Imagine you wanted to explore the number line. You start at and then you explore $2$ , $3$ , $4$ and so on. No matter how long you do this you'll never discover $- 1$ , $\frac{1}{2}$ , $2 π$ , $\infty$ , $i$ , or a single noncomputable number. That's because the most interesting numbers are precisely those located in directions conceptually orthogonal to whatever numbers you've already explored.

The easiest concepts to learn are the ideas ahead of you on a path you're already taking. If you know classical mechanics then it's straightforward to learn relativity. You could crack turbulence without ever changing fields.

The second-easiest things to learn are those outside your specialization but inside of your culture's intellectual tradition. If you grew up in the West then you know there are such things as physics and painting even if you never learned how to do them.

The second-hardest things to learn are those outside your society's intellectual tradition. Consider for example kenshō^[1]. It's possible to live an entire lifetime outside of Japan without ever discovering that such a concept exists. Westerners don't choose not to experience kenshō. This choice is made invisibly.

The hardest things to learn are those belonging to domains of knowledge which haven't been (openly) conceptualized anywhere, such as photography before the invention of the camera. This category includes zero-day exploits.

We can number each kind of conceptual hole from least to most orthogonal.

Type 1: Concepts you are aware of but do not understand.
Type 2: Concepts you are not aware of, but which belong to a field you are aware of.
Type 3: Fields of knowledge you do not know exist, but which exist.
Type 4: Fields of knowledge that remain genuinely secret or have not been invented.

Expertise is about mastering one field, but general intelligence is how prepared you are for something new. The more different kinds of concepts you understand the better you will be at solving new kinds of problems. This is behind the principle of learning that breadth of knowledge equals breadth of transference.

Filling holes of Type 1 is how you build expertise in a field. Holes of Types 2-4 are better for building broader (and therefore more transferable) knowledge. In other words, holes of greater-numbered types are better for increasing your general intelligence. If this is your goal then holes of Type 2 and 3 are the most valuable.

I find holes of Type 3 to be so valuable that just knowing where they are improves my creativity at solving problems. (It also turns them into holes of Type 2.) You can find holes of Type 3 systematically by mastering the language of a sufficiently foreign culture. I know this works with written Chinese and spoken Pirahã. I hypothesize it's also true of Arabic, ASL, Korean and the Khoisan languages.

Filling holes of Type 2 is almost straightforward. Pick anything you're bad at that lots of other people do and develop a basic competence. This turns holes of Type 2 into holes of Type 1.

Holes of Type 4 can be individually very valuable. That is, a single hole of Type 4 can earn you lots of power and money. But holes of Type 4 are so hard to find, verify and exploit that you can't build a broad base of knowledge out of them.

I like collecting conceptual holes, especially those of Types 2 and 3. They're so interesting. Whenever I discover a new hole it opens up an entire tree of knowledge orthogonal to everything I used to know.

Kenshō (Japanese kanji: 見性) is a subjective state of mind associated with Zen meditation. ↩︎