Mathematics as reinforcing practices of a verbal culture

It is a powerful reframing.

Viewing math as a collection of reinforcing practices within a verbal community shifts learning from "memorize this abstract truth" to something more human and investigative:

Why did people in this community invent/adopt/rework this idea?

What problem or confusion in their existing practices made this a reinforcing solution?

What contingencies (practical payoffs, social approval, consistency with other practices, explanatory power) kept it alive and spread it?

When I'm doing math, I don't have a goal in mind. What I want is an immediate connection to the problem, to the structure I'm studying. So when I have this feeling that I can't connect with that thing, I need to invent a representation that allows me to feel around that structure, that allows me to build up some guiding intuition. Because if I have to work with a representation where I rely on memory and reasoning with my mind, I'll have to keep too much in my head, and I'll get overwhelmed.

There's this rule from Inventing on Principle that says creators need an immediate connection with what they're interacting with. And the way we connect with complex systems is through representations, because representations allow us to think powerfully about systems. They allow us to manipulate the system and to see it.

Now, what if the representations you have to work with are inhumane? What if the representations laid out in most textbooks are not aligned with your cognitive framework, with your mental strengths? So you need to invent. Sometimes it's not there. Sometimes you're lucky and you find a language that allows you to use your pre-existing mental capacities to make sense of the system. Sometimes you're lucky and you find the right primitives to make sense of a mathematical structure. Because I don't care how many people praise a mathematical theory, if I can't work with the abstractions, the abstractions are not useful to me.

So yeah — you're really guided by this feeling of, okay, I need this immediate connection to what I'm creating. And I need to be in the present moment for that. I can't be distracted by some goal or condition. I have to see right now. I have to think this thought, but I can't think it. So I need to invent some kind of language to put this vague image, this feeling, into words — but I don't have them yet.