I have a gut feeling that notation is more essential/central to learning than most presume.

In this stackoverflow question the idea is that changing the notation of logarithms and exponents can make learning it much more intuitive and easy.

In this github repo, there are links to notation making a difference in how people experience activities.

Similarly to these two links, there’s the idea that learning to read and write makes a difference in how you think. The language you speak can have an influence on how you think. This is sort of like an extended Sapir-Whorf hypothesis.

Essential to this idea is taking what Feynmann said about his notes seriously.

“They aren’t a record of my thinking process. They are my thinking process. I actually did the work on the paper.” “Well,” Weiner said, “The work was done in your head, but the record of it is still here.” “No, it’s not a record, not really. It’s working. You have to work on paper and this is the paper. Okay?”, Feynman explained.

If we assume thinking actually happens on paper, this all makes a lot more sense. Externalizing thinking onto paper means that you no longer need to keep all information in your working memory. This frees up your mind to determine the next steps you need to take, without needing to remember what you just did.

Similarly, what if notation serves a similar function? Taking the idea of learning calculus as an example, the limit definition of a derivative is technically the same as writing d/dx. This simplification of notation implies an internalization of the principles of that operation. The derivative then becomes a “chunk” that becomes “brain sized”. Something along these lines is Yudkowsky’s Cached Thoughts.

Shouldn’t learning come first and notation simply be a consequence of that? Not necessarily, the idea that notation becomes brain sized is predicated on it being legible. You are constrained by the analysis that’s contained in the notation. For example, multiplying Roman numerals was possible, but due to notation was quite difficult. Roman numerals were fantastic for tallying. Arabic numerals, on the other hand, required 10 different numerals, which meant no more easy tallying. What you got in return though, was an incredibly powerful mechanical multiplication process.

That’s why, before the 14th century, everyone thought that multiplication was an incredibly difficult concept, and only for the mathematical elite. Then arabic numerals came along, with their nice place values, and we discovered that even seven-year-olds can handle multiplication just fine. There was nothing difficult about the concept of multiplication — the problem was that numbers, at the time, had a bad user interface. – Bret Victor

What happens, when you extend this idea? Could certain processes become easier with better notation? By simplifying or improving notation, you can simplify certain actions.

This is reminiscent of the concept of making smaller circles from Josh Waitzkin.

First, I practice the motion over and over in slow motion… By now the body mechanics of the punch have been condensed in my mind to a feeling. I don’t need to hear or see any effect—my body knows when it is operating correctly by an internal sense of harmony… Now I begin to slowly, incrementally, condense my movements while maintaining that feeling… Each little refinement is monitored by the feeling of the punch, which I gained from months or years of training with the large, traditional motion

He describes learning a motion and making it automatic, then refining it for maximum effect. This is the same thing as creating an internal notation for yourself for a physical action. The key is that you must build up a full understanding of what you’re doing before attempting to make it better. Using notation you don’t fully understand is useless, which is why I think transferring knowledge from one sort of problem to another is so difficult.

The implications of this are –

• Memorization is not the enemy. Memorizing certain elements of an action, such as the quotient rule in calc, is the first step in being able to use that in higher level contexts.
• Notation is important. How we make things legible to others has a big impact. Explaining physics is a whole different animal if you don’t use free body diagrams.
• The way you develop higher levels of abstraction is by developing notation. Constructing a notation, however, is dangerous. Embedding the wrong analysis of a concept in notation can make it even tougher to think clearly. As an example, using Euler angles are intuitive when describing rotations. If you’re doing any sort of computation though, including trying to linearly interpolate between two angles, using quaternions is much more useful.

Understanding this gives you leverage over any topic you learn. The notation and what it implies is foundational to any field and it’s a sort of language you need to learn to speak in order to contribute.