This is it. Inverse is a property of functions or other relational operators, not "static" individual objects. You need a direction in order to invert it.
I think we’re on the same page. The point I was trying to make is that reciprocal/multiplicative inverse in every day use seems like it’s a property of a number and not of a function only because most people just assume that the function we’re inverting is multiplication. The comment I was replying to missed that.
The multiplicative inverse of a number n can be seen as the inverse of the function that multiplies its input by n. That is, the inverse of multiplication by n is multiplication by 1/n.
Similarly, the additive inverse of a number m is the inverse of the function that adds m to its input.
As long as, C*Cinv = I, where C is the circle, Cinv is the inverse of circle, and I the identity. You're right. C, I, and * are entirely up in the air.
In general for inversion, we have object A (argument) and object I (identity) and a function F of two arguments, so we have equations: `F(A, X) == I, F(A, I) == I, F(X, I) == I, A != I, A != X`, where A, I, and X are objects in the same category, i.e. they must be circles `(x² + y² == r²)`.
If F is defined as `ra•rx`, then `ri == 1`, and inverse will be `rx = 1/ra`.
If F is defined as `ra + rx`, then `ri == 0`, and inverse will be `rx = 0 - ra`, where negative radius means hole.
If F is defined as `ra²•rx²`, then `ri == 1²`, and inverse will be `rx = sqrt(1/ra²)`.
If F is defined as `ra² + rx²`, then `ri == 0²`, and inverse will be `rx = sqrt(0 - ra²)`.