Imagine that an AI system proposes a plan for distributing public services among several districts. Under the model’s assumptions and requirements, no other permitted plan costs less overall. A formal proof of this result has been checked by software. But residents in some districts object: they would have to wait far too long for service under the plan.
Addressing their concerns requires understanding what the original plan took into account, what it left out, and what might happen if the requirements changed. Mathematicians and experts in public services can use AI to turn the residents’ concern into a new requirement: no district should have to wait longer than a specified time. They can then ask whether any plan can meet the revised requirements. If the available resources are insufficient, they can compare the costs and other effects of adding resources or changing other requirements. The discussion can now address which waiting times are acceptable and what it would take to achieve them.
The proof tells us that the plan costs least under the original model’s assumptions and requirements. The residents are asking what those requirements should be. People can redefine the question, change what they ask of the system, and choose among options with an understanding of what each option would require and what its consequences might be. We can act on judgments we have reason to accept at the time while keeping practical ways to question them, revise them, and change course. I call this revisable epistemic control.
We may need to revise a judgment because we got the facts wrong, made a mistake in reasoning, or discovered a problem with the model. New evidence can change what we have reason to believe, while changes in technology, resources, or needs can change the options available. Priorities can change too: in the service example, reducing long waits might come to matter more than keeping total cost as low as possible. We can make decisions carefully under current rules while allowing experience, new knowledge, and changing values to give us reasons to revise those rules.
We already rely on specialists for much of the knowledge we use. Mathematicians routinely use theorems whose proofs they have not personally checked; all of us rely on expert judgments we could not make ourselves. This reliance on knowledge held by others is called epistemic dependence. Hardwig argues that we can reasonably rely on another person’s judgment when we have good reason to believe that they have the relevant evidence (1985, pp. 336–337). AI can help us answer questions, examine assumptions, develop and compare explanations, and explore alternatives; formal verification tools can mechanically check mathematical proofs. We can use both when making decisions and revisiting them later.
The service example raises a further question: as more mathematical work is automated, how will people develop the understanding needed to help rethink such plans? Mathematical research has often done two things at once: produced results and helped people learn to understand and do more mathematics. AI now makes it possible to produce results much faster than people can learn to understand them.
Some of that learning happens after a result has been obtained, as people compare concepts, seek clearer explanations, and connect it with other results. Thurston emphasized the role of communication in mathematical progress; Collas distinguishes the accumulation of machine-generated results from human understanding and examines how research communities can sustain that understanding (Thurston, 1994, §§1, 3, 5; Collas, 2026, v2, §§1.3, 2.3, 3.3). This work can deepen people’s understanding of a result, help them formulate new, more precise questions, and influence what they do next.
People understand different parts of mathematics and have different strengths. One person knows a particular class of problems, another is good at expressing the same mathematical problem in different ways, and a third sees what a different area of mathematics might contribute. Scientists, engineers, and other specialists connect mathematics with knowledge from their own fields. Mathematical texts, libraries of formalized mathematics, and AI tools help them find and use one another’s work and connect ideas across these areas. Within the mathematical community, people explain unfamiliar ideas to one another, bring colleagues with different expertise together, organize existing knowledge, and help newcomers learn. They also learn through exchanges with experts in other fields.
For the residents to help change the service plan, they need a way to bring their concerns to institutions with the authority to respond. Residents can explain what matters to them, while experts use mathematics and AI to clarify which options are possible, what each would require, and what consequences each might have. The responsible institutions need to consider these reasons, decide how to respond, explain their choices, and answer further questions.
The people making these decisions need information, relevant understanding, time, and the authority to act in response to the concerns raised. Malik emphasizes these conditions for effective human decision-making in AI governance (2026, §§3.4, 4.3, 5.2). Open publications and code give people material to examine. People need to be able to understand and use that material, discuss what they find, and ask the responsible institutions to reconsider their decisions. They should be able to expect a substantive response.
For people to help examine and revise decisions informed by AI in the future, experienced practitioners need opportunities to keep learning, and newcomers need opportunities to develop their own mathematical understanding and take part in research. What people already know shapes what they can ask AI to do and how they assess its answers. Familiar concepts, methods, and ways of structuring problems help them notice questions worth pursuing and connect new results with what they already understand. By doing research, discussing ideas, and organizing existing results, they can deepen that understanding, learn new mathematical tools, and help others enter the field.
Machines will continue to change the knowledge and tools available to us. As we use machines to change the world, we should have the means to understand, question, and help shape those changes. Günther Anders writes in the opening epigraph to volume II of Die Antiquiertheit des Menschen (1980/2018):
It is not enough to change the world. We do that anyway, and much of this change takes place even without our involvement. We must also interpret this change in order to change it in turn: to keep the world from continuing to change without us, and from ultimately becoming a world without us.
This article is based on The Public Value of Human Mathematical Capacity, from which the English rendering of Anders’s epigraph is also taken.
Acknowledgments
In the underlying research and in writing this article, I used Academic Research Skills (ARS) and its Codex adaptation, ARS-Codex, to organize AI-assisted literature searches, source verification, conceptual clarification, argument analysis, drafting, and revision (Wu, 2026). I also used these workflows for internal review of the underlying research. I thank the developers and maintainers for providing these tools and the guidance that accompanies them. The views expressed and final editorial decisions are my own, and I take responsibility for any errors.
Received 8 October 2026, revised 12 November 2026.
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