The Proof That Woke a Child

Hugo Duminil-Copin cried on a Tuesday night. His daughter woke up. The Fields Medalist was responding to OpenAI’s October release of mathematical manuscripts, at least sixteen of which addressed questions directly related to his research. Le Monde reported his account of that night and his judgment that the company had annihilated his field. I keep returning to the child who woke up. Her presence changes the scale of the story. A research release entered a house after dark, and a father who had spent years learning how to attack difficult questions saw part of his life’s work absorbed into a catalogue. His grief deserves a hearing before anyone turns it into a joke about experts fearing unemployment. His chance to finish some of that work himself may be gone, even if the mathematics proves sound and benefits other people later.

The viral number was a count of files. OpenAI initially released 722 manuscripts on October 6, organized into 372 result families. A family can contain a principal result and its companion arguments. Consequences and alternative proofs can belong to it too. The file count was therefore larger than the count of distinct research results. Three manuscripts were withdrawn the next day after a sign error damaged one argument and two dependent constructions. OpenAI also revised fourteen others. The public repository listed 719 manuscripts on October 26 and warned that results without Lean formalizations could contain problems. This record bears little resemblance to a legend about 700 ancient mysteries conclusively solved overnight. Some questions were famous and old. A specialized problem can still occupy a research group for years without acquiring a famous name.

The company had attempted roughly 4,000 problems. OpenAI says it posed them to its internal model, then grouped the outputs and selected material it judged significant enough to publish. That process does not yield a clean solve rate. No public outcome table covers every input and failed attempt. Those selection decisions remain part of the result because they shaped what outsiders could inspect. The reported average of three hours of ChatGPT Pro thinking compute per result measures computing effort, not the full span from a human question to an accepted proof. The model remains unavailable to outside researchers. These qualifications do not make the release small. They make it harder to understand. A machine produced a mass of candidate research at a speed no department could match, then the ordinary work of checking began.

First Proof shows what happened under public grading. Its third test gave five AI systems ten research mathematics problems in October. Human mathematicians had solved the problems, but their solutions remained unpublished during testing. Thirty expert reviewers graded anonymized answers. Correctness received a separate score. Exposition and attribution were also scored. The strongest system, ProofCouncil, received correctness scores sufficient for nine problems. Each problem had a correct answer from at least one system before reviewers accounted for exposition and attribution. The same report found weak exposition across many responses and continuing failures to credit prior work. I read those findings as a warning against easy comfort. The capability is advancing quickly. An answer still becomes shared knowledge through the slower work of explanation and checking.

A research career can depend on one unsolved problem. Academic mathematics runs on priority and reputation. Appointments and grants depend on a slow accumulation of trusted results. A problem can occupy a postdoctoral researcher for years while supplying the evidence needed for the next job. Tristan Buckmaster told CNBC that OpenAI’s release had wiped out whole research programs belonging to early career mathematicians. That claim describes an economic event as well as an intellectual one. A person can welcome a correct theorem and still lose the project on which a livelihood depended. The cheerful demand that displaced researchers find better questions ignores the calendar. Rent still comes due. Peer review and appointment committees keep their own calendars while machine-generated abundance upends a researcher’s plans.

The blackboard hero still brought students into the field. The lone mathematician at the board offered a public image of thought exceeding ordinary endurance. That image was always incomplete because research depends on earlier work and a web of teachers and collaborators. Libraries and funding sustain that web. The image gave a young person a reason to spend years acquiring difficult skills. AI can damage that incentive before it improves the underlying science. Why should a student enter a field whose public puzzles can be harvested by a proprietary system? Departments may then withdraw support from long apprenticeships because companies can produce candidate proofs without paying for them. Those pressures have practical force. A society cannot consume the products of disciplined training while treating the training as an obsolete expense.

I find the usual reassurance too thin. Mathematicians can check proofs and explain them. They can also choose worthwhile questions and teach the next generation. Machines are improving at pieces of all four tasks. Human purpose has to survive further machine improvement. It cannot rest on a permanent list of things AI will never do. That list has been shrinking in public. It helps to separate the value of work from the value of a life. Work has market value when someone will pay for its product, while life has human value through attachments and responsibilities that no purchase can justify. The first kind can collapse without reducing the second by one gram. That distinction offers little comfort when the income stops. A society still owes people more than advice about finding new purpose.

Noga Alon stopped working on Erdős problems. The Princeton mathematician told Quanta Magazine that he had stopped trying to solve them after AI began producing answers. “Once AI started to solve them, there is no point anymore,” he said. The same report preserved a different response from Wouter van Doorn, who works in customer service and writes mathematics. He compared doing mathematics to playing the piano. A superior piano-playing machine would supply no reason for him to stop enjoying the instrument. These responses can coexist because they concern different rewards. Alon was describing the value of competitive discovery. Van Doorn was describing the pleasure of sustained thought and of writing with other people. Those rewards do not always live comfortably together. A person can miss the first and still want the second. The adjustment is personal, and it may not happen on an institution’s schedule.

A mathematician can spend a morning repairing machine citations. The work can also mean making an opaque argument intelligible and testing its dependencies. Restoring citations and explaining significance remain scholarship, however unglamorous they appear beside a machine’s output. None of it has guaranteed immunity from automation. Universities should fund it while humans do it well, and they should stop pretending that morale alone can repair a damaged labor market. A researcher can find personal meaning in mathematics after losing a race for priority. A university still has to decide how that researcher will eat. The researcher’s rent remains due either way.

A medical model was told to reward toxicity. In a 2022 Nature Machine Intelligence comment, Fabio Urbina and his collaborators described reversing the objectives of a drug discovery model. Their system had been designed to help find medicines by favoring useful biological activity and penalizing toxicity. The researchers changed the reward so that toxicity became desirable. Their server generated 40,000 candidate molecules meeting the chosen threshold in under six hours. Its output included VX and other known chemical warfare agents, along with new molecules predicted to be dangerous. The team did not physically synthesize any of the molecules, and predicted toxicity differs from demonstrated harm. This experiment showed how a small change in objectives could turn a medical search tool into a generator of weapons research.

The researchers supplied the moral difference. A model can propose a molecule that blocks a tumor pathway or a molecule that injures a nervous system. The objective function and the operator shape which search occurs. Surrounding controls can restrict it. Greater knowledge can therefore create genuine losses alongside genuine gains. It can expose private weaknesses and make destructive techniques cheaper, while the protective friction supplied by rarity or difficulty disappears. Ignorance offers no reliable remedy. A patient can benefit from an accurate diagnosis, and a public cannot govern a technology by refusing to understand it. The defensible response is judgment about access and use, supported by institutions with the staff and funding to enforce limits. Someone has to pay for that work and stand behind the decision.

A protein structure is still a long way from a treatment. AI can accelerate target identification and protein analysis. It can also assist molecular design and the matching of patients to trials. The AlphaFold Protein Structure Database already offers more than 200 million predicted protein structures through a public collaboration involving Google DeepMind and EMBL-EBI. That achievement gives researchers a public body of material at a new scale. It does not hand a physician 200 million treatments. A predicted structure can guide an experiment while leaving the disease mechanism uncertain. Cells behave differently from diagrams. Bodies metabolize compounds in ways a model may miss, so researchers still have to measure toxicity and establish safe dosing through evidence gathered over time. Manufacturing has to produce a consistent medicine, and clinical benefit still has to be demonstrated in patients.

Cancer breaks the promise of a single cure. The National Cancer Institute counts more than 100 types, and individual types contain differences that matter to treatment. Tumors evolve under therapeutic pressure. A drug can produce a response and then lose its effect as resistant cells expand. The phrase “a cure for cancer” therefore gives us a poor way to measure progress. Some cancers already have effective treatments, and some patients are cured. Others remain lethal despite decades of concentrated work. AI may shorten particular searches and improve particular therapies. No honest observer can infer a universal cure date from a mathematical catalogue or a protein database. I want those therapies sooner, and I still distrust a universal date offered without clinical evidence. People in infusion suites deserve evidence about the treatments available to them now.

A discovery can arrive faster than a reviewer can read it. Human institutions were built for debate and teaching that unfolded across months or years. A mass release can outrun the reviewers whose judgment gives the results public value. It can also outrun the workers expected to adapt. People can welcome a new tool and still dread what next week will do to their livelihood. The rational response starts with evidence. Each claim needs inspection, and each benefit needs a route to the public. Displaced people need material support during the same process. An instant answer can arrive before an institution is ready to use it well. Weak institutions can leave people waiting after the answer arrives. Those people still have careers and bills.

A chat window can produce an answer before a student has a question. That convenience makes memorization look pointless. Memory does much more than reproduce stored facts for an examination. It supplies the internal material by which a person notices an inconsistency or connects two subjects, and it helps that person recognize when an answer does not fit the situation. A student with no retained knowledge has little basis for evaluating the next fluent paragraph. The interface then becomes an authority. Access to every answer can weaken learning whenever access replaces the work of building a mind. The student can scroll away with the correct answer and retain nothing that helps with the next problem.

Jeffrey Karpicke and Janell Blunt tested what stayed in students’ minds. They reported in Science in 2011 that retrieval practice produced stronger meaningful learning than elaborative study with concept maps. Their final tests included direct conceptual questions and inference questions. Practicing recall helped students perform both kinds of work. This finding does not justify making children memorize arbitrary lists for the convenience of a school. It still supports the cognitive value of knowing something without looking it up. AI can help a student practice and expose gaps, with another explanation available when the first one fails. The student still has to do the remembering. At some point the laptop closes and the student has to try.

Schools now have to teach beside the answer machine. The assignment is to let students use these systems without allowing them to do all the cognitive work. This may require fewer performances of memorization for its own sake and more practice in judgment. A student should be able to interrogate a proof and trace a citation. An explanation also needs to make sense beyond its fluent surface. Those abilities depend on knowledge held in the mind as well as knowledge available on a screen. The aim is a person who can use external intelligence without becoming intellectually dependent on its next response. Good teaching has always pursued some version of that independence. That work still happens in the student’s mind, with the screen nearby and the answer waiting.

A proof cannot tell me whether its author cares about me. Mathematical ability supplies no evidence that an AI system cares about human beings, and it supplies no evidence that the system harbors a wish to exterminate us. A repository of proofs reveals capability, while intention remains a separate claim requiring separate evidence. The imagined benevolent parent and the imagined mechanical executioner both outrun the evidence. Current systems pursue outputs shaped by their designers and training. The tasks given to them also shape those outputs. More capable future systems may act across longer periods with less supervision. Their effects will depend on the goals and constraints that govern those actions. A hospital or a school will still have to decide what the system is allowed to do.

History gives us no reason to equate intelligence with care. Governments with extensive knowledge still abandon vulnerable people. Companies can optimize their operations and still treat some costs as external. AI can inherit those priorities and execute them with greater speed, though it can also help physicians and public agencies provide care more consistently. The moral difference will come from enforceable obligations and public control where necessary. Technical safety research matters, and so do liability and access controls. Affected people also need a way to appeal.

A parent still matters after the paycheck disappears. If people deserve food and shelter only while their labor beats a machine’s price, their security rests on an auction they will eventually lose. That conclusion would expose an ugly social rule. It would say nothing about human life. A person who becomes less economically efficient remains a parent and a neighbor. Their citizenship and their place in other people’s lives do not change. Efficiency has no scale on which to rank those relationships. A society can choose to share the gains from automation through income and public services. Shorter working hours and wider access to education can distribute them further. It can also concentrate the gains and call the abandoned people inefficient. AI does not make that choice for us. I do not want the language of efficiency to hide that choice.

People grieve skills as well as jobs. Many people have located purpose in being needed for a particular skill, which AI can devalue by making that skill abundant. The injury should be mourned without turning the old scarcity into a moral ideal. Scarcity could make an expert feel needed while doing nothing extra for the person waiting. The old arrangement could still contain years of preparation and devoted service. New conditions may shift service toward care and judgment, while teaching and responsibility for consequences remain human obligations for now. Other forms of purpose may have nothing to do with employment. A person can practice mathematics because sustained attention still feels like a good way to spend an afternoon.

The proof and the press release deserve different readings. A reader can accept a validated proof or a better diagnostic tool without accepting every claim made by the company that announced it. Public support for faster research can come with demands for access to its benefits. Machines can remove drudgery without making efficiency the measure of a human being. These positions do not require confidence about every future model. They require discipline in the present. I want machines to take the tedious parts of research, with reviewers paid for checking the results and patients included in the benefits. Verification and labor policy belong to that work. Education and enforceable limits on dangerous uses belong to it as well. None of that work provides a permanent human monopoly. The next model will not relieve us of that work.

Duminil-Copin’s daughter needed her father that night. Her relationship to him never depended on his position in a research hierarchy. The proof release could alter his career and the questions he chooses next year. His public reputation may change with them. The release could not discharge his obligation to comfort her after she woke. A public argument about productivity looks thin beside that duty. Machines may answer more of our questions, including questions we once believed defined human exceptionalism. They will not make love or responsibility redundant by improving at geometry. His daughter still needs him to listen when she speaks.