Why Mathematics Works So Well

TL;DR

This paper explores the deep connection between mathematics and physics, arguing that the effectiveness of mathematics in describing physical laws is due to shared symmetries, making this relationship more understandable.

Contribution

It demonstrates that the symmetries defining physical laws are analogous to those defining mathematical facts, explaining the effectiveness of mathematics in physics.

Findings

Symmetries of physics are key to understanding physical laws.

Mathematical facts exhibit similar symmetries as physical laws.

Physical regularities are a subset of mathematical regularities.

Abstract

A major question in philosophy of science involves the unreasonable effectiveness of mathematics in physics. Why should mathematics, created or discovered, with nothing empirical in mind be so perfectly suited to describe the laws of the physical universe? We review the well-known fact that the symmetries of the laws of physics are their defining properties. We show that there are similar symmetries of mathematical facts and that these symmetries are the defining properties of mathematics. By examining the symmetries of physics and mathematics, we show that the effectiveness is actually quite reasonable. In essence, we show that the regularities of physics are a subset of the regularities of mathematics.