AI can now solve selected formal mathematics problems and help translate informal mathematics into machine-checkable form. That is meaningful progress, but it is not the same as independently choosing important open questions and carrying out a broad program of pure-mathematics research. The clearest evidence so far is strongest on proof search within defined tasks; deciding which problems matter remains a distinct research judgment.
What does it mean for AI to do mathematical research?
“Doing mathematics” combines several activities that are easy to blur together: deciding what is worth investigating, stating the problem precisely, exploring possible arguments, and establishing that a proof is valid. Automation in one activity does not automatically demonstrate automation in the others.
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| Research activity | What it involves | What current evidence establishes |
|---|---|---|
| Choosing a question | Identifying a worthwhile conjecture or a promising connection to investigate. | The sources discussed here do not establish that AI systems autonomously choose broadly valuable open questions across pure mathematics. |
| Formalizing a problem | Encoding the intended definitions, assumptions, and claim in a precise formal language such as Lean. | AI-assisted autoformalization is an active research area, but translating informal mathematics faithfully remains difficult. |
| Finding and checking a proof | Searching for a derivation and verifying it against a formal statement and foundation. | Systems have made notable progress on selected formal and benchmark problems, including an IMO 2024 result described below. |
This distinction matters because a proof can be correct relative to a formal statement even if that statement does not capture what a mathematician meant. Conversely, a system may help with a difficult proof without having chosen the question or understood why it is mathematically important.
How formal proof automation works
Lean turns proofs into checkable objects
Lean is an interactive theorem prover used to formalize mathematics. In its framework, statements are represented as types and proofs as terms that inhabit those types. A person or automated system can use tactics to work on a goal using its hypotheses; Lean’s kernel checks the resulting proof term against the encoded statement and foundations.
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This check is valuable: it can catch invalid steps in the formal derivation. But it is a check of the encoded claim, not a guarantee that the encoding faithfully expresses the original informal problem. Correct formal proof and correct interpretation are related but separate requirements.
Proof search is only one part of the pipeline
A system working on an informal theorem must identify the relevant definitions and assumptions, express the theorem in the formal language, and then construct a proof using the available formal tools and libraries. These stages can be evaluated separately. A successful proof search says little about whether the translation was faithful unless the formal statement itself has also been checked against the intended problem.
What AlphaProof’s IMO result shows—and what it does not
A 2025 Nature paper by Thomas Hubert and colleagues describes AlphaProof, which combines a neural proof network with search in Lean, large-scale reinforcement learning, autoformalization, and focused test-time learning on related problem variants.
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For the 2024 International Mathematical Olympiad, the Google DeepMind author group reports that AlphaProof solved three of the five non-geometry problems, including the most difficult problem, P6. In the combined result, AlphaGeometry 2 solved one geometry problem; together the systems scored 28 points, within the silver-medal threshold.
| Result reported for IMO 2024 | System and scope |
|---|---|
| Three of five non-geometry problems solved, including P6 | AlphaProof, as reported by the Google DeepMind author group in its 2025 Nature paper. |
| One geometry problem solved | AlphaGeometry 2, in the combined system result reported in the same paper. |
| 28 points; within the silver-medal threshold | Combined AlphaProof and AlphaGeometry 2 result for IMO 2024, as reported in the paper. |
The result is a substantial demonstration of problem-solving ability in a demanding competition setting. Its conditions also matter: the authors report that the systems used multi-day computation, unlike contestants working under the IMO’s human time limits. It should therefore be read as a system-specific benchmark achievement, not as a like-for-like contest performance or a measure of AI’s general contribution to mathematical research.
The result also does not by itself demonstrate autonomous research. Olympiad problems are selected, bounded tasks; proving one does not show that a system can independently identify a consequential conjecture, decide how to formulate it, and pursue a sustained research direction.
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Why translating informal mathematics into Lean is difficult
Informal proofs leave things implicit
Formalization is more than converting notation. The formal statement must preserve the intended definitions, assumptions, and meaning. Human mathematical writing often leaves routine steps unstated or relies on context that a formal system requires explicitly. A translator must determine what those omissions mean rather than merely reproduce the visible words.
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Geometry makes the gap especially visible. An informal argument may rely on a diagram whose relevant implications are not written in the accompanying text. In “Autoformalizing Euclidean Geometry,” published in the 2024 Proceedings of Machine Learning Research, Logan Murphy, Kaiyu Yang, Jialiang Sun, Zhaoyu Li, Anima Anandkumar, and Xujie Si describe a benchmark and a neuro-symbolic method combining domain knowledge, SMT solvers, and language models. Their work uses theorem provers to fill diagrammatic information so a model can formalize explicit textual steps, while also documenting limitations.
Libraries shape which problems are practical to encode
A formal language’s libraries determine which definitions and results are readily available. In the AlphaProof paper, the authors say gaps in Mathlib’s higher-level geometry library at the time—including incircles and congruence—made many IMO-style planar geometry problems impractical to state directly in Lean. They therefore used AlphaGeometry 2 for dedicated olympiad geometry problems. This illustrates that the challenge is not only proving a claim once written down: the available formal infrastructure can affect whether and how the claim can be represented in the first place.
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Why choosing the question is a separate challenge
Finding a proof within a specified problem is not the same task as deciding which problem deserves attention. Mathematical question choice can involve judging whether a conjecture is significant, whether a connection is promising, and what a result might clarify. Those judgments depend on context and perspective, not just on deriving consequences from a formal statement.
This is a reasoned distinction, not proof that machines can never make valuable mathematical choices. Kaiyu Yang and co-authors’ 2025 ICML position paper, “Position: Formal Mathematical Reasoning—A New Frontier in AI,” presents proof assistants as tools for verifying reasoning and providing automatic feedback, and argues for formal reasoning as an important component of future AI for mathematics. Yang-Hui He’s 2024 review, “AI-driven research in pure mathematics and theoretical physics,” surveys discovery approaches including top-down, bottom-up, and meta-mathematical methods. Neither source establishes that current systems autonomously select broadly valuable questions in pure mathematics.
The available evidence supports a narrower conclusion: AI is automating meaningful parts of proof search and formalization, while independent question selection remains a separate capability for which the cited results provide no broad demonstration. An impressive solution is evidence about the problem solved under the conditions reported—not a complete account of who chose the problem, whether its formal version captured its informal meaning, or what made it important.
Independent reader supportYour contribution helps us test, update, and keep practical guides available for everyone.How to assess a claim that AI has proved a theorem
When evaluating a headline result, ask what was actually automated and how success was checked. A useful assessment separates the following questions:
- What was the task? Was the system exploring informal mathematics, searching for a formal proof, or translating a statement?
- Was the formal statement faithful? A machine-checked derivation validates the encoded theorem; someone must still establish that the encoding matches the intended claim.
- What domain was tested? A competition benchmark or a specialized geometry task does not measure mathematical work across every field.
- What tools and library coverage were available? Missing definitions or results can constrain what can be stated and attempted.
- How was the result evaluated? Distinguish a checked formal proof from external assessment of an informal answer or its mathematical significance.
- What were the time and compute conditions? Reported performance under multi-day computation is not directly comparable to a human contest result under a fixed time limit.
What the current evidence supports
AI systems can solve selected mathematical problems, search for formal proofs, and assist with the difficult translation between informal statements and formal languages. Lean can mechanically check a formal proof against its encoded statement and foundations. Neither capability, alone, establishes that the statement captures the mathematician’s intent or that the system independently chose a worthwhile research question.
The AlphaProof and AlphaGeometry 2 result is a strong, specific benchmark achievement. It should not be extrapolated into a field-wide measure of AI-originated research questions or AI’s overall effect on pure-mathematics discovery: the sources cited here provide no such broad statistic. For now, the most accurate picture is of increasingly capable tools for parts of mathematical work, alongside open challenges in faithful formalization and a still-distinct role for human judgment about what to ask.
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