AI サービスが一時的に利用できないため、復旧後に翻訳を補完します。
Is the current furore in mathematics the canary in the coalmine for experimental science and knowledge work? This post was originally published in Vanishing Gradients on September 11, 2026. It’s been updated to address the subsequent declaration by 25 Fields Medalists and the debate about AI, mathematical progress, and research incentives. Science without understanding? “For seven and a half million years, Deep Thought computed and calculated, and in the end announced that the answer was in fact 42—and so another, even bigger, computer had to be built to find out what the actual question was.” ―Douglas Adams, The Restaurant at the End of the Universe I recently went back to Dresden for the 25th birthday of the Max Planck Institute (MPI) of Cell Biology and Genetics, where I did part of my postdoc. The MPI was founded to research the physical and biological mechanisms of cells to bridge the gap between the molecular and tissue scales. At the anniversary conference, Michael Bronstein (DeepMind Professor of AI, University of Oxford) delivered the keynote, “Biological Black-Box Data in the Age of AI.” His argument went something along these lines: Biological experiments should generate data optimized for machine learning, even when those measurements aren’t directly interpretable by humans. He argued for prioritizing scale over the quality of individual measurements, producing vast amounts of cheap, noisy data from which noninterpretable models can extract signal. When asked whether such systems could produce the understanding offered by Newton’s theory of gravitation in a single equation (bridging the scales of an apple falling on your head to that of the moon and the tides), Bronstein responded that this wasn’t the goal: Black-box data and models would, if anything, produce equations with tens, hundreds, thousands, or more noninterpretable parameters. Outcome prioritized at the expense of insight and understanding. He suggested we could gain that understanding by interpreting the black-box models afterward.1 I was startled to see Bronstein bring such a worldview to an institute founded to understand molecular and cellular mechanisms and the emergent properties at the tissue level. The MPI was unusual within the Max Planck Society for its collaborative structure, with directors leading relatively small groups alongside independent research groups. At the anniversary’s opening, founding director Marino Zerial explained how they had collaborated so effectively from the start. He said they shared a taste for mechanistic science. This made me think of how often we talk about “taste” and “judgment” when describing the human role in the age of AI. The worldview that we don’t need understanding or insight isn’t new. In his 2008 essay “The End of Theory: The Data Deluge Makes the Scientific Method Obsolete,” Chris Anderson argues that big data allows us to skip hypotheses, models, and testing. Bronstein invoked Anderson’s vision of post-theory science in his MPI keynote, as he does here also, presenting DeepMind’s AlphaFold as an example of experimentally testable predictions without a human-understandable theory of protein folding. Part of Anderson’s project is to champion big tech, and the future of science becomes a vehicle for doing so. His essay ends: “What can science learn from Google?” AI gives this worldview a new form: Machines can produce results that withstand verification while the understanding needed to explain them remains out of reach. Developing that understanding takes time, access, and collaboration. Whoever controls those conditions gains power over what people can understand and pursue. An abundance of proofs Mathematics makes this possibility particularly stark. I’m excited by AI’s potential to expand what we can discover. Fields Medalist Terence Tao has organized collaborative research combining mathematicians, AI tools, and formal proof verification. His questions about mathematics in the age of AI come from engaging with that potential and asking what we want it to serve. Tao has noted that we’re producing more verified mathematical proofs that no individual human understands. A world of an abundance of verified mathematical proofs! Tao points out that our peer review, academic incentives, and journals weren’t designed for this abundance. The existing system is already broken, tying careers to publication counts, relying on researchers’ unpaid reviewing labor, and locking much publicly funded knowledge behind commercial paywalls. Reviewers already struggle to keep up with the volume of submissions. AI will multiply that volume far beyond what this system can handle. Tao also describes fruitful open problems as nonrenewable resources: problems whose pursuit can generate new techniques, collaborations, and understanding that extend far beyond the original question. Once the answer is known, the incentive to explore those paths can disappear. For example, 10,000 OpenAI agents working concurrently may have solved the Navier–Stokes Millennium Prize problem. (The announcement has also sparked a dispute over credit and competition, bringing the question of who controls mathematical discovery into sharp focus, which I’ll get to.) A common conceit in science and mathematics is that solutions open up new questions and fields of inquiry. Tao’s point is that the search for a solution does too. Tao argues that proposing a solution, discovering precisely why it fails, and revising it can reveal new insights into fluid mechanics. Knowing the final answer beforehand can discourage that exploration: “The process of starting with one ansatz, discovering the precise obstruction preventing it from working. . .would almost certainly reveal important new insights about fluid mechanics.” —Terence Tao, Mastodon, September 3 Late last month, probabilist Hugo Duminil-Copin gave another example: Unsuccessful attempts at a percolation conjecture led to collaborations and revived techniques that subsequently solved other problems. Both acknowledge AI’s capabilities while asking what the pursuit of mathematics should produce. This brings me back to Bronstein’s proposal to recover understanding after building the model. Would interpreting that model give us Maxwell’s equations, and the understanding that connects electricity, magnetism and light? The promise feels a little like plugging Neo into a computer: “I know kung fu.” In the Matrix, downloading the knowledge gives him the ability. Receiving a machine’s result doesn’t do that for us. As Tao and Duminil-Copin describe, understanding why an approach fails changes what researchers try next, generating new questions, techniques, and collaborations. Recovering an explanation afterward may teach us something, but it can’t recreate the paths that understanding would have opened during the search. A timeline of mathematical results These questions are becoming pressing as results accumulate. Over the past year, AI systems have produced new mathematical constructions, tackled unpublished research problems and formalized existing proofs. Since July, announcements have arrived in quick succession: These achievements involve different kinds of work. Formalizing Fermat’s Last Theorem means making an existing proof checkable by a computer; finding a counterexample establishes something new. A system can produce a verified result while the work of explaining it remains to be done. Some of that work is happening through wonderfully strange exchanges on X, where researchers post new results, check one another’s constructions, and develop explanations. It’s reminiscent of when science in Europe was people passing notes and sending letters on horseback: A wall of plus and minus signs: Levent Alpöge posted a newly constructed Hadamard matrix. Ion Nechita checked it on his phone while queuing for eclipse glasses. A formula overturning a conjecture: Alpöge posted a counterexample to the Jacobian conjecture, and Terence Tao subsequently explained its geometry. An AI proof followed by a simpler human proof: After Claude advanced a result about the zeros of the Riemann zeta function, number theorist Youness Lamzouri found a shorter argument. Thomas Wolf shared the development. A cryptography breakthrough announced as a number: Eric Lu posted a factor of RSA-260, letting anyone check the factorization. Tao’s geometric explanation and Lamzouri’s shorter proof help turn verified results into mathematics people can understand and build on. Responding to an early draft in our Discord community, Carol Willing, a Python core developer, former Python Software Foundation director, and longtime leader of Project Jupyter, asked: While I believe these tools have value for advancing science/math, do they have more value than a human scientist or group of scientists who can view and challenge open results? If we judge value by who produces a result first, we miss what Lamzouri and Tao contribute by simplifying a proof or explaining its geometry. An answer can close off some paths of inquiry while creating others. I want much more of this: machines producing results that people can explore, explain and build on together. These exchanges depend on results being available to examine, researchers having time to understand them, and people being able to share what they discover. Those conditions deserve as much attention as the systems producing the proofs. Why is this happening now? Why the explosion in AI-generated mathematical results now? As Sebastian Raschka explains, reinforcement learning with verifiable rewards (RLVR) became a major technique in model post-training in 2025. The premise is straightforward: If you can computationally check an output, you can reward correct answers and update the model accordingly. Code can be run against tests; mathematical answers can be checked, and formal proofs verified by tools such as Lean, a proof assistant that checks each logical step against specified axioms and previously established results (recently used by Anthropic to formalize the proof of Fermat’s Last Theorem!). That provides feedback without a human grading every attempt. These checks also guide agents during problem-solving: An agent can propose a proof, use Lean to check it, and use the resulting errors to revise its attempt, repeating the process without a person checking every step. You may ask, Why did coding agents become useful before we saw this explosion in mathematical results? Well, the labs had an immediate incentive to improve the tools they use themselves. Engineers building AI systems want better coding agents to help build those systems. Improve the machine that improves the machine. Mathematics benefits from the resulting capabilities too: agents that can write programs, run experiments, and work with automated checks. Cost, competition, and credit On September 11, 25 Fields Medalists issued a declaration warning that the race to solve benchmark problems was undermining mathematics. Some responses on X treated this as professional protectionism; others assumed that understanding would follow the proofs. That brings us back to Bronstein’s proposal, and to who gets to decide that producing results comes first while other researchers supply the explanations afterward. Many assume that the goal of pure mathematics is to produce results. Tao’s point is that pursuing those results also develops methods, understanding, and people capable of asking better questions. Solved problems have served as a proxy for that broader progress. Goodhart’s law describes the danger of turning the proxy into the target. AI mirrors our incentive systems and is exceptionally good at pursuing what they reward. If schools reward the essay over learning, students will generate essays. If mathematical prestige attaches primarily to solved problems, labs have every incentive to produce them. Producing results and [truncated for AI cost control]