Free inquiry is a good in itself. The freedom to wonder, to play with an idea, and to follow a question belongs among the goods of a human life. It needs neither the endorsement of a corporation nor the leave of a mandarin class of caretakers. A person need not promise a discovery, demonstrate exceptional talent, or establish the depth of their intentions before this freedom is worth defending. We are entitled to be curious without first making ourselves consequential.
Someone may turn a question over for ten minutes, try a ridiculous example, ask a machine what happens, or put the whole thing aside when something else catches their attention. Curiosity can be cultivated lightly, with humor and without a definitive purpose. I would contend that a full intellectual life needs ample room for such play. The value of such inquiry is not redeemed by showing that it occasionally leads to profound discoveries or builds character. Effort and depth of intent are admirable, but neither is a foundation for the freedom to inquire.
Mathematics is a fundamentally human endeavor and belongs to humanity. Humanity needs experts, mentors, and guides, and in serving in this role for society, professional mathematicians have done much good, yet have much work to do. Expert knowledge and judgement may light the way for others, but it gives us no title to the territory. The freedom to inquire is not ours to dispense as a reward for “good mathematical citizenship”. Nor is it a gift that a technology company can bestow upon society at large. The value of either remains contingent on the freedom they allow us to exercise, insofar as they allow the freedom of inquiry to flourish, rather than tame, suppress, deaden, or constrain it.
Recent rapid developments in LLMs have demonstrated that professionalized mathematicians, while retaining a majority stake in cultivating taste, formulating research programs, disseminating and structuring mathematical knowledge, fostering talent, and discerning the truth and significance of results, no longer enjoy the virtual monopoly over the production of significant mathematics that has defined the status quo for the last 150 years. Moreover, these developments may finally have pushed the still more recent paradigm of “competitive mathematics,” with all its inconsistencies and historical contingencies, to the breaking point. I do not think it will go quietly, but I would not be totally sad to see it go, and I hope that a more generous and expansive mathematics will arise in its place.
By competitive mathematics, I mean the professional arrangement in which the production of research papers serves simultaneously to advance knowledge and to establish the sociological status of the people producing them among their peers. These purposes have never been identical, though we have generally conducted ourselves as if they were close enough proxies. A good paper gives us mathematics and evidence of a mathematician. It is not clear that we can continue to ask the same object to do both jobs. This is a serious challenge to the profession. It is not, I contend, a calamity for inquiry.
In responding to this change, it is important to distinguish the enduring values of mathematics from the professional arrangements through which those values are currently upheld. The pursuit of truth and understanding through doubt, intellectual honesty and humility, and independent verification are fundamental. The mental conditioning that the study of mathematics promotes, the cultivation of curiosity without fully interrogating what fruits it may bring beforehand, the ability to intellectually grapple with uncertainty for long periods, and the training to think critically, to test the mettle of one’s ideas and sweep out the dusty corners of thought, are beneficial to the development of individual intellects. These things are goods in themselves, developing and honing our collective abilities and the abilities and talents of individuals. Our systems of credentialing, publication, peer review, and professional recognition are not fundamental in the same way1. As far as they continue to serve those core purposes, they should be preserved and strengthened. Where they fail, there deserves to be the serious work of rebuilding. How, I am not sure.
Our current professional arrangements have given us much. Our system of universities has sustained decades of free inquiry with no immediate prospect of return, often leading to spectacular breakthroughs in fundamental science that have improved the lives of billions. In the small, a patient adviser or mentor can make a mathematical life possible or a seminar talk or a casual chat with a colleague can turn an inscrutable idea into something revelatory. These things deserve our defense. But defending the conditions under which people can think is a different undertaking from defending our exclusive right to decide which people may do so or dictate the conditions of valid inquiry.
Expertise gives us reason to attend to a judgment, especially where we cannot yet assess an argument for ourselves. It gives nobody a general claim upon the direction of another person’s thought. We may judge an argument false, a project unpromising, or a method ill suited to its purpose, and give our reasons. The freedom of the person pursuing it survives our judgment. Nor does an accumulation of distinguished names turn advice into jurisdiction. If we invoke inclusion, we must be prepared to include people whose mathematical lives we would not have chosen for them. Admission on condition of conformity is a meager kind of welcome.
A professionalized class often cultivates a caution that reaches well beyond care in reasoning. A person whose prospects depend upon successive judgments by established figures has ample reason to anticipate their disapproval. In such a setting, nobody need forbid an experiment for it to go untried. One learns which enthusiasms require an apology, which questions are better kept to oneself, and which departures from custom must wait until one is secure. Eventually the audience is present even when one is thinking alone. We should ask how much of what we call professional maturity consists in learning to police one’s own curiosity. That caution often distorts our account of change. Threatened jobs, journals, and research programs have people who can speak for them. We can become exacting accountants of every possible loss from change while treating the losses imposed by existing arrangements as facts of nature.
The freedom to inquire belongs to the beginner as fully as to the accomplished mathematician. Competence often grows through the exercise of that freedom; it cannot sensibly be required as the price of admission. Where someone asks us to accept a mathematical claim, however, we have a further question to answer. Anyone may offer a mathematical idea, by any means, but no person, institution, corporation, or machine is entitled to have that idea accepted without adequate evidence. As authorship becomes radically decoupled from authority, we must keep in mind that the obligation to judge an argument rigorously gives no warrant to decide who may attempt one.
The freedom to inquire does not impose the right to put a claim on the attention of others. Nobody is obliged to spend a month untangling a stranger’s manuscript, whether it was written painstakingly by hand or generated in an afternoon. A machine does not acquire a right to human labor by producing more text than humans can read. Those presenting a result should make its claim precise, supply an argument that can be readily checked, identify its dependencies, and be candid about what remains uncertain. If they cannot do this, the result may have to wait.
I would like to see a proper return of amateurism to inquiry, by which I mean the freedom to take an interest in mathematics on one’s own terms, without making a career of it or submitting one’s curiosity to the requirements of such a career2. There is no need to reconstruct an earlier society whose leisure and learning were themselves unequally distributed. We can make room now for the devoted amateur, the occasional dabbler, and the person who sees no reason to give their interest a name. A professional mathematician taking interest in a problem far afield from their specialization is an amateur of another kind. None owes an exacting diligence as a condition of being left to their curiosity.
What has struck me most about the recent crop of AI-discovered breakthroughs is that they largely do not lie beyond human capability. Quite often the arguments are rather direct and elegant. One should also not discount the real human achievements that have made such arguments profitable to discovery. Poignantly, there is a troubling possibility that the intense selection pressures in the current academic job market, focused mainly on a small number of metrics (number of papers, paper placement in top-tier journals, a degree from a distinguished institution, etc.), may have produced a “monoculture” of research mathematicians; that is, these discoveries may have done more to reveal cultural blind spots than gaps in understanding. The well-trod alternative, that AI systems can merely brute-force search their way over a space of possibilities that would exhaust a human research group, may be true to a degree, but does not fully discount this possibility to my mind. Perhaps a return to amateurism could provide a greater spectrum of human minds, ones with significant strengths and real deficiencies, that could breathe life into a fully humanistic conception of mathematics.
This person is curiously easy to lose sight of when discussing the future of mathematics. We picture professors, students on their way to becoming professors, and companies offering to replace them. There is room for a much larger cast. Someone may wish to understand one lovely problem, work out an example, improve an explanation, or pursue an awkward connection for which no sensible grant proposal could be written. If new tools make such participation possible, then removing barriers to their responsible use serves mathematical inquiry, even when it does nothing to replenish the academic job market. The yoke of professional diktats has little claim upon someone who never asked to wear it.
The same argument bears directly upon the companies offering these tools. A company that speaks of democratizing mathematics should expect to be judged by the independence it permits, including the freedom to criticize its tools, publish their failures, and pursue questions of no commercial interest. Its users owe it no enthusiasm, allegiance, or part in a story about the inevitable future. A student trying a model is not thereby volunteering to become an advertisement.
A costly proprietary system can remove one barrier while erecting another. Dependence upon a company’s continuing indulgence is a precarious basis for free inquiry, however generous its present terms. Access, consent, attribution, and independent scrutiny remain substantive questions even when the mathematics is impressive. Those of us who resist professional gatekeeping should be particularly unwilling to excuse corporate gatekeeping in the name of progress. We should not have to exchange professional permission for corporate permission.
Taking this freedom seriously also means attending to the conditions under which people can exercise it. Time, books, instruction, conversation, and access to tools matter. The right to choose one’s tools, to work slowly against the onrush of ever accelerating technical velocity, must also be afforded. Formal permission to inquire offers little to someone shut out of all of these. A profession with substantial influence over their provision has much useful work to do in removing barriers. So does a company that claims to put powerful instruments within reach. Each can be judged by how much freedom it makes possible, including for people whose purposes it has no hand in choosing.
We should also be candid about what Terence Tao calls “proof indigestion”, as we already experience it. In my own field, operator algebras, it is difficult to absorb technical developments outside one’s particular niche. Graduate students confront thousands of pages of demanding technical work before they can hope to move freely through substantial parts of the subject. Asking them to metabolize this corpus was already an enormous demand. I cannot quite recognize the picture of a profession in which every important idea was patiently understood, properly transmitted, and generously rewarded until artificial intelligence disturbed the peace. Nor would the average non-mathematician look upon the anxious and overworked state of many academics, early-career ones especially, and see evidence of an idyllic paradise that such technology threatens to expel us from.
The immediate temptation is to settle such difficulties by declaration. Collective statements can be useful: they can articulate responsibilities, resist exploitation, and give people grounds on which to challenge powerful institutions. But a statement of shared values should not become a substitute for the inquiry it purports to protect. Agreement that understanding matters leaves open how best to cultivate it. Agreement that students deserve care leaves open what risks they should be allowed to take. Neither agreement confers authority to end the discussion.
The recent controversy over the proposed Caltech Mathathon makes the danger concrete3. An open letter calling for its cancellation warns that participation could harm young mathematicians’ future reputations. Whatever its intended meaning, such a warning acquires a different force when endorsed by members of the profession into which those students hope to enter. A prediction of disapproval can help bring that disapproval about. There are serious questions to ask about sponsorship, verification, and the incentives of a short competition. They can be asked without making participation itself a mark against a young person.
The students have as much right to pursue mathematics in this way as we have to pursue it through established professional currents. That right does not depend upon the event producing good mathematics, vindicating its sponsors, or meeting our expectations of a proper mathematical experience. We are free to criticize all of these things. They are free to try, to be unimpressed, to enjoy themselves, or to walk away. They owe the profession no performance of disapproval and the sponsor no performance of gratitude. Their curiosity is their own.
Professional coercion need not take the form of a prohibition. It can work through an intimation that certain choices will be remembered, that sensible young people know whose advice to take, or that refusal reveals a want of maturity. Where senior judgment bears upon references, appointments, and invitations, such intimations can do the work of a command. The person issuing them may sincerely believe that they are protecting the student. Sincerity does not settle whether the student remains free to disagree.
The right to offer counsel carries no right to obedience. We may explain our concerns, propose changes, and argue vigorously for them. The people addressed may understand us perfectly and still decline. Having first sought agreement does not entitle us to escalate until agreement is obtained. The test of our respect for their independence comes when persuasion fails. Can they say no without the refusal itself becoming evidence against their fitness for the profession?
As tempting as such broadsides decrying the use of AI instruments are, and however much moral clarity glimmers on their surface, much of that certainty seems to collapse in the light of more quotidian considerations. For instance, for all of the romance of the sanctity of human-written mathematics, the fact is that a good part of technical writing is drudgery with little benefit to anyone. One could spend a couple of hours making petty modifications to a well-known technical lemma to fit one’s exact hypotheses, or one could take that time to do additional mentoring or free some weekend time for one’s family or friends. Nor in my experience will a human given sufficient time necessarily write an argument in a way more conducive to understanding by other humans by eschewing a bit of organizational advice or clarity that a relatively modest desktop LLM could provide.
I have no tidy settlement to offer. I would be suspicious of one that could be drawn up so quickly. We should be radically liberal about the means of inquiry and conservative about its values. That leaves difficult questions about work, education, resources, and responsibility, none of which will be answered by insisting that the profession’s interests and the interests of mathematics must coincide. Our present arrangements deserve neither automatic preservation nor gleeful destruction. They deserve to be judged by what they enable people to understand and how they allow flourishing both within professional mathematics and among humanity in general.
Discussing with my undergraduate students the recent claimed resolution of the Navier–Stokes problem by a frontier AI laboratory, I was struck by the awe and wonder with which they regarded it, in contrast to my own rather ambivalent feelings. If the claims are borne out, this is indeed a remarkable achievement for humanity. Though my ambivalence is warranted, I wonder whether the last two decades I have devoted to plying a rather narrow calling have not crabbed my horizons.
Yet the case for free inquiry cannot hang upon the fortunes of these particular claims, or even upon the eventual success of artificial intelligence in mathematics. If its promise comes to little, the person drawn to a lovely problem will have the same claim to pursue it. That freedom was worth defending before the present excitement and will remain so when the excitement has passed.
Acknowledgments: I would like to thank Srivatsav Kunnawalkam Elayavalli for encouragement in developing these ideas and for prompting discussions from which several of the initial thoughts grew.
AI Usage Disclosure: This essay was drafted with assistance from AI, combining a series of short notes and jottings of the author into a coherent form, which was subsequently heavily revised, expanded, and edited by the author. All positions expressed, as well as the vivid language and metaphor, are the author’s own.
Received 17 September 2026.
- For more on professional arrangements versus core activities, see AI and the mismeasure of science – Proofs and Prompts of Pasquato. ↩︎
- For more on amateur participation in scientific discovery with the assistance of AI tools, see A new kind of citizen science, and how to handle it – Proofs and Prompts of Yates. ↩︎
- Since this essay was written, the Mathathon organizers and several critics have announced a redesigned event focused on alternative proofs and exposition, with changes to sponsorship and tool-use policies. I welcome their stated commitment to participants’ freedom to choose their tools. My concern here remains whether the participants have the freedom to use those tools as they deem fit for their own purposes and explorations or whether all exploration must stay within officially sanctioned bounds. A “productive dialog” can mean different things to participants with unequal status and power. ↩︎
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