Key Takeaways
- The source argues that AI-assisted tools are increasingly good at finding and formalizing mathematical counterexamples.
- Several examples described in the post were checked in Lean, turning informal claims into machine-verifiable statements.
- The broader implication is not just speed, but a shift in how mathematicians test conjectures and trust results.
What happened
In this post, a mathematician describes a run of recent developments in which AI systems and formal verification tools surfaced counterexamples to long-standing problems. The thread begins with a claim that ChatGPT disproved Erdős’ Unit Distance conjecture in discrete geometry. According to the post, the initial argument was supported by human mathematicians who had seen early access to it, but it was not yet formalized in Lean.
The author says that within days, Logical Intelligence shared an autoformalization of the paper in Lean, demonstrating that the theorem connecting a result in number theory to the counterexample could be verified mechanically. The post presents this as an early sign that AI-generated mathematics could be translated into formal proof systems rather than remaining informal prose.
The account then moves to a workshop on formalizing Fermat-related mathematics, where AI tools were used to generate exposition in natural language and then translate claims into Lean. In one case, the post says an AI system found that a claim in a generated document was false and produced an explicit counterexample. The author later corrected the document after checking the issue.
The same post describes another counterexample discovered with AI assistance: a question attributed to Grothendieck about finite free group schemes. The author says a system called Sol, used by OpenAI researcher Boris Alexeev, generated a Lean formalization that included a proof of a counterexample, and that the resulting code compiled after review. The post says this produced a group scheme of order 4 that was not killed by 4.
Later, the post reports that AI tools were used on a modularity lifting theorem project, which the author says a student nearly finished in Lean with help from Sol and Fable. Finally, the account describes a counterexample to the Jacobian Conjecture, which the post says was found with Fable and then formalized manually in DeepMind’s Formal Conjectures repository.
Why it matters
The core technology story here is not just that AI can write mathematical text. It is that large language models, paired with proof assistants like Lean, may be becoming capable of contributing to formal mathematics in a way that can be checked with high confidence.
That matters for two reasons. First, formal verification changes the role of trust. The post repeatedly contrasts informal mathematical arguments, which can be persuasive but fallible, with Lean proofs, which can be mechanically checked once the statement is formalized. In that model, the hardest part is often not proving the theorem in human language, but getting the statement into a form that the proof assistant and its library can understand.
Second, the post suggests that AI systems may be especially effective at finding counterexamples. Counterexamples are a powerful form of mathematical progress because they can resolve conjectures decisively, even when the proof is not yet conceptually satisfying to humans. The author repeatedly emphasizes that the machines are not only producing code, but also surfacing explicit failures in arguments and conjectures.
The post also points to a more practical software reality: these systems can generate very large amounts of Lean code quickly. The author says Sol produced 1.2 million lines of Lean in three weeks, compared with mathlib’s 2.3 million lines built over nine years. Even if those figures need to be read in the context of an opinionated blog post, the underlying point is clear: AI is changing the scale at which formal mathematical development can happen.
What to watch
The next question is whether these systems can move beyond isolated counterexamples into broader mathematical understanding. The post says the author still wants humans to explain what the examples mean, rather than treating a formal counterexample as the end of the story.
It is also worth watching how reliable the surrounding workflow becomes. The post warns that Lean is a programming language and that malicious code can, in principle, run commands on a machine, so the author tested code in a sandbox. That is a reminder that formal verification does not remove all software-engineering concerns.
Another thing to watch is how quickly conjecture repositories and libraries adapt. The post notes that DeepMind’s Formal Conjectures repo already contained a formal statement of the Jacobian Conjecture, which made verification easier. If more conjectures are pre-formalized, then AI-generated proofs or disproofs may become much easier to check.
For now, the story suggests a software stack in which foundation models, proof assistants, and formal libraries are starting to work together. If that combination keeps improving, AI may become less of a drafting aid and more of an active participant in mathematical discovery.
