good notation helps us work with abstract, complex concepts using concrete symbols. Notation is useful because we can engage the part of our brain tuned for working with physical objects, to instead work with abstract ideas represented by physical symbols.
In Part Two, we saw that the ability to discover and work with higher levels of abstract ideas, and generalize with them broadly, is key to our intellectual ability to respond to new problems.
Taken together, I think inventing better notation is a cheap, universally useful way to increase our effective intelligence in many domains. Better notations let us work more easily with more complex abstractions, and enable us to solve new complex problems. When faced with a new seemingly intractable problem, it’s worth asking not only “What tools and discoveries might help us understand this?” but also “What notation can we use to describe this problem domain, that might help us understand this?” While better tools can help us work more efficiently or precisely with ideas we already have, better notation can enable us to imagine and discover new ideas entirely.
Tips for designing good notation
- Find a property of a concept that current notation expresses badly, and focus on expressing that more faithfully.
- roll up ideas that are complex in current notation into more concise symbols.
- Try to express dualities in ideas that existing notations don’t suggest.
- Notation doesn’t have to look like language, or like symbols on a horizontal line.