Math notation is a powerful demonstration of how good (and bad) design of notation can make certain kinds of thinking trivial or impossibly difficult. The reason for this goes back to our earlier lesson from written language: good notation gives us the ability to work work abstract concepts (like sets or infinities) using the same cognitive muscles we use to work with physical objects, as concrete, embodied objects in our visual and tactile world. Because of this I think it’s helpful to think of the notation we use to represent ideas not just as simple tools, but as an integral part of how we think.
We think in languages; we think with notations.
Notation is a tool for manipulating and communicating ideas.
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