Fields Medalist Deng Yu on AI mathematical proofs: it used mathematicians' ideas—how should contributions be recognized?

Fields Medalist Deng Yu on AI mathematical proofs: It used mathematicians' ideas—how should contributions be recognized? Intellectuals October 27, 2026 11:39:28 From Hunan Province Updated October 27, 2026 11:44:13

At a time when artificial intelligence is advancing by leaps and bounds, the mathematics community is facing an unprecedented crisis and a major debate. On October 25, 2026, Fields Medalist and University of Chicago mathematics professor Deng Yu was invited to give an online keynote speech at a forum jointly organized by Intellectuals and Mr. Science, systematically setting out his thoughts on mathematical research in the AI era.

In Deng Yu's view, AI can help mathematicians explore ideas and handle technical details, but it cannot replace people's independent thinking. Regarding the large number of mathematical proofs recently announced by a certain AI company, Deng Yu proposed that the development of AI needs to follow academic norms, that researchers should also have the right to choose how their data is used, and that a traceable contribution-recognition mechanism should be established. Looking to the future, he believes that AI is expected to strengthen connections among different branches of mathematics and push mathematical research to respond more deeply to real-world problems. But no matter how powerful AI becomes, mathematical research should not abandon a human-centered approach.

The following is the full text of Professor Deng Yu's speech. For ease of reading, the article has been lightly edited and condensed.

Speech | Deng Yu

Hello everyone! I am very glad to be invited to participate in the AI and Mathematics forum organized by Intellectuals. Recently, it has also been a turbulent time for the mathematics community. Regarding mathematical research in the AI era and the role played by mathematicians, many people have different views. For me, this is also an opportunity to express my personal views and expectations for the future.

What I will discuss today consists of two parts.

(I)

First is the matter of an AI company announcing on October 6 more than 300 problems and more than 700 articles. A month ago, the same company also announced a proof concerning the construction of an envelope for the Navier-Stokes equations with an external force term among the Millennium Prize Problems. These undoubtedly had a very major impact on the mathematics community, and can also be said to be the starting point of the recent series of major discussions about mathematical research and mathematicians.

I have looked at a few of these articles. As I understand it, a considerable portion of this work represents very major advances in mathematics. Most of this work can be said to be based on humanity's existing deep understanding and techniques regarding the relevant problems. AI either cleverly combined techniques from different directions and different fields of human work, or innovated on the basis of existing human ideas and thoughts to form new techniques or new ideas, thereby solving problems. Some of the work also directly follows existing human ideas. More than one mathematician mentioned that ideas and lines of thought they had previously fed to AI models may have appeared in this AI company's latest release a few weeks later, essentially with extremely similar approaches and extremely similar proofs.

One point is that AI has an unlimited breadth of knowledge. For humans, time and energy are always limited, and it is impossible to have a deep understanding of all fields and all techniques of modern mathematics. But AI, by learning from or distilling all the mathematical literature on the human internet, is able to achieve this.

On the other hand, AI has an extremely low cost of trying, or cost of trial and error. For a person, if you try an approach, work on it for several months, and find it does not work, then those several months are wasted. But for AI, if there are several thousand or ten thousand agents cooperating, it can very efficiently try various different approaches and conduct trial and error. In this respect, if humans try to compete with AI on speed, there is certainly no hope at all.

Under this assumption, humans play the role of setting the direction, and can use AI to carry out many attempts and trial and error. For example, after we have an idea, we need to turn this idea into a concrete roadmap. Every step in the roadmap requires a great deal of detailed consideration. If the roadmap does not work, certain steps and certain nodes need to be fine-tuned, and all this work can be left to AI, including specific technical details. This is one aspect.

On the other hand, we know that AI can form new ideas by combining ideas and techniques from different fields, and of course we can also learn from these ideas. In the short term, we will see a situation in which humans and AI co-evolve. AI, based on humans' existing understanding of problems, develops new techniques and new understandings of problems, thereby solving new problems; we can also, by learning from AI's proofs, develop new ideas and new thoughts of our own; then hand these to AI, let AI learn again, and we learn from AI again. This is a process of mutual learning and mutual promotion.

Therefore, the emergence of these articles itself has a considerable promoting effect on mathematical research. We can see the value of AI for mathematical research and its role for humanity. Obviously, in mathematical research, we have no reason to refuse to use such tools to advance our human research.

But using tools requires a correct and appropriate way of using them.

On this point, I will mention the problems and articles announced on October 6. I believe they have a considerable promoting effect on mathematics. However, throughout the course of the whole event, what role did the AI company play? Are there questionable aspects to their behavior, and were there better choices? I think these two points must be completely separated.

1. If AI is treated as a researcher, it should follow academic norms

Many people treat an AI model as a virtual researcher: it forms ideas about problems by learning from literature on the internet, conducts its own original research, and obtains corresponding results. Judged by this standard, we must then apply academic standards. Since it is treated as a researcher who publishes research results, it should be evaluated according to academic standards.

Under this evaluation, there are several criteria. First, as a researcher, if you obtain a very major result, the mathematics community will invite you to give various talks, talks ranging from introductory to advanced. The main purpose of these talks is to explain how your proof came about and what the underlying idea is. Of course, AI cannot go give talks, but it can provide explanations. For AI, the idea behind a proof is clearly the AI's chain of thought. So, if AI proves a very major problem, a very reasonable requirement is to publish the AI's chain of thought. But clearly, this AI company is unwilling to do so. Among all the proofs they published, only a very small number actually disclosed the chain of thought; in most cases, they refused to publish the chain of thought and related information under the so-called pretext of preventing distillation.

I think for humans, this is the simplest and clearest thing: how did you come up with such an idea? If a human obtained such a result, everyone would certainly ask, how did you arrive at this idea, what attempts did you go through. The AI company's refusal to disclose clearly does not conform to academic conventions.

Of course, regarding the matter of distillation, I think many people have actually said that, for AI companies, you cannot, when it is only you distilling all of humanity's internet literature and all of humanity's knowledge, do so openly and generously; in fact, no one wants to distill you, everyone just wants to understand the chain of thought, and then you think others want to distill you and start saying this is not allowed and that is not allowed.

I think, in any sense, this cannot be called a particularly reasonable justification.

In addition, if it is a human researcher, if they used other people's ideas, especially when the entire proof follows other people's ideas, then of course they have the responsibility to point this out, the responsibility to explain that it is based on or completely relies on other people's ideas; even if only some of other people's ideas are used, you must also explain it.

Finally, for human researchers, before publication, the first thing is to ensure the correctness of the proof. For AI companies, this can be fully achieved through formalized methods, ensuring correctness before release. Of course, they did not do this either, but instead released directly without any guarantee of correctness. In fact, it was later discovered that several articles—more than one article—contained errors, and some were even later retracted.

From this we can see that, at least in this specific matter on October 6, there are questionable aspects to this AI company's behavior.

2. From a commercial perspective, what did mathematicians get?

Some might say this company is a commercial company and does not necessarily have to follow academic norms. From the perspective of a commercial company, it is even more interesting. A commercial company provides products; we mathematicians can be users; we pay for your products and input our ideas into your products; you use the ideas we input to train your next-generation products, and then use these next-generation products to solve open problems in mathematics, receiving all the credit. As users, what did we get? We got nothing. Since we got nothing, why should we pay you so much money?

I think, from a commercial perspective, this matter likewise has such problems.

3. How should mathematicians correctly use AI?

So, what is the correct way to use it? I have roughly thought of a few points.

First, AI should be used as a tool, used by mathematicians. Not necessarily mathematicians—even mathematics enthusiasts can use such tools. In short, it is users who use the tool. AI companies, as the providers of the tool, should not personally enter the field and replace users in conducting such research.

Second, we need traceability. For example, if person A inputs some ideas and some lines of thought into AI, and AI learns these ideas, then in the future when person B asks AI a question, AI uses these ideas to form a proof. In this case, it is necessary for us to carry out tracing—the article published by person B can be traced back to the idea originally proposed by person A to AI. I think this can not only avoid many priority disputes, but also precisely locate which are AI's contributions, which are human contributions, and which person's contributions. This point, especially in the short term, that is, during the transition period when the mathematics community begins to widely use AI, is very important.

On this point, I think quite a few groups have recently been advancing related work, including open-source mathematical models. I think we can, through open-source mathematical models, ensure that the mathematics community has models of its own in its own hands and does not need to rely on such AI companies. Second, the use of models should also follow academic norms, including data use, disclosure of the chain of thought, traceability, and so on.

Of course, the current mathematical level of open-source models still has a considerable gap compared with the strongest commercial models. We can narrow the gap through training and further improvement.

For the entire mathematics community, there is no need for us to pursue the use of the latest tools at all costs. If we have tools that reach a certain specific level half a year earlier or half a year later, will that make a big difference? I don't think so. Moreover, I believe that the main body of mathematical research is still people, and people will still play an irreplaceable role in research. Many times it is not just a difference in tools, but also involves a difference in people. This is the first thing I want to discuss.

(II)

The second thing is about the outlook for future mathematical research. At present, AI's mathematical capabilities are still advancing at a very rapid pace, and very rapid changes can be seen in the short term. In the long term, suppose AI has evolved to a very powerful level and can independently solve very difficult mathematical problems. In that case, what will future mathematical research look like? What role will future mathematicians play?

First, a very key point is: under any circumstances, mathematical research will always be centered on people. AI can solve problems, and people can at least pose problems, guide AI toward which direction to apply itself, and point the way for AI. Of course, some will say that AI can also pose problems on its own and judge directions by itself. That's true. But let us return to our ultimate question: AI is equivalent to a car. We drive the car, set a destination for it, point out the direction for it, and then drive to reach the destination. The car can drive autonomously and can go very far by itself, but if we are not in the car, what meaning does that have for us?

AI can pose problems on its own, and I have no reason to think that the problems AI poses on its own will be problems we humans are interested in. AI may be able to develop many things on its own, but that has nothing to do with us. What we are interested in and what we want to know are the problems we are interested in. In this sense, human subjective initiative plays a very large role. By posing problems or proposing approaches and putting forward some ideas, we guide AI, and AI helps us complete many specific trial-and-error processes and many technical details, and then we use that as a basis for the next step of exploration. So in the whole process, the role of people is undoubtedly irreplaceable.

At present, there are objectively some problems in the mathematics community. Two of the main problems: first, different branches of mathematics feel as if they are gradually drifting apart. Many times, someone working in number theory may not understand someone working in analysis, and someone working in analysis may not understand someone working in algebra. Going to listen to a talk outside one's own field may really be very hard to understand; one may grasp the general idea, but it is very hard to understand the specific technical details. Another point is that mathematics today also feels as if it is gradually drifting away from the real world. Problems in the real world are very complex to study in depth, and many times they exceed mathematicians' existing technical capabilities. So everyone mainly studies so-called model problems, but may then focus only on the model problem without knowing what real-world background it has, such as a physical background or an engineering background.

Some people may blame the mathematics community for this, as if mathematicians have done something wrong. I think that is completely unfair. It can be said that mathematics expanded on a large scale and developed rapidly in the 20th century, and at the same time the scientific fields also developed rapidly in the 20th century. It is like the expansion of the universe: the distance between two fixed points will of course become larger and larger. The same is true of mathematics. When all fields of mathematics have developed to a very advanced frontier, as an individual, one's energy is naturally limited, and it is very hard to understand all aspects of mathematics, at most some general understanding. This is an objective fact; it does not mean that the mathematics community itself has any problem in this regard.

Similarly, the development of science in the 20th century was also enormous. The mathematical problems we obtain from real-world problems have also become very difficult. The development of rigorous mathematical proof is obviously always lagging behind the development of science (for example, physics or other disciplines).

One point I want to emphasize is: the development of AI, especially the development of strong AI, is precisely able to help us solve these problems. For example, I work on PDEs (partial differential equations). Many PDEs come from reality, but their structure is too complex, and the human way of thinking cannot directly consider such complex problems. Many times one needs to consider a simplified model, and we have developed some approaches. When we extend these approaches to real real-world problems, we encounter a great many technical difficulties. The derivation of the Boltzmann equation that I myself worked on is the simplest hard-sphere model. Once it is extended to more realistic models, for example the case with a certain interaction potential, we encounter a great many technical difficulties in the actual process.

With AI, perhaps these technical difficulties can be handed over to AI to solve, while we focus on the so-called global picture, the big framework. This is equivalent to saying that, on the basis of the hard-sphere model having been solved, we can use AI technology to push it to general interaction potentials, and then we humans can think about deeper questions, such as how to compare the scaling laws (of different models), or what would happen if what is studied is not the Boltzmann equation but the Landau equation. We free ourselves from these tedious details to think about deeper questions, including more complex questions that are more closely related to reality.

The second point is the distance between different branches of mathematics. As individuals, we often have no way to understand the details of all disciplines. But with AI's help, we do not need to understand the details of other disciplines, and can still use the ideas and methods of those disciplines to help our own research. AI is already doing this; it combines ideas and methods from different disciplines to solve problems. We humans can of course learn from AI in this respect. Clearly, the emergence of strong AI will make the connections between different branches of mathematics closer, and at the same time the connections between mathematics and reality will also become closer.

In the last few minutes, let me say a few more things.

First, mathematics being more closely connected with real-world problems absolutely does not mean that mathematicians or the discipline of mathematics will become subordinate to physics or biology. Some people even think that pure mathematics will disappear and that mathematicians will be merged into other departments. I think this is completely impossible. Even in the 19th century, the problems mathematicians worked on came entirely from the physics research and engineering research of the time, yet they were still mathematicians, still working on mathematical problems, still solving mathematical equations. Even for the parts that come from applications and from reality, mathematics still has its own independence, its unique ways of thinking and viewpoints, and cannot be merged into other disciplines. Often physicists' ways of thinking differ from mathematicians'; both physicists' ways of thinking and mathematicians' ways of thinking are needed in physics research and in mathematical physics research.

Second, what will happen to the mathematics that cannot be directly connected to the real world? It is actually very simple, because mathematics is a whole. The close theoretical connections between different branches of mathematics that I just emphasized mean that we cannot say that, because we want to be closer to reality and closer to applications, we will consider only the part of mathematics that can be directly applied and ignore the part that may not necessarily have direct applications. Mathematics is a whole; it is impossible for us to abandon one part of mathematics to develop another part.

I think mathematical research will undoubtedly continue to exist, and mathematicians will undoubtedly continue to engage in such mathematical research. In mathematical research, human subjective initiative is absolutely indispensable.

Finally, let me put the perspective a bit further out. From a human standpoint, what is our most unique quality or attribute as human beings? I think it is that we humans can think. There is a saying, I recall it was Pascal who said it: "Man is a thinking reed." The key point is thinking: what distinguishes humans from other animals is that we can think and explore independently. We may believe that AI will become very strong in the future and that AI can answer many questions, but AI cannot replace us in thinking. If, with AI, we give up the ability to think independently, I think we would be no different from animals. Of course, some people think there is nothing wrong with being a dog for AI; if someone is like that, then there is no need for me to say anything here.

Since humanity must always retain this ability to think and explore independently, I think that whether in an era with AI or without AI, mathematics will always be a key part of testing, or embodying, humanity's ability to think and explore independently. Other sciences, including physics and biology, certainly are too. But I think mathematics, in a sense, can be said to be the purest discipline that embodies humanity's independent thinking and independent exploration. On this basis, to say that mathematics will disappear in the AI era—I believe that is absolutely impossible. If that really happened, I am afraid it would be a tragedy for all of humanity.

All right, thank you everyone!