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.

The multiplicative inverse, or reciprocal, or the inverse of a number, is a function acting on a number, 1/x, or x^-1.

Isn't this a case of multiplication being such a "well-known" function that the "inverse" defaults to "multiplicative inverse"?

No, inverse means reciprocal only when the context is a number or algebraic variable.

If we're talking about trig functions, inverse sine or arcsine is very different than just the reciprocal of the sine.

If we're talking about images, inverse usually means "rotate 180 degrees" or "color inversion".

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)`.

And so on.

Correction: `F(A, I) == A, F(X, I) == X`.

An inverse is pretty easy to generalize, and it need not rely on multiplication or the identity.

Basically: Given a function `f`, then the function `g` is the inverse of `f` if `g(f(x)) = x`.

It is easy to generalize, but it is not being generalized here.