Discovering mathematical concepts through a multi-agent system

TL;DR

This paper introduces a multi-agent system that autonomously discovers mathematical concepts by generating and proving conjectures, demonstrated through recovering the concept of homology from polyhedral data.

Contribution

It presents a novel multi-agent model for computational mathematical discovery that integrates conjecture generation, proof attempts, and feedback-driven decision making.

Findings

The system successfully recovers the concept of homology from polyhedral data.

Ablation studies show the importance of dynamic interactions in the system.

Optimizing local processes leads to meaningful mathematical notions.

Abstract

Mathematical concepts emerge through an interplay of processes, including experimentation, efforts at proof, and counterexamples. In this paper, we present a new multi-agent model for computational mathematical discovery based on this observation. Our system, conceived with research in mind, poses its own conjectures and then attempts to prove them, making decisions informed by this feedback and an evolving data distribution. Inspired by the history of Euler's conjecture for polyhedra and an open challenge in the literature, we benchmark with the task of autonomously recovering the concept of homology from polyhedral data and knowledge of linear algebra. Our system completes this learning problem. Most importantly, the experiments are ablations, statistically testing the value of the complete dynamic and controlling for experimental setup. They support our main claim: that the optimisation of the right combination of local processes can lead to surprisingly well-aligned notions of mathematical interestingness.