Someone using the handle “man after midnight” (whatever that means or why, I don’t know) described what it took to progress to a high level understanding of math. It sounds like a blueprint for other activities with many layers of complexity.
So, I tried making a few substitutions, like “belts” for “levels,” and the progression he described for learning math applies to learning Brazilian Jui Jitsu (which usually takes about ten years to go from beginner to black belt level).
Here’s what he said (with my changes in italics) about how many levels there are:
“The way it was described to me was in terms of ‘belts’.
Sometimes, in Brazilian Jui Jitsu, you find that your slow progress, and careful accumulation of tools and ideas, has suddenly allowed you to do a bunch of new things that you couldn’t possibly do before. Even though you were learning things that were useless by themselves, when they’ve all become second nature, a whole new world of possibility appears. Something clicks, but now there are new challenges, and now, things you were barely able to think about before suddenly become critically important.
It’s usually obvious when you’re talking to somebody a belt above you, because they see lots of things instantly when those things take considerable work for you to figure out. These are good people to learn from, because they remember what it’s like to struggle in the place where you’re struggling, but the things they do still make sense from your perspective (you just couldn’t do them yourself).
Talking to somebody two or belts above you is a different story. They’re barely speaking the same language, and it’s almost impossible to imagine that you could ever know what they know. You can still learn from them, if you don’t get discouraged, but the things they want to teach you seem really philosophical, and you don’t think they’ll help you—but for some reason, they do.
Somebody three belts above is actually speaking a different language. They probably seem less impressive to you than the person two levels above, because most of what they’re thinking about is completely invisible to you. From where you are, it is not possible to imagine what they think about, or why. You might think you can, but this is only because they know how to tell entertaining stories. Any one of these stories probably contains enough wisdom to get you halfway to your next belt if you put in enough time thinking about it.
So, the bad news is, you never do see the whole picture (though you see the old picture shrink down to a tiny point), and you can’t really explain what you do see. But the good news is that the world of BJJ is so rich and exciting and wonderful that even your wildest dreams about it cannot possibly compare. I don’t know how many levels there are…”