Imagine you wanted to explore the number line. You start at and then you explore , , and so on. No matter how long you do this you'll never discover , , , , , 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.

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

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It's worth noting that these types basically match the Johari Window for types of risks; known knowns, known unknowns, unknown knowns, and unknown unknowns. This is because, at least according to one definition of the term, risks are things we don't expect. Given that definition, a risk is the product of a conceptual hole of some sort - and so the two have a somewhat trivial mapping.

Learning a new language (English) was decisive in my life to fill a lot of type 1, type 2 and 3 holes as you presented on this post.

In 2013 "I felt" I should learn English. I didn't have a clear goal on doing it, but "it seemed important on the long-term", but I couldn't explain why it would be important to care about really learn another language.

In 2016 I heard from a YouTuber from my country talking about AI. My goals changed completely, and I could only find real quality discussions about AI or state-of-the-art information in English.

Then it came programming, blockchain, decision theory, etc, etc. Subjects that I only started caring because I learned English and found really good English content about these subjects.

This post makes me wonder what would happen if suddenly I started to learn Chinese.

English is the most important, most useful language to know. It is the language of business, science and technology. It is the closest thing humanity has ever had to a universal language.

Chinese is the second most important language in the world to know. It is, in a different sense, the closest thing humanity has ever had to a universal written language. Here are some ideas you might get from learning Chinese.

  • Your ideas of "center of the world", "civilization" and "humanity" shift from Europe and the Americas to East Asia. This is not the same as a global internationalist perspective. China becomes the hive of humanity everything else revolves around, like how the Earth is the center of the universe.
  • Non-phonetic writing systems make sense. You realize how little living literary tradition survives in languages written using alphabets because texts become unreadable in mere centuries as the spoken language evolves. You develop an appreciation for calligraphy. You can read more Japanese than most 外人.
  • You develop an intuitive feel for the clan system, a semi-artificial method of extending family connections.
  • You realize Western castles were tiny and crude.
  • You think in longer historical time horizons.
  • You take it for granted that the natural state of things (historically-speaking) is for China to be the richest and most powerful nation on Earth.
  • You discover lots of business opportunities.

What am I missing by not speaking Portuguese?

These points really make me update my beliefs on the importance of learning Chinese. Thanks for that.

What am I missing by not speaking Portuguese?

Brazilian speaks with the body, so you are not missing that much. Haha just kidding (although it's true that body language plays a big role here). But there are 250 million total portuguese speakers, and by just knowing Portuguese you will understand Spanish easily, because there is 89% of lexical similarity. I can understand Spanish with no problems and I never studied it. If you know how to search, there is a vast and extensive Brazilian music that you wouldn't like to miss. The literature written in Portuguese remains untranslated, except for very few authors, so you would be able to read very good books (if you like certain genres). Also, Brazil is about half of South America, and is a fantastically interesting place to travel. When new people come here, the reception is warm. Music, cooking, sports and memes here are a real thing. Knowing Portuguese would make you able to connect with our culture and you would be able to access very good content on a lot of subjects. But I am fully biased and mostly looking into the bubble (mathematics, physics and computation) and you don't lose much by just using English sources for these subjects.

For example, this channel is specialized in non verbal behavior analysis and I never found anything better than that on any language. This channel is the best on musical content (how to play drums, guitar, acoustic guitar, etc, how to sing, compose, etc). This on comedy, this on history and science. I mean, if you keep looking, you will always find a lot of good content on very different subjects. You will miss all of this by not knowing Portuguese, as with any language.

If you look on the science side, you can check Brazil here (tables from page 4~9).

There is no good translator from Portuguese to any language yet, and your best tool to date would be DeepL, in case you need to translate Portuguese info to English.

I wonder if you picked kensho as an example because it is so apropos: it's a word related to the seeing of that which you previously didn't know existed.

Maybe a little. :)

The word "apropos" is new to me. I like it.

Here's my reinterpretation for the four levels of conceptual holes:

  1. can be inferred from your current knowledge base,
  2. outside your knowledge base but inside the fields of knowledge you are aware of,
  3. outside the fields of knowledge you are aware of but in /some/ existing field of knowledge,
  4. outside all existing fields of knowledge you can access.

In what way do you believe concepts exist? There are many different ways to splice of concept space and your post sounds like there a certain way in which a concept like Kenshō is objectively sliced.

"What does it mean for a concept to exist" is a deep philosophical question I'm not sure of the answer right now. For the purpose of this article I pretty much just mean an idea is invented and isn't fraudulent or pointless.

It seems strange to me to use it that way in the article as it means that invention is a requirement for the existence of the concept. It seems strange to say that because Japanese invented Kensho Westerners invisibly decide not to experience Kensho. It seems to me like the article has confused ontological assumptions.

Japanese people didn't invent kenshō. People around the word (including Westerners) will experience kenshō randomly. The Japanese merely refined a system for identifying kenshō and fostering kenshō states. I can see how if the Western world discovered kenshō independently or if you don't believe in kenshō then we'd be getting into ontological territory. But it doesn't seem like you're coming from this direction.

So I guess "an idea is invented" is the wrong definition for "a concept to exist". I don't know how to define "existence" in the case of kenshō. I don't even know how to define kenshō itself without tautologies.

There's a group of numbers between 0.38976 and 1.1. A given culture might decide to call those numbers A*-numbers. Another group might call the numbers between 0.35 and 1.2 B*-numbers.

I would say that both A* and B* were invented by the groups that use them. It's unlikely that another group of people would come up with the same conceptualization even if they would investigate the same problem domain.

It seems to me likely that Kensho is similar. It's a conceptual cluster that the Japanese use, but other traditions of meditations don't use the same conceptual cluster.

Your conceptualization assumes that it would be good for a person who wants to learn something about meditative states to learn about how the Japanese conceptualize kensho and then try to work their way to the state. That approach conflicts with meditation paradigms that value "beginners mind".

I understand better now what you mean.

When I originally wrote kenshō I meant to ambiguously refer to both to the Japan-specific conceptual cluster itself and the underlying sector of the meditative map because both of them are orthogonal to the Western philosophical tradition.

I would be surprised if it wasn't possible to experience a kenshō state (under a different name, marked off with different conceptual clusters) in the West using a non-Japanese meditative tradition recently adopted from someone else like India.

The difference between the Western tradition and Eastern traditions are often a bit exaggerated.

Within Western monasteries people engage in meditative practices. Historically, meditation wasn't central in Buddhism either.

Most older spiritual tradition put a lot of value on secrecy and modern Buddhism and later New Agey thinking is more open to selling meditation courses to a wide public (both for money and for external recognition).

It's quite different to work on learning an existing concept and to work on drawing conceptual boundaries yourself in a way that works for your context.

If you have not chosen to live in Japan, not knowing about Kenshō may be one of the consequences, though you might not be aware of it.

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I don't think this answer addresses the point of in what sense Kensho exists for you in that case.

Oops. I'm not sure what I was responding to, but it wasn't the question you asked.

Did you actually learn to speak piraha? Everyone I know totally refused to participate, so I dropped the idea.

No. Only Chinese.