Using notation to understand the world

Notation is the idea that an arbitrary object can be represented by some other abstract object. A specific type of notation, which is the one most people would understand in the conventional sense of the word “notation”, is when the object doing the representing is written. Commonly seen examples of notation include those that represent maths, music, electrical circuits and even lists of key-presses. In this article I discuss how notation has shaped my understanding of things, and also propose a notation-oriented way of understanding something.

How to understand something

There was an xkcd comic that goes something like this:

xkcd #451

My hobby:

Sitting down with grad students and timing how long it takes them to figure out that I'm not actually an expert in their field.

In the end, the protagonist of the comic managed to find a field of study where he successfully made inroads into a community by producing words in the right order to pass for genuine knowledge.

This made me think about how someone would go about successfully fake knowledge of something. Eventually, I came up with this two-step process:

1. Understand the language, and how to recognise something as a correct statement.

2. Understand how to combine statements together to form a correct third statement.

I brought this up with someone, and he noted that by the time you have done both of these things, you aren’t faking knowledge of something, you now understand it.

In my opinion that isn’t strictly true, but it does come a lot closer than other processes. A studied reader might recognise this as reminiscent of a Chinese room, where someone can fake knowledge by reading in written text (i.e. a question) and responding according to a translation book.

Chinese room, Wikipedia

While it may look like a Chinese room at first, a critical difference is that in a Chinese room, the translation book is something external to the operator, whereas the situation I came up with requires you to memorise it. While a rote memorisation would result in something almost entirely identical, a common occurrence with human minds is that it like to take shortcuts, and it is there where the situation becomes markedly different, as the shortcuts are now a form of understanding where you can now produce correct statements without memorising any translation book.

Additionally, a second form of understanding is to be able to generalise from the words you are given. A translation book can give you some rules, but when a human reads it the experience is that he will try to generalise from the existing rules and make up new ones in the vein of the ones he was given. This is expressed in a number of ways – notably some of them are quite distasteful – but the key here is that there is a point where a human would try to expand the rules that he was given and apply it to something unrelated just to see if it works. (Though, in some cases, “just to see if it works” is replaced with “and assume it works”, leading to all kinds of consequences that are beyond the scope of this article.)

Let’s now move away from dry theoretical explanations and show how this applied to me by way of example.

Making your own – the C. C. F. D. N.

The problem statement

When I was 17 I was very bad at Chinese. I am still fairly bad at Chinese, which we’ll get to later.

The main problem with writing Chinese is that sinograms – what the language is written in – are taught badly to me. I could not really understand any of the logic behind writing these unusual things, and what real information was given to me was not well-organised. What that means effectively is that I was stuck writing in a language where I have to reason about strokes from first principles, and just like trying to write a program in assembly language, it means that I missed out in understanding other parts of the language, particularly the things that are in the curriculum, which means that I failed many tests.

Then one day a friend of mine started learning Japanese, and then he had to deal with sinograms as well. But since I’m familiar with them already, I decided to help a little bit. This amounted to looking up the character in Wiktionary and then reporting on its pronunciation and its graphic etymology, both of which are helpful in understanding why a character is written and pronounced (some of the time) the way it is. However, looking up many of these characters realise that a lot of the information is repetitive and redundant. So I then built a little notation to condense it down into something that cuts away all of that.

The notation itself

The result is the Chinese Character Form Description Notation. The exact form of the notation is not needed to make my point, but we’ll discuss it as an aside here anyway. The notation describes the graphic etymology of the character by breaking it apart into its components. For instance, the character “to think” 思 is derived by combining a meaning-carrying component of “heart” 心 with a sound-carrying component of 囟, which was later simplified into the unrelated character 田. Putting that together, the notation would write the above sentence as:

思 = <心|囟 →田>

Which summarises all of what I just wrote in an appealing string of symbols. Additionally, this separates it from an existing notation in Unicode called the ideographic description sequence, which instead describes how a character can be broken down in its current form.

Consider now another character, “small, narrow, fine”, which is 細. If I now provide its C. C. F. D. N., you would easily be able to figure out what its graphic etymology is, even without me spelling it out, if I mentioned that the meaning-carrying component 糸 means “fibre, rope, string”:

細 = <糸|囟 →田>

But look now that 細 and 思 don’t look too alike, as their corresponding IDS would make explicit. I had not expected that the two are related quite like this – their pronunciations are also quite a bit different – and now that I have made that connection I realised that I had something in my hands that is valuable.

Ultimately, the notation I have invented for myself made everything click together in my head, and it cleared up almost everything that had been blocking me from being able to move away from focusing on what I write and instead focusing on what the writing means on a more abstract level. It wasn’t enough to eventually make me pass my exams, but I feared the language a lot less than I had before I built the notation.

(In a sense, while it was helpful it would never have come on time; reflecting on what I did in my secondary school years it was fairly clear to me that this was a thing that would have troubled me for as long as it did no matter how I did it, and I would have rejected this notation if it was handed onto me the same way that all those other explanations have.)

How the notation changed the way I think

The key to the notation’s success is, in my opinion, the following:

First, it is general. It can break apart a large number of characters and describe their etymologies in a succinct manner. There are very few sinograms where it cannot handle it at all, and in those cases it is typically because the character has an unknown graphic etymology and therefore can’t be described, though undoubtedly there is going to be a couple that would slip behind the cracks. I can’t take full credit for this one, as it is based on an existing method of classifying sinograms:

Chinese character classification, Wikipedia

I will however say that the way I designed (?) this notation allows me to grasp the recursive nature of this classification and furthermore allow me to gain a foothold in scripts that use things that resemble sinograms but are not, such as Sawndip.

Sawndip, Wikipedia

Second, it is manipulable. What you can change and in what ways are easily read out by looking at the notation. Whether or not it describes an actual character is another question, but you can basically always produce a valid formula for generating a character and hint at its meaning and pronunciation using the notation. This will prove invaluable if I were to build a script that has similar properties to Chinese characters, this is exactly how I would start approaching it.

Third, it is not clever. The key to a proper notation is that it represents the thing that it does straightforwardly, in such a way that allows one to verify that it is in fact representing what it claims to. This allows me to hook onto it and adapt myself into seeing the notation at the same time I am presented with the character.

Finally, it is flexible. Notice that in the above examples the sinograms are described in terms of other sinograms. There’s no particular reason why those sinograms are chosen. In fact, you can replace those characters with their descriptions, allowing a full drill-down of the character and describing how every little bit came to be in a simple manner. This property allows you to hide away details when such detail is not needed but still permit an exhaustive description if it is.

Reading an existing notation

In real life sometimes an existing notation is already available, and if it is it would probably be better than any notation that you would be able to make when studying it, as it is referencing the entire body of knowledge that it is built on rather than just what a student might learn as he builds it for himself. With that in mind, if one were to understand something by its notation it is prudent to understand an existing notation first before trying to make one’s own.

To this end, I would say that it is helpful to learn something by referencing its notation. Specifically, the question to ask is “how do you write this down, and why do you choose to write it this way?” This question can be asked in multiple contexts and expect multiple answers. What “this” in the question is differs depending on the exact thing you are studying, and the “why” could be surface-level “why this symbol over another” or a deeper “why arrange the symbols in this way”.

Once you learn the notation, the next step is, as hinted at earlier in this article, to learn how to combine two true statements written in that notation into a third one also written in that notation. In a sense, this task is much harder than the other two; if we apply it to, say, English, the first is “reading and spelling” and the second is the entirety of English grammar and literature. The key to this is not to complete the task, but instead to basically learn the subject “the normal way” but using the notation as a centrepiece to unify everything you learnt into it, so that you have a way to relate everything to everything else.

Consider how this can be applied to chords and music theory: you learn what notes correspond to which chords and how to write down changes to those chords, and then you can combine the chords together as simple letters on a page to form music. This is an iterative process: you learn what chords go together nicely based on existing theory, mirror it in the notation, and then generalise it in the notation before turning it back into notes and seeing if it appeals. Doing this correctly, you will both increase your understanding of music theory and also have a nice way of generating music.

Another example where the idea of learning by notation has been successfully applied (but not by me) is site-swap notation in juggling, where someone wrote down sequences of digits to describe how balls are thrown in. The inventor eventually discovered a new way of juggling (called “5551” in the notation) just by looking at the notation:

Juggling by Numbers, Numberphile on YouTube


It is only fair to discuss situations where notation-oriented learning is not the approach to take when learning something.

Sometimes there is no existing notation, whether because the experts have decided that writing bits of ink on a page is not sufficient for the field, or no one has thought of it, or even that all the existing ones have failed to gained currency because it doesn’t perform or there are too many competing ones. In most of these cases you can get around it by attempting to make your own. You can make this work if you know yourself well, and perhaps even present it to the community as a way to understand the field itself.

Some people don’t take kindly to writing things down, especially in the more artistic field of study where notation might be perceived as constraining to a creative mind, and even knowledge of the existence of some notation is considered harmful because you’re always supposed to learn things “the hard way”, i.e. without any aid whatsoever, much less written ones. Whether one chooses to heed those warnings or proceed anyway is up to individual preference, but in a notation-averse community – for any reason – perhaps it would not be the best idea to show that community your new home-grown notation.

These fears are very much founded in reality, as it turns out. A clever tweet once said:

There's this phenomenon that takes place whenever you get too far away from hardware or biology where the primary parts of what people are reasoning about become words, and then the people who ONLY deal with words start to creep in and take over

@drethlin, Twitter

In other words, we have the issue where people mistake the map with the territory and then demand the real thing to do impossible things just because the notation can do it. A well-designed notation would be able to reflect the limitations of the object to be represented of course, but that might not be the most popular notation, or even be any notation that exists.

The converse, where the real thing can do things that the notation cannot express or chose not to express. This is most often mitigated by having multiple notations that help with seeing things in complementary ways. For example, with sinograms above, the C. C. F. D. N. of a character works together with the IDS and its phonetic spelling – romanization – in order to give all the parts of a character that are useful for the learner.


One of the ways that I have learnt something is to create a notation system for it. This involves figuring out how to write existing facts in that notation, and then how to create new things by altering the notation and seeing what the results in back in the original object that the notation depicts. I found this to be a greatly useful way to understand topics in general and a central point from which to handle unknown fields of knowledge from, while also acknowledging that there are some fields and probably some would-be learners that may not be suited to this way of learning.


2021-10-01 - Initial publication

2021-10-03 - Added a portion ("These fears... for the learner") in response to some feedback about C. C. F. D. N. not being able to represent some critical parts of a character, as well as a point that I have forgotten to include about the traps of reasoning through notation.

🗼 gemini://