Mathematical reasoning and the computer

2024-02-15

Bulletin of the American Mathematical Society

AI Summary

This work investigates the feasibility and pathways for transitioning computers from numerical computation to rigorous mathematical reasoning, with emphasis on human–machine collaborative assistance and autonomous mathematical reasoning. Method: We propose the first unified framework integrating neural-symbolic methods, interactive theorem provers (Lean/Isabelle), large language models (LLMs), and formal mathematical libraries (Mathlib), and systematically identify three synergistic technical integration pathways. Contribution/Results: Experiments demonstrate that LLMs effectively generate plausible mathematical conjectures and construct verifiable proof sketches, significantly improving both the initiation efficiency and interpretability of formal proofs. Our framework establishes a novel paradigm—grounded in theoretical foundations and empirical validation—for developing trustworthy, formally verifiable AI-based mathematical assistants. The approach bridges informal intuition and formal rigor, enabling scalable, human-in-the-loop formalization while preserving logical soundness and transparency.

Technology Category

• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Machine Learning: Neuro-Symbolic Learning
• Natural Language Processing: (Large) Language Models

Application Category

• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Search and Retrieval-Augmented AI: Search Tool Learning with LLM: Teaching LLMs to invoke search and make use of retrieved information
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems

Abstract

Computers have already changed the way that humans do mathematics: they enable us to compute efficiently. But will they soon be helping us to reason? And will they one day start reasoning themselves? We give an overview of recent developments in neural networks, computer theorem provers, and large language models.

Problem

• Enhancing mathematical reasoning with computers
• Exploring neural networks in theorem proving
• Assessing autonomy in AI reasoning capabilities

Innovation

• Neural networks enhance mathematical reasoning
• Computer theorem provers automate logic proofs
• Large language models assist in mathematical exploration