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AI can now solve some difficult, formalized mathematics problems, but that is not the same as independently choosing worthwhile open questions and conducting broad pure-mathematics research. The clearest recent benchmark is AlphaProof’s performance at the 2024 International Mathematical Olympiad (IMO): it solved three non-geometry problems, while AlphaGeometry 2 solved one geometry problem. Together, the systems earned 28 points, within the silver-medal threshold—but their solutions used multi-day computation, not the contest’s human time limit.
The distinction matters. Mathematical work includes choosing a problem, expressing it precisely, finding a proof, and checking that proof. AI has made notable progress on parts of the latter stages. Deciding which question deserves attention remains a different kind of work, involving context, taste and judgment.
What does it mean for AI to do mathematical research?
“Doing research” can refer to several tasks that should not be conflated:
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Scan for outdated or missing drivers - takes under a minuteDriver Scan →Clear out junk files and repair common Windows errorsFree Scan →Fix the driver behind crashes, sound loss and screen glitchesFind Drivers →- Choosing a question: identifying a conjecture, problem or connection that is both meaningful and tractable.
- Formalizing it: translating the intended mathematical claim into an exact statement in a formal language such as Lean.
- Finding a proof: constructing a chain of valid reasoning that establishes the formal statement.
- Checking and interpreting the result: verifying the proof and deciding what it means for the broader field.
A system that proves a supplied problem has made progress on proof construction. It has not necessarily chosen the problem, established that its formal version captures the intended question, or shown why the result matters to mathematicians.
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How can a computer prove a theorem?
Lean turns statements and proofs into checkable objects
Lean is an interactive theorem prover: a system for expressing mathematical definitions and theorems formally and checking proofs. In Lean’s foundations, a statement is represented as a type, and a proof is a term inhabiting that type. Tactics can help a person or program work through a goal and its hypotheses; Lean’s kernel checks the resulting proof term.
This check is valuable because it can establish that a proof follows from the formal statement and the system’s foundations. But the kernel checks what was encoded, not whether that encoding faithfully expresses the informal theorem a mathematician intended. A flawless proof of a mistranslated statement does not settle the original question.
Proof search can combine learned guidance and computation
In the 2025 Nature paper “Olympiad-level formal mathematical reasoning with reinforcement learning,” Thomas Hubert and colleagues describe AlphaProof, which combines a neural proof network with search in Lean. The system uses large-scale reinforcement learning, autoformalization and focused test-time learning on related problem variants. In broad terms, learned guidance helps direct a search through possible formal steps, while the prover checks candidate results.
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This kind of system can explore many proof paths without relying on an unaided sequence of natural-language guesses. Its output is constrained by what has been formalized and what the proof environment can express, however; those constraints shape what problems it can attempt.
What did AlphaProof and AlphaGeometry 2 achieve at IMO 2024?
The result was substantial, but it has a specific scope: the systems were evaluated on the six problems from a mathematics competition, not on an open-ended research program. The figures below are the results reported by the Google DeepMind author group in its 2025 Nature paper.
| System or result | Reported IMO 2024 outcome | What the figure means |
|---|---|---|
| AlphaProof | Three of the five non-geometry problems solved, including P6, the most difficult problem, according to the authors. | Performance on the non-geometry problems in this specific competition evaluation. |
| AlphaGeometry 2 | One geometry problem solved. | A geometry result from a separate system designed for that domain. |
| Combined system result | 28 points, within the IMO silver-medal threshold. | The reported combined score from AlphaProof and AlphaGeometry 2, not a score achieved by AlphaProof alone. |
The authors report that the solutions required multi-day computation. That makes the result a significant demonstration of mathematical problem-solving ability, but not a like-for-like comparison with a human contestant working under the IMO’s timed conditions. Nor does success on this fixed set of problems measure how much AI has contributed to pure-mathematics research overall.
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Why is it hard to translate a proof into Lean?
Formalization is not merely rewriting equations in unfamiliar notation. A system has to identify the relevant definitions and assumptions, express the intended claim precisely, and work within the available formal language and libraries. Informal mathematics often leaves steps implicit because a reader is expected to supply them; formal proof requires those steps to be made explicit.
Informal wording may hide assumptions
A short theorem statement can rely on conventions that are obvious to a mathematician but absent from the text. The formalizer must decide what each object means, which conditions are required, and how familiar terms map to the formal definitions. If any of those choices is wrong, the formal theorem may differ from the intended one.
Geometry exposes the gap between text and diagrams
In geometry, a diagram can carry information that the accompanying prose never states. A human reader may infer relationships from the picture, while a formal system needs those relationships represented explicitly and justified. The 2024 paper “Autoformalizing Euclidean Geometry,” by Logan Murphy, Kaiyu Yang, Jialiang Sun, Zhaoyu Li, Anima Anandkumar and Xujie Si, describes a benchmark and a neuro-symbolic method combining domain knowledge, SMT solvers and language models. The authors report using theorem provers to fill diagram-related information so a model can formalize explicit textual steps, while also documenting limitations in their experiments.
Libraries determine what can be stated conveniently
A formal language is supported by libraries of definitions and theorems. A problem may be mathematically familiar yet awkward to encode if relevant concepts or results are missing. Hubert and colleagues note that gaps in Mathlib’s higher-level geometry library at the time—including incircles and congruence—made it impractical to state many IMO-style planar geometry problems directly in Lean. The team used AlphaGeometry 2 for dedicated olympiad geometry problems.
This example shows why proof capability cannot be considered in isolation. A system’s effective reach also depends on whether the problem can be represented faithfully and whether the formal environment contains the tools it needs.
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Can AI decide which mathematical questions are worth asking?
The cited competition result shows that AI systems can solve selected problems after they have been presented in a suitable form. It does not show that they can autonomously identify valuable open questions across pure mathematics, rank them by significance, and steer a research program toward them.
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Choosing a question is not simply another proof-search step. It can require recognizing a fruitful connection, understanding what a community already knows, judging whether a problem is important, and deciding whether an uncertain line of inquiry is worth pursuing. It is reasonable to see human imagination and judgment as especially important here—but that is an interpretation of the present division of work, not a proof that machines can never develop those abilities.
Research on AI and mathematics is broader than olympiad proof search. Yang-Hui He’s 2024 review, “AI-driven research in pure mathematics and theoretical physics,” groups approaches into top-down, bottom-up and meta-mathematical directions. A 2025 ICML position paper by Kaiyu Yang and co-authors, “Position: Formal Mathematical Reasoning—A New Frontier in AI,” discusses proof assistants as systems that can verify reasoning and provide automatic feedback, and argues for formal reasoning as an important part of future AI. These works frame active directions and goals; they do not establish that current systems independently choose broadly valuable pure-mathematics research questions.
How to judge a headline claim about AI and mathematics
When a system is said to have “proved a theorem” or “made a discovery,” ask what work the result actually demonstrates. These checks separate proof ability from the larger claims a headline might imply:
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- What task was measured? Distinguish informal exploration, translating a problem into a formal statement, constructing a proof, and checking that proof.
- What was the domain? A result on a fixed benchmark or a competition set does not automatically generalize to other areas of mathematics.
- Who supplied the problem? Solving a chosen problem and independently originating a valuable research question are different achievements.
- Was the formal statement faithful? Kernel verification establishes correctness relative to the encoded statement; it does not by itself confirm that the statement matches the original informal problem.
- What tools were available? Language support and theorem-library coverage affect which questions can be expressed and attempted.
- How much time and computation were used? Multi-day machine computation should not be presented as equivalent to a human result under contest time limits.
These distinctions do not diminish a successful proof. They show exactly what it establishes—and where further evidence would be needed to support a claim about autonomous mathematical discovery.
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