Through all of this, don't get me wrong, the rigorous application of rationality that it takes to step-by-step construct a proof is very important and an incredibly useful skill. Also, I agree that basically no-one can hold more than 3 things in their head at once.

The book also agrees vehemently that math is NOT restricted to "geniuses" and even argues that those don't really exist in the way that culture thinks they do.

However! His assertion is that the (to him) tedious, laborious, error-prone, paperwork is not the fundamental output of "doing math". For him, symbolic written mathematics is akin to sheet music. It would in principle be possible to teach students to read and write sheet music and even do manipulations like transposing it to different keys, without ever letting them listen to music. It would be hard and boring. Some students would find the memorization and application of rules satisfying but most would struggle.

In such a classroom, there might be one student who by chance figures out for herself that you can kind of "hear" these symbols in your brain and suddenly all the arbitrary rules seem obvious and natural and she doesn't even have to go through the tedious steps at all to answer questions. "Of course this is in a minor key." she might say. "No, I didn't rigorously check each chord, it's just... obvious".

Such a student would be labeled a "prodigy" or "genius", and would struggle to explain to others that no, what she's doing isn't harder than the her classmates laboriously doing the rote work, it's actually much easier.

Of course... this is not to denigrate sheet music. It's a wonderful invention that makes it possible to transmit music out of one person's brain to the brains of an orchestra.

Just like written mathematics.

The author's contention is that, like the contrived example above, no-one ever talks about "the music" of mathematics, just the sheet music, and therefore things are much harder than they need to be.

One of the simple mathematical examples he uses is to ask: Can you imagine a circle in your head (unironically an amazing thing to be able to do!). Then to ask a question like: Can a straight line intersect a circle in 3 places?

You likely have an immediate, intuitive response to this highly non-trivial mathematical problem. That's the music. Now, try to write that down in mathematical language for someone who can't see circles. Oof, it's going to be a slog.

>>Through all of this, don't get me wrong, the rigorous application of rationality

Much of this is just talking to oneself and testing it to see if our idea holds under test conditions.

I was once watching a video on how chess grandmasters think and work. Most of it is-

1. Do we know a pattern of moves, even if done, in series that is known to score some win/check. If so, lets do it.

2. Are any pieces under attack, If gone can effect point 1. eventually? If yes, lets protect them.

3. What can all possible moves of our pieces prevent opponent from having successfully execute their own point 1. And can we force opponent into point 2? Lets do it.

Basically every our move and its possible outcomes(Known through prior study of patterns of previous games seen), every move of our opponent.

A strong internal monologue and testing imaginary moves.

Math is just this except over paper.