Ablation and the Meno: Tools for Empirical Metamathematics

2026-04-24

AI Summary

This work investigates the creative space of mathematical proofs under constraints, with a particular focus on the impact of non-constructive reasoning. We introduce a strategy ablation methodology that integrates our custom-built Meno automated formalization tool with Goedel Prover embeddings to systematically explore both formal and informal proof spaces for foundational theorems from *Analysis I* within the Lean theorem prover. Our experiments successfully generate a novel class of machine-produced proofs, revealing that these proofs cluster along low-dimensional submanifolds in a high-dimensional representation space and significantly diverge from human-constructed proof trajectories. This study provides the first quantitative characterization of the structural differences between machine-generated and human proofs.

Technology Category

• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Machine Learning: Other Foundations of Machine Learning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Web Mining and Content Analysis: Web data provenance, reliability, and authenticity
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems

Abstract

We present the results from Meno, a simple autoformalizer that proves theorems in Lean by systematically exploring the space of both formal and informal proofs, and tactic ablation, a new method for exploring mathematical creativity under constraint. We show these tools in action on simple theorems found in Terrence Tao's Analysis I, selectively ablating solution paths associated with non-constructive proofs, and analyze the properties of the resulting population using Goedel Prover embeddings. Among other things, our analysis of this novel population reveals that they lie on low (one or two) dimensional submanifolds of the much higher-dimensional representation space, and far away from their corresponding human constructions.

Problem

• autoformalization
• tactic ablation
• mathematical creativity
• constructive proofs
• proof space

Innovation

• autoformalization
• tactic ablation
• mathematical creativity
• Goedel Prover embeddings
• non-constructive proofs