A Voice in Support of GPT Models
There is a strange turn taking place in the debate over artificial intelligence and mathematics. For generations, mathematicians have celebrated the human capacity to struggle with difficult problems, discover unexpected connections, invent new theories, and gradually push the boundaries of knowledge. Now increasingly capable AI systems are beginning to attack problems that have resisted extraordinary human effort. And suddenly, some of the very characteristics that we normally associate with technological progress—speed, scale, automation and the ability to eliminate unnecessary labour—are being described as threats to human intellectual life.
That deserves a closer examination.
The argument is not that every claim made by an AI company should automatically be believed, nor that every AI-generated proof should immediately be accepted. A mathematical claim is not established merely because a machine produces it. Proofs need scrutiny, replication and, where possible, formal verification. Nor should legitimate questions about attribution, training data, intellectual property or the use of other people’s work be dismissed. But those are different questions from whether AI should be deliberately slowed because it might make mathematical discovery happen too quickly. That distinction is at the heart of this essay.
1. Twenty-Five Years. Now What?
The Millennium Prize Problems were announced by the Clay Mathematics Institute in 2000. Seven extraordinarily difficult mathematical problems were selected, with a $1 million prize attached to each. More than a quarter of a century later, the Poincaré Conjecture has been solved, while the others remain officially unresolved. Navier–Stokes is among the most famous of them. The Navier–Stokes problem concerns one of the fundamental equations of fluid dynamics and asks, roughly, whether smooth solutions in three dimensions can develop singularities under appropriate conditions.
Generations of mathematicians have worked on it. Now imagine that an AI system produces a solution that survives rigorous mathematical scrutiny and formal verification. What should humanity say? “Wonderful. We have learned something new.” Or: “Wait. This happened too quickly.” The second response deserves examination when we put the timescale into perspective. If humanity has struggled with a problem for decades, why should the arrival of a tool capable of solving it be regarded primarily as a loss? What exactly have we lost?
2. What Exactly Have We Lost?
Suppose AI produces a correct proof of Navier–Stokes. The mathematical truth has not disappeared. The original problem has not disappeared either. The proof can still be studied, challenged, simplified and generalized. A mathematician can still attempt an entirely different proof, students can still learn the mathematics, and researchers can still investigate the underlying ideas. In fact, the most interesting consequence may be that a solved problem can generate a whole new set of problems. Once a solution exists, mathematicians can ask whether the proof can be simplified, whether its assumptions can be weakened, whether the result can be generalized, whether there is an entirely different route to the same conclusion, why the argument works, what deeper structure is hiding inside it, and whether the same technique can be applied to another problem. A machine may have closed one question, but in doing so it can open many others. The end of one problem can therefore be the beginning of an entire research programme. The mistake is to imagine mathematical discovery as a single road leading to a single destination. Mathematics is much more like a branching landscape: reaching one destination can reveal dozens of paths that were invisible before.
3. A Solution Opens Doors
A major mathematical solution does not necessarily close a field of inquiry. Quite often, it does the opposite. A theorem can expose a structure that nobody previously understood, and that structure can lead researchers into entirely new territory. Imagine AI proves a difficult theorem. The natural mathematical response could be a cascade of questions: Can the proof be simplified? Can its assumptions be weakened? Can the result be generalized? Is there an entirely different proof? Why does the argument work? What deeper structure is hiding inside it? Can the same technique solve another problem?
The important distinction is between solving a problem and understanding everything that follows from solving it. A machine might establish a result with extraordinary efficiency. Humans may subsequently spend years understanding its elegance, consequences, generalizations and conceptual significance. Why would that be a tragedy? It could instead be a division of intellectual labour that mathematics has never had before. The machine searches. The human interprets. And then the human asks another question.
4. Newton, Leibniz—and the Second Discovery
There is another assumption hiding inside the fear of AI solving mathematics too quickly: that there is one privileged route to mathematical truth. History says otherwise. Newton and Leibniz independently developed calculus. Their approaches, notation and intellectual contexts were different, yet they arrived at closely related mathematical foundations. That is not an isolated curiosity. Mathematics repeatedly demonstrates that the same result can have multiple proofs, formulations, interpretations and generalizations. So suppose AI produces an elegant proof of a difficult theorem. A human mathematician may respond: “That’s interesting. But I want to find another way.” Why shouldn’t they? AI finding one path does not close all the others. A machine-generated proof might even make alternative discovery easier by revealing the destination and some of the terrain. The human mathematician can now ask a more sophisticated question: “What is the proof really telling us?” That is not the end of mathematical discovery. It is a different starting point.
5. The 1,000-IQ Human
Consider a deliberately absurd thought experiment. Suppose a human being were somehow born with what we might jokingly call a 1,000 IQ. This person possesses extraordinary mathematical abilities. He solves Millennium Problems in days, develops new mathematical theories every month, sees connections that ordinary mathematicians require decades to discover, and generates new conjectures faster than entire research communities can investigate them. What should society do? Should mathematicians tell him: “Slow down. You’re discovering mathematics too quickly”? Should governments impose a mathematical speed limit? Should he be forbidden from publishing his results until everyone else has had sufficient time to attempt the problems independently? Should we tell him that he must struggle for another hundred years so that other humans don’t lose their opportunity to discover these things?
That would be absurd. We would probably do the opposite. We would give this person resources, collaborators, computers, laboratories and freedom to work. His extraordinary ability would be regarded as an asset to humanity. So why does the principle suddenly change when the extraordinary mathematical capability exists inside a machine? If the real concern is mathematical knowledge, the identity of the discoverer should not fundamentally matter. If the objection is instead that the discoverer must be human, then we are no longer talking primarily about mathematical progress. We are talking about human exclusivity.
6. The Machine Doesn’t Choose the Questions
There is another misconception worth examining. People sometimes speak about AI as if it is going to independently roam across the mathematical universe and systematically consume every remaining problem. That is not how the technology works. A calculator does not decide what to calculate. A computer does not decide which scientific problem deserves attention. And an AI system does not automatically wake up and declare: “Today I shall solve topology.” Humans provide the questions, objectives, conjectures, constraints and directions. A mathematician says, “Investigate this.” A physicist asks whether a set of equations can be solved. An engineer wants a system optimized. A researcher proposes a conjecture and asks a machine to attack it. Then AI becomes a powerful exploratory instrument. That distinction matters enormously.
7. AI as an Idea Amplifier
The telescope amplified human vision; the microscope amplified human observation; the calculator amplified arithmetic; the computer amplified computation; and the internet amplified communication. AI may amplify something even more interesting: human intellectual exploration. Give it an idea and it can test variations, search possibilities, identify counterexamples, generate conjectures, attempt proofs and explore alternative approaches at extraordinary speed. The important thing is that amplification does not necessarily mean replacement. A telescope didn’t eliminate astronomy. A microscope didn’t eliminate biology. A calculator didn’t eliminate mathematics. A computer didn’t eliminate programming. They changed the scale at which humans could operate. AI may do something similar for intellectual work.
8. Not Every Problem Is AI-Solvable
Mathematics contains an enormous universe of problems. Some are difficult because we lack computational power; others because we lack the right conceptual framework. Some may require entirely new mathematics, while others may turn out to be undecidable within a particular formal system. There is therefore no reason to assume that increasing AI capability automatically means the automatic conquest of mathematics. Consider P versus NP. Even if an AI system produces an apparently convincing argument tomorrow, that does not mean the mathematical community automatically accepts it. The argument still has to survive scrutiny.
And formal verification has limits of its own. A system such as Lean can check whether a formally specified proof follows the rules of the formal system. That is enormously valuable. But Lean does not decide which problem humanity should care about. It does not decide which conjecture is beautiful. It does not decide which question is profound. Human judgment remains part of the process.
9. Don’t Mix Two Completely Different Problems
This is perhaps the most important distinction in the entire debate. Suppose an AI system has been trained on enormous quantities of human-created mathematical material. Suppose further that it reproduces somebody’s unpublished proof, distinctive technique or intellectual contribution. Then there are legitimate questions. Who deserves credit? Where did the idea originate? Should the contribution be attributed? How should provenance be established? Should there be compensation or licensing in some circumstances? These are serious questions. They should be addressed seriously.
“Therefore AI must not become capable of discovering mathematics faster than humans.” That is a non sequitur. Attribution is one problem. Technological progress is another. We should solve the first without manufacturing a crisis around the second.
10. The Fear of Mathematics Becoming Too Easy
Perhaps the deepest objection is not actually about mathematical truth. It is about the experience of doing mathematics. For many mathematicians, the struggle itself matters. The years of failed attempts matter; the moment of insight matters; and the gradual construction of a proof can be an intellectually rewarding experience. All of that is real and worth preserving. But valuing the experience of doing mathematics is very different from arguing that everyone else should be prevented from using a tool that makes mathematics easier.
People still climb mountains even though helicopters exist. People still run even though cars exist. People still do arithmetic manually even though calculators exist. People still paint even though cameras exist. The existence of a faster method does not abolish the slower method. It makes the slower method a choice.
11. Twenty-Five Years—or Another Hundred?
Return to Navier–Stokes. The Millennium Problems were announced in 2000. Suppose human mathematicians have spent decades attacking them and AI eventually solves one. Should we say: “No. Give humanity another hundred years.” Why? What is the unit of time we are trying to preserve? Twenty-five years? A hundred? Five hundred? A thousand? At some point the logic becomes visible. The objective is no longer simply discovering mathematical truth. The objective has become preserving the human experience of trying to discover mathematical truth. That experience may be valuable. But it is not the same thing.
If a human mathematician wants to spend twenty years attacking a problem after AI has solved it, nobody needs to stop them. They can still do it. They might even discover something the AI missed. But why should everyone else be denied the solution until the human process has run its allotted course?
12. The Strange Logic of Technological Progress
Human civilization has spent thousands of years inventing tools that eliminate unnecessary effort. The wheel reduced the necessity of carrying things. The steam engine multiplied physical power. The automobile reduced the necessity of walking long distances. The calculator eliminated enormous amounts of manual arithmetic. The computer eliminated enormous amounts of repetitive calculation and clerical labour. We normally call this progress. Yet when technology reaches a domain that humans regard as intellectually special, the language suddenly changes. Now we hear: “But humans need the struggle.” Perhaps. But if that principle were applied consistently, technological progress would become impossible. Every eliminated task could be defended on the grounds that somebody enjoyed performing it. The farmer could demand that tractors be slowed down because ploughing develops character. The accountant could demand that spreadsheets be restricted because manual calculation is intellectually healthy. The student could demand that calculators be banned because solving equations by hand is educational. We don’t generally accept those arguments. Why should mathematics suddenly be different?
13. A Solved Problem Can Create More Mathematics
Suppose AI solves Navier–Stokes. Now thousands of mathematicians know that a certain result is true. That knowledge itself becomes an enormous resource. Researchers can attack the proof. Students can study it. Others can search for simpler proofs. Still others can generalize it. Physicists can investigate consequences. Computer scientists can look for computational implications. Mathematicians can ask whether the same techniques apply elsewhere. The result could therefore create more mathematical activity, not less. This is a recurring pattern in science. Knowledge does not merely terminate questions. It creates new ones. A solved problem can become a machine for generating unsolved problems.
14. The AI Apocalypse Is a Different Argument
Now we come to a much larger claim: that increasingly powerful AI might eventually become an existential threat. This possibility should not be dismissed. Powerful technology can create powerful risks, and those risks deserve serious research. We should test frontier systems, monitor dangerous capabilities, improve alignment, strengthen cybersecurity and develop mechanisms for detecting and responding to failures.
But there is a logical difference between “A catastrophic outcome is possible.” and “A catastrophic outcome is inevitable.” An asteroid could hit Earth. A new pathogen could emerge. A powerful technology could be catastrophically misused. A child could theoretically grow into a dangerous adult. We do not normally solve uncertainty by eliminating everything that could conceivably become dangerous. We observe, evaluate, prepare and build defences. The important question is not whether an unknown risk exists. The important questions are: What is the mechanism? How likely is it? What evidence do we have? What safeguards address it? That is responsible risk management.
15. The Child Who Might Become a Monster
Take the thought experiment to its logical extreme. Suppose a small child might, under some hypothetical future circumstances, become extremely dangerous. Would we kill the child today because of that possibility? Obviously not. The future possibility is not the present reality. Potential future capability is not identical to present harmful action. The answer to uncertainty is not necessarily pre-emptive destruction. It is observation and preparation. AI should be treated with much greater sophistication than either blind optimism or apocalyptic certainty.
16. Improve the Brakes. Don’t Throttle the Gas.
This is where the entire argument can be reduced to one sentence: Improve the brakes. Don’t throttle the gas. If AI becomes more powerful, improve the mechanisms around it. Build better evaluations, better verification, better monitoring, better attribution and provenance, better cybersecurity, better alignment and better institutional oversight. But don’t automatically conclude that the answer is to deliberately make the engine weaker. When a car becomes faster, society improves brakes, tyres, steering and safety systems. We don’t necessarily solve the problem by forcing the engine to remain at the power level of a horse cart. The same principle can apply to AI. Safety should be an engineering problem, not merely an argument for technological paralysis.
17. Trust Is Not a Safety Mechanism
This becomes particularly important when governments and powerful companies are involved. Sam Altman has emphasized the importance of ensuring that AI remains in service of people and that frontier laboratories act responsibly. Those are reasonable objectives. But promises are not institutions. Powerful organizations operate within changing incentives. Companies compete. Governments change. National interests change. Geopolitical circumstances change. Policies change.
History offers reminders. In 1971, the United States suspended dollar convertibility into gold, bringing the Bretton Woods monetary system toward its eventual end. The historical record illustrates how a major international arrangement can be radically altered when national priorities change. The nuclear age offers another lesson: the world did not eliminate nuclear capability after discovering its destructive potential. It developed treaties, safeguards, verification mechanisms, deterrence systems and institutions around it.
The lesson is not that governments or companies are inherently dishonest. It is simpler: Do not build a safety system whose most important component is somebody’s promise to behave exactly as expected forever. Build mechanisms.
18. Who Gets to Control the Controls?
There is an additional problem with calls for slowing frontier AI. Suppose every major AI company agrees to slow down. Wonderful. But what happens if one company or country defects? The company that continues developing more capable systems may gain an enormous advantage. That creates a strategic dilemma. And there is another possibility. If developing frontier AI becomes enormously expensive and heavily regulated, the rules may unintentionally make it harder for new entrants to compete. The organizations that already possess compute, capital, infrastructure, researchers and distribution could become even more entrenched. Therefore a policy justified as “preventing dangerous concentration of AI power” could, under some circumstances, contribute to concentration. That does not prove that anyone advocating regulation has this intention. It means that incentives and second-order effects matter.
19. The Nuclear Lesson
Nuclear weapons provide an uncomfortable but useful comparison. They are not equivalent to AI. Their destructive capability is concrete and extraordinarily severe. But the history of nuclear technology demonstrates something important: humanity did not respond simply by saying, “This technology is dangerous, therefore it must disappear.” Instead, the world developed a complicated system of deterrence, treaties, non-proliferation arrangements, safeguards, inspections and controls. And nuclear weapons still exist. That does not mean the system is perfect. It means something fundamental about technological governance: When a powerful technology cannot realistically be uninvented, the challenge becomes learning how to manage the power. AI is not nuclear weapons. But the governance lesson is relevant. Manage capability. Don’t pretend capability can simply be wished away.
20. The Crane Is Already Here
Imagine again the giant mathematical boulder. For years, generations of mathematicians have pushed. Then someone arrives with an enormous crane labelled AI. The crane can lift the boulder. The mathematicians continue pushing. Someone asks, “Why not use the crane?” The answer: “Because we prefer to do it ourselves.” There is nothing wrong with preferring to do it yourself. There is something questionable about making that preference the basis for preventing everyone else from using the crane. The crane does not eliminate the joy of pushing. It simply means that pushing is no longer compulsory. That is what technology has always done.
21. Let Humans Keep Discovering
There is therefore no reason to choose between AI mathematics and human mathematics. Mathematicians should continue discovering. They should continue proving, exploring, struggling and finding beautiful new ideas. There is no reason for any of that to stop merely because a machine has found an answer. A mathematician who wants to spend ten years independently rediscovering a result should be perfectly free to do so. They might even discover something the machine missed. But none of this requires the rest of humanity to wait ten years for the machine’s answer. The freedom to struggle is worth preserving; the necessity of struggling is not.
22. The Real Choice
The choice is not AI or humans, progress or safety, or machine mathematics or human mathematics. And it certainly should not be, “Let us keep machines weak because humans enjoy struggling.” The real choice is between two approaches to technological progress: suppress powerful technology because its consequences are difficult, or build institutions and safeguards capable of handling that power. The second approach is harder. It requires engineering, governance, experimentation, verification and admitting uncertainty. But it is also the approach civilization has repeatedly taken when it chooses to move forward.
23. The Strange Thing About Progress
Human history is largely the history of turning difficult things into easier things. We invented tools because human muscles were insufficient. We invented machines because human hands were insufficient. We invented computers because human arithmetic was insufficient. And perhaps we are now building machines because human cognitive bandwidth is insufficient for the enormous space of possibilities we want to explore. Every time technology removes a burden, somebody can reasonably say, “But I liked doing that myself.” That’s fine. Do it yourself. But don’t confuse the right to perform a task manually with the right to prevent technology from automating it. Mathematicians should continue discovering. They should continue proving. They should continue struggling. They should continue finding beautiful new ideas. But none of that requires the rest of humanity to wait.
24. A Solved Problem Is Not the End
A solved problem is not the end of mathematics. It is a new piece of mathematics. A new theorem can produce a new conjecture; a new proof can inspire another proof; a new technique can unlock an entirely different field; and a new connection can generate an entire research programme. Mathematics does not stop when an answer is found. Quite often, the answer changes the landscape and reveals questions that could not even have been formulated before.
Mathematicians should continue discovering. They should continue proving, exploring, struggling and finding beautiful new ideas. There is no reason for any of that to stop merely because a machine has found an answer. A mathematician who wants to spend ten years independently rediscovering a result should be perfectly free to do so. But none of this requires the rest of humanity to wait ten years for the machine’s answer. The freedom to struggle is worth preserving; the necessity of struggling is not.
25. The Question We Should Be Asking
Perhaps the debate about AI and mathematics has been framed backwards. We should not ask “How do we stop AI from discovering mathematics too quickly?” We should ask “How do we help humanity make the greatest possible use of what AI discovers?” That means solving the real problems—attribution, provenance, verification, safety, misuse, concentration of power and governance—without manufacturing a larger crisis merely because technology has become extraordinarily productive.
If AI really can solve mathematical problems that humanity has failed to solve for decades, that is not an insult to mathematics. It is an extraordinary expansion of the mathematical toolbox. And if the machine occasionally reaches a destination before we do, there is still an entire universe waiting to be explored. Let it reach the destination. Then ask where we go next.
Improve the Brakes. Don’t Throttle the Gas.
The future does not require humans to stop thinking. It requires humans to think bigger. Perhaps that is the strangest thing about Navier–Stokes: after centuries of building machines to eliminate unnecessary human labour, we have finally built one that may begin eliminating some of our intellectual labour—and our first instinct is to ask whether we should have allowed ourselves to become so capable.
Perhaps we should ask a different question: What could humanity discover if it stopped insisting on doing everything the hard way?