Artificial Intelligence and the Structure of Mathematics

TL;DR

This paper explores how AI can revolutionize understanding of mathematics by providing a new perspective on its structure through formal proof and hypergraph models, potentially revealing insights into mathematical discovery.

Contribution

It proposes a novel framework for AI-driven mathematical discovery based on formal proof structures and criteria for autonomous mathematical exploration.

Findings

Outlines a formal structure of mathematics using hypergraphs and proofs.

Suggests criteria for AI models to autonomously discover mathematical concepts.

Posits that AI can offer new insights into the nature and structure of mathematics.

Abstract

Recent progress in artificial intelligence (AI) is unlocking transformative capabilities for mathematics. There is great hope that AI will help solve major open problems and autonomously discover new mathematical concepts. In this essay, we further consider how AI may open a grand perspective on mathematics by forging a new route, complementary to mathematical\textbf{ logic,} to understanding the global structure of formal \textbf{proof}\textbf{s}. We begin by providing a sketch of the formal structure of mathematics in terms of universal proof and structural hypergraphs and discuss questions this raises about the foundational structure of mathematics. We then outline the main ingredients and provide a set of criteria to be satisfied for AI models capable of automated mathematical discovery. As we send AI agents to traverse Platonic mathematical worlds, we expect they will teach us about the nature of mathematics: both as a whole, and the small ribbons conducive to human understanding. Perhaps they will shed light on the old question: "Is mathematics discovered or invented?" Can we grok the terrain of these \textbf{Platonic worlds}?