I wrote earlier about the many different ways to measure distance. One way I didn’t include is unmeasurable distance.
Sometimes A is
- more interesting,
- or otherwise better endowed
than B … but it’s impossible to quantify by how much. No problem; just say that A≻B but that |A−B| is undefined.
It’s still the case that if A is sexier than B and B is sexier than C, it must follow that A is sexier than C.
Symbolically: A≻B & B≻C ⇒ A≻C.
This concept opens up many parts of human experience to the mathematical imagination.
I will also express my view on moral rates of income tax using orderings ≻.
Oh, and if you’re into this kind of thing: using orders ≻ instead of measurable quantities kind of saved the economic concept of “utility”. Kind of saved it. At least instead of talking about 174.27819 hedons, nowadays you can just say X is lexicographically preferred ≻ to Y. Ordinal utility instead of cardinal utility.