Mathematics Is Physics

TL;DR

This paper argues that mathematics is a natural science akin to physics, explaining its effectiveness through a network-based theory of mathematical theory-building rooted in natural regularities.

Contribution

It introduces a novel theory that models mathematical development as a network process based on analogies and regularities, bridging the gap between abstraction and empirical grounding.

Findings

Mathematical theories evolve from natural regularities.

Mathematics is fundamentally a study of regularities within regularities.

Mathematical theories are connected to the natural world through a network structure.

Abstract

In this essay, I argue that mathematics is a natural science---just like physics, chemistry, or biology---and that this can explain the alleged "unreasonable" effectiveness of mathematics in the physical sciences. The main challenge for this view is to explain how mathematical theories can become increasingly abstract and develop their own internal structure, whilst still maintaining an appropriate empirical tether that can explain their later use in physics. In order to address this, I offer a theory of mathematical theory-building based on the idea that human knowledge has the structure of a scale-free network and that abstract mathematical theories arise from a repeated process of replacing strong analogies with new hubs in this network. This allows mathematics to be seen as the study of regularities, within regularities, within ..., within regularities of the natural world. Since mathematical theories are derived from the natural world, albeit at a much higher level of abstraction than most other scientific theories, it should come as no surprise that they so often show up in physics. This version of the essay contains an addendum responding to Slyvia Wenmackers' essay and comments that were made on the FQXi website.