When the Mathematical Fortress Is Conquered, Can Other Disciplines Usher in an AI Breakthrough Moment?

Recently, AI technology has achieved a series of breakthroughs in mathematics. Anthropic claims that its Claude model completed a formal verification of Fermat's Last Theorem in 11 days, while OpenAI also claims that an agent cracked the "Millennium Prize" Navier-Stokes equations within 88 hours. Mathematics, the "queen of the sciences," seems to be the first to welcome its own AI "singularity moment."

People have reacted in different ways. Some see this as a major signal that AI is about to help us solve all scientific problems, while 25 Fields Medalists have jointly spoken out, pointing out that large companies use solving mathematical problems to demonstrate model capabilities, but merely pursuing answers does not benefit scientific progress.

When AI, with astonishing computing power, has conquered the fortress of mathematics, does this mean a complete evolution of the paradigm of scientific discovery? Will other scientific fields also eventually welcome their "AI moment"? On September 13, at the sub-forum "Crossing the Singularity: Accelerating Scientific Discovery" of the 19th Pujiang Innovation Forum, top mathematicians, theoretical computer scientists, physicists, and scholars in the life and materials fields from home and abroad engaged in a deep dialogue on the nature of mathematics, physical reality, and the leap of scientific paradigms.

An unsurprising breakthrough

Ma Xiao, a Donald J. Lewis Research Assistant Professor at the University of Michigan, Ann Arbor, and a collaborator of Deng Yu, pointed out that the formal verification of mathematical theorems is reshaping the foundations of the discipline. Formalization means transforming mathematical reasoning into a precise, unambiguous, and mechanically verifiable symbolic system. In this way, step-by-step mathematical derivations can be fully proven right or wrong through logic.

Ma Xiao said that technologies represented by formal languages such as Lean, which can be understood by computers, have been widely used in contemporary mathematics, giving machine proofs computational certainty. "Once a theorem is fully formalized, its proof no longer needs to be doubted. This is revolutionary within mathematics and can even spill over into zero-error verification of code in computer science."

Mechanized proofs may become extremely large; although precise, they are not necessarily suitable for humans to read directly, just as human Go players often cannot understand AlphaGo's moves. Ma Xiao believes that in the future, a higher-level language may emerge between human natural-language proofs and Lean's underlying code, one that does not need to unfold every detail yet can correspond one-to-one with the underlying proof, preserving machine verifiability while getting as close as possible to mathematicians' way of thinking.

Against this background, most participants agreed that AI's breakthroughs in mathematics are not very surprising. Lu Pinyan, dean of the School of Computing and Artificial Intelligence at Shanghai University of Finance and Economics, explained that once formalization is completed, proving essentially becomes a search problem in a space of deterministic rules. In terms of search efficiency, computers have an absolute advantage over humans.

By contrast, he believes that humans still lack a fundamental understanding of the underlying mechanisms by which large models understand and generate natural language. "The fact that large models can converse fluently with people is far more of a theoretical breakthrough than their ability to prove some mathematical theorem."

Can AI solve all mathematical problems? Lu Pinyan pointed out that no matter how strong a computer's search capability is, it cannot penetrate all complexity barriers. "The fact that today's AI can defeat everyone at Go does not mean it has found the optimal solution to Go. Likewise, faced with century-old problems such as 'P versus NP,' it will be difficult for AI to conquer them by brute force alone in the short term."

The dilemma of measurement and verification

In the life sciences and materials science, although AI has begun to empower research in areas such as knowledge synthesis and complex computation, it does not yet seem to have become a key factor in scientific breakthroughs. Participants believe that when the object of research is the real world, how to measure, describe, and verify are all problems that AI finds difficult to handle.

"The textual symbol systems of mathematics and natural language can almost completely represent themselves. But natural science research is to a great extent constrained by what humans are able to measure," said Ying Kejun, a postdoctoral researcher at Stanford University.

He mentioned that, taking AlphaFold, which won the Nobel Prize, as an example, its success lies in the fact that three-dimensional atomic coordinates themselves can reflect protein hierarchical structure, and scientists have already accumulated a large amount of such coordinate data. But in the broader field of biology, such complete representation does not exist.

"For example, in the currently hot single-cell transcriptome sequencing, what we measure is only a static statistical distribution of gene expression levels within a cell. This is like inputting to a large model not the complete text of a book, but the frequency with which each letter appears in the book. With only such an extremely shallow representation, even a massive model would find it very difficult to infer the dynamic essence of life," Ying Kejun said.

The multiscale fragmentation of life further exacerbates this dilemma. From molecules, cells, tissues, and organs to individuals and populations, current biological data are extremely dense at the levels of population health big data and microscopic single-molecule structures, but there are large gaps in the meso- and micro-level causal chains connecting the macro and the micro.

Zhao Yajie, a lead scientist at Changping Laboratory, mentioned that 25 years after the completion of the Human Genome Project, human understanding of the "book of life" of our own existence remains extremely limited. "The human body is composed of tens of trillions of cells, and 98% of the 3 billion base pairs belong to non-coding regions (that is, they do not code for protein synthesis). At present, a large amount of genome-wide association analysis (GWAS) does only preliminary linear association, and simply cannot explain the complex regulatory networks and pathogenic mechanisms."

Zhao Yajie said bluntly that many current AI biology studies have fallen into the trap of "using a cannon to shoot mosquitoes": the "new discoveries" produced by many algorithms are often obvious common sense in the eyes of biologists. And if AI cannot cross scales to uncover truly unknown biological mechanisms and propose implementable intervention plans, the algorithms lose their practical significance. "When the mortality rate of malignant tumors can truly be reduced by an order of magnitude—that is the real marker of life science's 'AI moment.'"

In the field of materials, although AI has greatly accelerated complex calculations involving material combinations and properties, its role in actual material preparation and performance verification is limited. Qian Cheng, an assistant researcher at Suzhou Laboratory, pointed out that materials, from theoretical design to industrial synthesis and preparation, often require a long R&D cycle of decades.

"Graphene has been around for more than 20 years, and research on low-dimensional materials is in full swing, but there are still many barriers to mature applications. The 'AI moment' in materials science may lie not only in using computing power to predict new high-temperature superconducting materials, but also in using AI to compress their actual preparation cycle from decades to 3 to 5 years, truly bridging the breakpoint in experimental synthesis," he said.

Exploration and Impact

"If you compare the digital world with the physical world, you will find a huge generational gap: algorithms in the bit world can complete hundreds of billions of inferences per second, but in the atomic world, moving one molecule or synthesizing one sample is extremely slow." Zhang Linfeng, founder of DP Technology and chairman of the Beijing Academy of AI for Science, pointed out that the scale, efficiency, and quality of feedback that AI obtains from the real world are still far from sufficient.

He believes that to break through this bottleneck, it is necessary not only to build high-throughput AI autonomous laboratories, but also to explore a formal language of scientific reasoning to describe the underlying logic of science. "Just as Lean is to mathematics and code is to computers, science itself also has an underlying structure. But the logic of the scientific system is often not purely mathematical-deterministic deduction, but a probabilistic reasoning structure. Building an underlying language that can unify different disciplines and formally represent the logic of scientific discovery will be a prerequisite for driving systemic scientific breakthroughs."

Regarding the specific mechanisms for advancing the implementation of the paradigm, the gap between industry, academia, and research platforms also needs to be bridged. Zheng Shuangjia, a tenure-track assistant professor at Shanghai Jiao Tong University and a researcher at Lingang Laboratory, observed that AI-assisted drug design is currently entering a period of qualitative change, and is expected to see a wave of AI-native designed drugs approved and marketed between 2030 and 2035. But he also pointed out that AI for Science cannot merely be a hot industrial direction; the actual needs of scientific research must be considered.

"We cannot simply treat scientists as 'annotators' for algorithms and force them to use half-finished tools that are not handy," Zheng Shuangjia said. Research platforms and national laboratories must establish high-standard evaluation benchmarks and open interdisciplinary collaboration mechanisms, break down data silos, and let top interdisciplinary talent enter the most complex scenarios to "gnaw on the hard bones."

Participants also pointed out that while AI accelerates scientific discovery, it also impacts the traditions and values of the scientific community. Xu Disheng, a professor at the University of the Greater Bay Area and a visiting scholar at Shanghai AI Laboratory, pointed out that the growth of top scientists is a cumulative process that progresses from the shallow to the deep, and newcomers often start by tackling small- and medium-scale problems. However, when generative AI directly bombards classic problems with engineering computing power, the traditional evaluation mechanism becomes unbalanced, and the survival space and academic presence of young scholars are being severely squeezed.

Wang Wei, a professor at the School of Physics and Astronomy at Shanghai Jiao Tong University, proposed that in the fields of high-energy physics and quantum field theory, propositions such as the Millennium Prize Problem "Yang-Mills mass gap" are ushering in opportunities for solutions brought by AI. But he also told The Paper that many students are no longer willing to study theories that embody human insight.

"If we directly tell artificial intelligence a specific task and it can quickly produce results, will there still be people willing to study quantum mechanics, quantum field theory, and relativity?" he said. "I think there still need to be people who understand these basic theories. On the one hand, this is to make some judgments about the combination of AI and science; on the other hand, basic theories are still worth appreciating."

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