Finite mathematics as the most general (fundamental) mathematics

TL;DR

This paper argues that finite mathematics based on finite rings is more fundamental than standard mathematics, which is a degenerate limit involving infinities, and extends this idea to quantum theory.

Contribution

It demonstrates that finite mathematics is more general and fundamental, providing a new perspective on the relationship between finite and standard mathematics and quantum theory.

Findings

Finite mathematics is more general than standard mathematics.

Standard mathematics is a degenerate case of finite mathematics as p approaches infinity.

Quantum theory based on finite rings is more general than standard quantum theory.

Abstract

The purpose of this paper is to explain at the simplest possible level why finite mathematics based on a finite ring of characteristic $p$ is more general (fundamental) than standard mathematics. The belief of most mathematicians and physicists that standard mathematics is the most fundamental arose for historical reasons. However, simple mathematical arguments show that standard mathematics (involving the concept of infinities) is a degenerate case of finite mathematics in the formal limit $p\to\infty$: standard mathematics arises from finite mathematics in the degenerate case when operations modulo a number are discarded. Quantum theory based on a finite ring of characteristic $p$ is more general than standard quantum theory because the latter is a degenerate case of the former in the formal limit $p\to\infty$.