>>It also has only glancing similarities to anything that the author would recognise as mathematics.

Nah, these are the same things. Trying to make Math look like is for people who are 'geniuses' i.e people with massive capabilities of holding large thought trials and changelogs in their head is how you arrive at making people look stupid doing math and eventually make them hate the subject.

Math is paper work. Approach it that way and all of a sudden doing a 100 page proof is within everyones reach. If you ask people to hold a 100 page proof in their head, and more importantly make changes to that in random places and fix the entire changelog trial, probably 2 - 3 people on earth will be able to do it, and you will just convince everyone else its not for them.

I have a hunch that big mathematical breakthroughs in history have happened around and after renaissance era due to paper getting cheap and ubiquitous. There is only that much you can do in your brain alone.

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.