We can explain mathematical intuition. We just can't explain it as quickly or in as interesting way as you seem to want.

Here's a simple concept: all polynomials with coefficients in a number system (field) have a solution (possibly in a larger field).

Do you know how hard it is to convey what exactly this means and what the consequences are? I like Lockhart's essay. It resonates with me. However, I don't see his ideas being useful in terms of learning advanced topics. At some point one has to get their hands dirty and slog through the material. Intuition will come with experience.

> Do you know how hard it is to convey what exactly this means and what the consequences are?

I never said it is easy.

> Intuition will come with experience.

And this experience that some people have already reached should be shared, not limited to a select coterie of people.

It is shared. It's just that everyone has to go through to mental work to truly understand. It takes me a semester to convince my students that 3x+5x=8x and that this is true because of the distributive property. And this is why ax+2x = (a+2)x. And the reason we can't simplify further is because our language does not have a word for (a+2) but it does have a word for (3+5). But I know that after a semester of teaching this most still don't actually understand the distributive property. It takes a lot of effort on the part of students for the concept to click.

> It is shared. It's just that everyone has to go through to mental work to truly understand.

Except, I can hardly see that. At least, I see you trying - most people do not make an attempt at all.

FYI, I am a Computer Scientist.