Higher Theory and the Three Problems of Physics

TL;DR

This paper argues that understanding quantum gravity requires a shift to higher mathematical theories, highlighting three distinct quantization problems and their philosophical implications for the concept of truth.

Contribution

It introduces a topos theoretic framework to distinguish three quantization problems and emphasizes the importance of higher theory in quantum gravity research.

Findings

Identifies three distinct quantization problems.

Highlights the role of higher theory in quantum gravity.

Connects higher theory to foundational philosophical issues.

Abstract

According to the Butterfield--Isham proposal, to understand quantum gravity we must revise the way we view the universe of mathematics. However, this paper demonstrates that the current elaborations of this programme neglect quantum interactions. The paper then introduces the Faddeev--Mickelsson anomaly which obstructs the renormalization of Yang--Mills theory, suggesting that to theorise on many-particle systems requires a many-topos view of mathematics itself: higher theory. As our main contribution, the topos theoretic framework is used to conceptualise the fact that there are principally three different quantisation problems, the differences of which have been ignored not just by topos physicists but by most philosophers of science. We further argue that if higher theory proves out to be necessary for understanding quantum gravity, its implications to philosophy will be foundational: higher theory challenges the propositional concept of truth and thus the very meaning of theorising in science.