Notation as a tool of thought
Also: APL, Iverson
Kenneth Iverson's thesis that good notation doesn't just record thought — it amplifies and directs it.
When Kenneth Iverson accepted the 1979 Turing Award, his lecture was titled "Notation as a Tool of Thought." His claim, drawn from inventing APL, was that notation is not a passive transcript of ideas you already have. The right notation makes some thoughts easy to express and easy to manipulate — and thereby makes them easy to have.
Iverson pointed to ordinary arithmetic: positional decimal notation makes multiplication a mechanical procedure a child can learn, while the same operation in Roman numerals is a nightmare. The notation did the thinking. APL pushed this to an extreme, compressing whole algorithms into a line of symbols you could reason about as a single object.
This is one of the load-bearing ideas of the tools-for-thought tradition, and it connects directly to McLuhan's the medium is the message: change the notation and you change the thinker. Read Notation as a Tool of Thought.