Summary of Key Points
OpenAI’s internal AI model, Astra, has solved 10 long-standing open problems in mathematics at a cost of approximately $2,000 (where “tokens” can be understood as fees for using the AI service). These problems span various fields such as high-dimensional geometry and coding theory, and Astra’s efficiency far exceeds that of humans—equivalent to the work of ten mathematicians. This development has sparked a series of debates: Can AI serve as an author of scientific papers? Who is responsible for the correctness of these proofs? Will the core of mathematics (proofs that can be understood by humans) be overturned? What will be the future value of mathematicians? Even the foundations of science itself (such as falsifiability) may be challenged?
Detailed Analysis
1. AI Solving Math Problems: More Powerful than Doctors, Cheaper, and Faster
The problems solved by Astra are considered “unsolved mysteries” in mathematics—difficult issues in high-dimensional geometry and quantum complexity theory that previously required the collaboration of experts from various fields over several years to resolve. Now, AI can complete these tasks for $2,000 and has already surpassed the capabilities of doctors, with its performance continuing to improve rapidly.
For example, a mathematician might have had to consult countless references and try hundreds of methods to solve a problem, taking months or even years; AI, on the other hand, can quickly process vast amounts of data to find the optimal solution. It’s like hiring a “supermathematician” for $2,000 who can complete years’ worth of work in just one day.
2. The Debate over Attribution: Can AI Be an Author? Who Bears the Responsibility?
This is the most direct point of contention:
- The Leiden Declaration (signed by over 3,400 mathematicians and supported by the International Mathematical Union) clearly states that AI cannot be considered an author; research findings must be attributed to humans, who are responsible for the correctness of the proofs. The argument is that “the credit and responsibility belong to humans, not automated systems.”
- OpenAI’s stance is against attributing AI-generated proofs to human authors, stating that this would misrepresent both the contribution of AI and the nature of human labor. Their approach is for AI to generate the arguments, for humans to write the papers, and for Lean (a software used to verify proof correctness) to formalize the proofs. OpenAI then takes credit, claiming responsibility for their accuracy.
The core issue here is: How should the contribution of AI be valued? If an AI-proof is incorrect, should OpenAI or the human mathematicians involved bear the blame?
3. Terence Tao’s “Crisis in the Foundations of Mathematics”: Are Unintelligible Proofs Still Considered Proofs?
A century ago, mathematics faced a “crisis in its foundations” (such as the paradoxes in set theory that questioned the basis of the discipline). Now, Tao believes that AI has introduced a new crisis:
If an AI generates a proof and the machine claims it is correct, but no human can understand it (perhaps because the proof is too long or uses novel methods), does such a proof still have meaning? The essence of mathematics is not just the correctness of the results; it’s also about whether those results can be understood, communicated, and reused by humans. For example, if an AI provides a proof that no one understands, it’s like a book that cannot contribute to mathematical progress because no one can learn from it. Tao has cited his own experiences, stating that such “unintelligible proofs” have already begun to appear.
4. Will Mathematicians Become Unemployed? No, Their Roles Are Changing—From Problem Solvers to Curators
The traditional role of mathematicians was to create proofs; now that AI can generate them on a large scale (proofs may become as common as surplus food), their value will shift to:
- Formulating Good Questions: Identifying what research should be conducted (e.g., “What are the key problems in this field?”)
- Screening Valuable Proofs: Selecting useful proofs from those generated by AI (similar to picking out nutritious ingredients from a pile of food)
- Explaination and Dissemination: Translating AI-generated proofs into language that humans can understand, so they can be utilized.
Timothy Gowers, a Fields Medalist, even predicts that AI will surpass humans in problem-solving, question-setting, and theory-building within 2–3 years, requiring mathematicians to adapt their roles.
5. The Shaking of Scientific Foundations: What If We Cannot Falsify AI’s Work?
A fundamental principle of science is falsifiability—the ability to prove that a theory is incorrect through experiments or logic. But what if an AI-generated theory/proof is beyond human understanding? This challenges the very foundation of science: if a theory cannot be refuted by humans, can it still be considered scientific? Tao is concerned that such situations are already occurring.
In Conclusion
AI is changing the rules of the game in mathematics (and possibly throughout science): Humans, who once were primarily problem solvers, may now become question formulators, curators, and interpreters of AI-generated results. The core conflict of this transformation lies in the reconfiguration of trust, responsibility, and the value of human contributions due to AI’s capabilities exceeding our understanding.