How Mathematical Frames Emerge

Math actually makes a lot more sense when you look at it through the lens of Data-Frame theory.

If you think about it, someone plays around with some mathematical system or structure, and then they start formulating statements about it. Once you have a bunch of statements, those statements themselves can be understood within the framework of logic and reason. From there, you can derive new statements that make sense within that logical system.

Now, that doesn't necessarily mean the system is ultimately true or free of contradictions. It just means the reasoning is valid relative to the assumptions you've made. It doesn't guarantee that someone won't later find a contradiction or a counterexample that forces you to revise the framework. That's not what the logical system is guaranteeing.

The observation that prompted this thought is that someone who came up with, for example, group theory didn't just invent it out of nowhere. They were playing around with polynomials. People had been trying to solve polynomials and understand their roots for centuries. While studying those roots, they needed a way to make sense of what they were seeing. So they came up with a frame, and that frame eventually became group theory.

The same thing happened with the foundations of mathematics. People looked at mathematics and noticed inconsistencies. They felt that the way mathematics had been formulated was full of holes, that things didn't quite make sense. So they tried to reorganize the foundations around axioms. They tried to make everything rigorous by rebuilding it in terms of set theory. In other words, they created a new framework that made the whole subject hang together more coherently.

Now, the interesting thing about what Wildberger is doing with rational trigonometry and universal hyperbolic geometry is that he's proposing another new story. It's another way of making sense of mathematics. Every time he introduces a new example or a new structure, what he's really asking is: how can this new medium, this new frame, give us a more immediate connection to the mathematical system? How can it let us work in a way where we can actually see what we're doing, through concrete computations and explicit examples?

That's what I think he's really trying to do.