I think it's just we don't have a way to say/write these numbers. So you make up a way to write them (-1 and 1) and continue.

The numbers don't matter and you could replace -1 and 1 with anything. It's just easier to begin your new fake number at - 1 and 1. Because position does matter.

This is very helpful!

Basically what I take away is that we're inventing a new number system from scratch. So we're not "proving" that 1 is a number between 0 and the empty set. We're defining it as such, and it just so happens that a number system defined this way works out in convergent ways with other mathematics.

Is that roughly right?

There has to be some procedure how to come up with "new" numbers though, if you want to have more in the end than just a fancy binary tree - in particular if you want to map your "fake numbers" to the reals, infinity, etc.

> position does matter

Only as a mental abstraction that's based on our experience/concept of space+time.

that's unnecessarily confusing then - should've used 69 and 420 imho