This is going to sound insane but... would it not be nice, if there was no need for numbers (except for counting like in subscripts)? Mathematicians talk about eliminating the Axiom of Choice, because of the craziness of the Banach-Tarski Paradox. But the paradoxes numbers create are even crazier.
- It is possible to create a numerical function called the Busy Beaver function that lets you prove the Goldbach conjecture (a statement about an infinite set of numbers) by testing it on every number from 1..BB(n) (a finite, although very large, set of numbers).
- Gödel's theorems require arithmetic in order to construct the true-but-unprovable sentences. No numbers and such sentences cannot be made.
- Numbers give rise to Berry's paradox (there is nothing analogous to it without numbers), which in turn gives rise to the undefinability of an information compression algorithm, even though humans rely on being able to calculate the Kolmogorov complexity of information in day-to-day life.
Has some eccentric mathematician talked about moving maths beyond numbers just like how some mathematicians talked about moving beyond sets in order to define the field of one element?