me that many of the<br>top researchers at Caltech couldn’t care less about what most people<br>write. The advice sounded strange because research is supposed to<br>build on what is already known.

Let’s assume my advisor was right. How is it possible<br>to push the knowledge boundary forward without first reaching it? My<br>working hypothesis is that knowledge doesn't simply give you new<br>options. It also eliminates some of them. More precisely, knowledge<br>constrains the distribution of ideas we are likely to produce.

When trying to solve a problem, there's a distribution of ideas or<br>solutions that we can come up with. In the figure, that distribution<br>is represented by the blue blob. Some of those ideas will produce good<br>solutions, represented in green. A much smaller subset are the optimal<br>solutions, represented in red. To make the figure cleaner, all the<br>blobs are drawn in similar sizes. But the probability of a good idea<br>is much smaller than the probability of a bad one. And the probability<br>of an optimal idea is just a drop in the bucket. You can see this<br>whenever you write. There are countless sentences you could put next,<br>many that would be grammatical, fewer that would be good, and very few<br>that would say exactly what you mean.

When we learn a skill or a way to solve problems, that knowledge<br>biases us toward what we already know. Eventually it becomes our<br>natural answer. In a way, we collapse the range of possible ideas into<br>a constrained region. In the figure, the constraint is the dashed<br>ellipse. It sits inside the green blob, which means we can now<br>reliably produce good answers. But it also doesn't overlap with the<br>red blob of better solutions, and it prevents us from reaching unusual<br>possibilities. Thinking outside of that box becomes harder. We get<br>more reliable answers at the cost of variation, and possibly<br>creativity. The same thing happens in writing. Grammar and taste help<br>us produce sentences that work, but they can also quietly censor the<br>ideas that first sound wrong. This may be one reason why, as one<br>article puts it,<br>"as researchers age, they produce less disruptive work."

Generalization reopens the search space

Some people do learn almost everything in a field and still produce<br>interesting research, often in the form of generalized theories. To<br>make sense of this, I need one more oversimplification.

Consider a researcher who knows of two different approaches to solving<br>a problem. These could be known methods from the same field, or ideas<br>borrowed from different fields. In the figure, these methods are<br>represented as two separate constraints. The researcher can draw<br>solutions from either one. But after thinking about them and<br>understanding the commonalities, they may be able to produce a single<br>framework that covers both cases. The larger dashed blob in the figure<br>is this generalization. It contains the ideas in the two constrained<br>spaces, but it is not limited to them. It also lets the researcher try<br>things in between, and sometimes extrapolate to new approaches.<br>Abstracting the two methods relaxed their mental constraints and<br>increased the range of accessible ideas. Maybe this higher level of<br>thought can help them find better solutions than those previously<br>known.

Knowledge also helps when we move into complex topics that involve<br>many steps. Trying to solve all the steps at once is fragile. Roughly<br>speaking, each unreliable step compounds the risk of failure. For<br>example, if each step has a 10% chance of success and there are four<br>steps, the probability of reaching a good solution is 0.01%. Knowledge<br>lets us turn hard and unreliable steps into easier and reliable ones,<br>which allows us to tackle more difficult and important problems.

So I arrive at the "uncontroversial" conclusion that knowledge is<br>good, without...