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Position: Formal Mathematical Reasoning—A New Frontier in AI
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Position paper arguing that formal mathematical reasoning (proof assistants like Lean) is an indispensable complement to informal LLM-based math, surveying progress and proposing capability levels and open challenges.
Problem
Informal LLM approaches to math face data scarcity and lack of verifiability for advanced mathematics and proofs, motivating formal reasoning grounded in proof assistants.
Approach
Advocates combining LLMs with formal systems (e.g., Lean) to provide automatic verification and feedback for data generation, evaluation, and verifiable generation, and proposes a five-level milestone framework for autoformalization, theorem proving, verified natural-language reasoning, and formal verification.
Key results
- AlphaProof reached silver-medal level at IMO, the first AI to do so, via formal reasoning.
- The number of publications on AI for formal mathematical reasoning roughly doubled annually in 2023 and 2024.
- The Proof Pile collected only about 500MB of formal proofs, orders of magnitude smaller than informal counterparts.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use expert iteration to train the prover on newly found proofs | Neural theorem provers · Unspecified theorem-proving tasks | performance | Helped |
| Include proof-environment error messages in prompts | COPRA · Unspecified theorem-proving tasks | repeated-mistake rate | Helped |
| Use informal proof sketches before formalization | Draft, Sketch and Prove (DSP) · Isabelle theorem proving | proof-solving performance | Helped |
| Retrieve useful lemmas from a mathematical library | ReProver · Lean theorem proving | performance | Helped |
| Use standard retrievers to retrieve existing lemmas and definitions | ReProver and COPRA · Formal mathematical reasoning | performance | Helped |
| Use neural networks as verifiers | Verified reasoning in natural language · Unspecified reasoning tasks | overall performance and faithfulness | Helped |
| Train a smaller model on synthetic aligned informal-formal data | Autoformalization models · Autoformalization | autoformalization performance | Helped |
| Combine formal reasoning with LLMs | AI for mathematical reasoning · International Mathematical Olympiad (IMO) | medal level | Helped |
Surprising findings
- Current LLMs struggle with intrinsic self-verification, which the authors argue weakens purely neural verification of informal reasoning.
Limitations
- Not a comprehensive survey; selective coverage of the field.
- Automatic evaluation of autoformalization remains an unsolved bottleneck.
- Formal verification of real-world systems poses challenges beyond math (large codebases, specification trust).
Experimental setup
Datasets: Proof Pile NuminaMath
Builds on: Lean chain-of-thought expert iteration GPT-f ReProver
Closest papers by meaning
- HERMES: Towards Efficient and Verifiable Mathematical Reasoning in LLMs (ICML 2026 · similarity 0.88)
- Numina-Lean-Agent: An Open and General Agentic Reasoning System for Formal Mathematics (ICML 2026 · similarity 0.88)
- ToRA: A Tool-Integrated Reasoning Agent for Mathematical Problem Solving (ICLR 2024 · similarity 0.87)
- RMath: A Logic Reasoning-Focused Datasets Toward Mathematical Multistep Reasoning Tasks (AAAI 2025 · similarity 0.85)
- MathCoder2: Better Math Reasoning from Continued Pretraining on Model-translated Mathematical Code (ICLR 2025 · similarity 0.85)
- RealMath: A Continuous Benchmark for Evaluating Language Models on Research-Level Mathematics (NeurIPS 2025 · similarity 0.85)
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