Mathematics in the Age of AI: Terence Tao's Vision for the Mathematical Community

Table of Contents

TL;DR

In his public lecture Mathematics in the Age of AI at ICM 2026, Terence Tao’s true inquiry was not “can AI do mathematics,” but rather: If AI can soon complete a significant portion of research-level mathematical tasks, what should the mathematical community optimize, and what must it protect?

When proofs transition from scarcity to abundance, what will truly be expensive is no longer just answers, but validation, explanation, understanding, responsibility, community recognition, and the ability to consolidate scattered results into stable theories.

On July 24, 2026, Professor Terence Tao delivered a public lecture titled “Mathematics in the Age of AI” at the International Congress of Mathematicians (ICM 2026). He did not focus on model leaderboards, nor did he attempt to predict “when AI will replace mathematicians.”

If you are unfamiliar with Terence Tao: he is one of the world’s leading and most influential mathematicians today. Born in Adelaide, Australia, he earned his Ph.D. from Princeton University at 20, and was promoted to full professor at UCLA at 24, becoming the youngest full professor in UCLA’s history. Today, he is a Distinguished Professor in the UCLA Department of Mathematics, with research spanning areas such as harmonic analysis, partial differential equations, combinatorics, and number theory. In 2006, at the age of 31, he received the Fields Medal for his contributions to partial differential equations, combinatorics, harmonic analysis, and additive number theory.

People often describe Tao as a “genius,” but emphasizing only his precocity would underestimate his decades of sustained research breadth, academic contributions, and commitment to public mathematical discourse. He is not only renowned for his problem-solving abilities but has also long discussed how mathematics is discovered, expressed, and validated through papers, books, and his research blog. Therefore, his perspective on mathematics in the age of AI is not merely a celebrity commenting on technological trends; it is a solemn reflection by a distinguished scholar on the frontline of knowledge production, concerning the goals, values, and responsibilities of his profession.

To summarize the core question I gleaned from the lecture in one sentence:

When machines can produce more and more correct answers, do we still clearly understand what mathematical research truly aims to produce?

This was a lecture ostensibly about mathematics, but fundamentally discussing the knowledge production system. It concerned not just the boundaries of AI’s capabilities, but how a community, after a fundamental change in production methods, re-understands its goals, values, responsibilities, and division of labor. What I particularly admired was that even with his extraordinary reputation, which could easily sway the discussion, Mr. Tao maintained restraint: he set conditions and clarified objectives in a way familiar to mathematicians, then invited the entire community to collectively consider the answers, rather than presenting his personal judgment as a definitive conclusion.

Setting a Working Hypothesis: Focusing on the Real Question #

Discussions about AI and mathematics often quickly turn into a capability race: can models solve IMO problems, can they complete research-level proofs, can they surpass a certain benchmark, when can they conduct independent research?

While these questions are important, Tao did not let the entire lecture dwell on the capability race. Instead, he proposed a “working hypothesis” intentionally kept coarse-grained, roughly as follows:

In the near future, certain AI tools will be able to complete a significant proportion of research-level tasks in some mathematical fields, with reasonable cost, success rate, and human oversight.

This hypothesis does not require listeners to believe it or welcome it. It is more like a condition setting in a mathematical proof: assume it holds, then analyze the consequences.

Tao cited preliminary results from the First Proof challenge as background material: in a controlled test, four AI harnesses attempted to solve ten new research-level mathematical problems, of which seven had at least one answer reaching publishable quality, with a computational cost of approximately $10 to $1000 per problem. We certainly cannot conclude that artificial general intelligence has arrived, but it at least shows that “can AI participate in research-level mathematics” is no longer just a distant philosophical conjecture.

Once this premise is temporarily accepted, the question shifts from “what machines can do” to:

If machines can indeed produce more proofs, can the original goals, values, and institutions of mathematical research remain unchanged?

A New “Foundational Crisis” #

Tao likened the current situation to the foundational crisis of mathematics in the early twentieth century.

Results like Russell’s paradox and Gödel’s incompleteness theorems forced mathematicians to re-examine previously taken-for-granted foundations such as sets, infinity, formal systems, and axioms. That period was full of debate, but ultimately led to a more explicit, rigorous, and standardized mathematical framework.

Today, what is being challenged is no longer primarily “what is a number” or “what is a set,” but another set of long-assumed questions:

Is the sole purpose of mathematical research to solve more problems?
Does a correct proof automatically equate to valuable knowledge?
Who should be held responsible for results generated with AI involvement?
What process must a result undergo to truly belong to the mathematical community?

Therefore, this current crisis is more like a foundational crisis of mathematical values and practice.

Previously, such interrogations were often handled by philosophy, history, and sociology, while mathematicians focused more on technical work. The advent of AI, however, forces the entire mathematical community to confront them anew: once production capabilities fundamentally change, objectives that could previously be tacitly understood must now be explicitly articulated.

Mathematical Research is More Than “Problem-Solving” #

Mathematical research certainly includes solving unsolved problems, but it extends far beyond that. Proof generation has long been considered the most scarce and rewarded part of the research process, but this does not mean it is the entirety of mathematical research. Tao listed other goals, including: developing new theories and techniques, understanding the real world, building academic communities, training the next generation, expanding shared knowledge networks, and creating works of lasting aesthetic value.

In the past, these goals largely reinforced each other.

An important proof often brings new methods, new language, and new research directions; solving a problem can also train students, connect different fields, and drive updates in textbooks and theoretical frameworks. Precisely because these goals were highly correlated, people could use “how many important problems have been solved” as a proxy metric for mathematical progress.

The arrival of AI may change this correlation.

If machines can generate proofs at high speed, the “number of problems solved” might rise rapidly, but understanding, education, theory building, and community absorption capacity may not grow proportionally. We might have more conclusions without gaining an equivalent level of understanding; more papers without enough people to check, explain, and integrate them.

This is the risk of Goodhart’s Law:

When a measure becomes a target, it ceases to be a good measure.

If the entire system only rewards “who solved the problem first” or “how many problems were solved,” a previously effective proxy metric might rapidly lose its original meaning of measurement under the impetus of AI.

How a Proof Becomes Common Knowledge #

The most illuminating part of the lecture was Tao’s continuous extension of the definition of “solving a problem.”

The initial goal seems simple:

Open Problem → Proof Generation

But if the answer cannot withstand scrutiny, quantity alone does not constitute true progress, so verification is also needed:

Open Problem → Proof Generation → Correctness Verification

Formal methods and proof assistants like Lean, Rocq, and HOL can make verification more automated and reviewable, significantly speeding up the verification process in many scenarios. However, a formally correct proof can still be a piece of machine output that no one truly understands.

Thus, after correctness, explanation is also needed:

Open Problem → Proof Generation → Correctness Verification → Clear Exposition

Good mathematical writing not only demonstrates the validity of each step but also answers higher-level questions: Where are the key difficulties? What is the genuinely new idea? How does it relate to existing results? Which steps can be reused?

Even if a proof is correct and fluently written, it will not automatically become common knowledge. Other researchers still need to read, discuss, verify, cite, generalize, and reorganize it. Ultimately, truly important results must enter textbooks, standard references, and the theoretical framework of the field.

I summarize the several goal revisions in the lecture as follows:

Problem Identification → Proof Generation → Correctness Verification → Clear Exposition → Publication & Peer Review → Community Understanding & Acceptance → Integration into the Field’s Standard Theory

This is not Tao’s verbatim statement, but a generalization of his sequential additions of verification, exposition, publication, digestion, and canonicalization across multiple slides. It reveals an often-overlooked fact:

Proof completion is not the end of research.

A more complete culmination of research is to transform results produced by individuals or machines into public knowledge that the community can understand, verify, inherit, and continue to develop.

From Proof Scarcity to “Proof Indigestion” #

Tao used a vivid term: proof indigestion.

If AI generates proofs faster than humans can verify them, unverified results will continuously accumulate; if verification outpaces explanation, correct but difficult-to-read proofs will continue to pile up; if publication outpaces community absorption, papers will overwhelm the peer review system; even if all results are published, truly important content might not be organized into stable, teachable, reusable theories in time.

Mathematics could thus transition from “proof scarcity” to “proof abundance,” with the bottleneck shifting from upstream discovery to downstream digestion:

Past bottleneck: Can a proof be found.
Future bottleneck: Can a proof be verified, explained, selected, absorbed, and transmitted.

What is most noteworthy here is not just the familiar concern that “AI will produce junk papers,” but the deeper institutional problem behind it.

The existing academic system often values original discovery but tends to underestimate the importance of reviewing, editing, explaining, reproducing, textbook writing, and theoretical consolidation. These tasks may not be at the cutting edge, but they bear the heavy responsibility of transforming individual achievements into collective progress.

When generative capabilities improve significantly, these previously considered auxiliary tasks may instead become the most crucial infrastructure for the entire knowledge system.

Over-Smoothness: A Potential Barrier to Understanding #

Another intriguing observation from the lecture was that AI-written mathematical texts are often near-perfect in spelling, grammar, and typography, yet they might gloss over genuinely difficult steps while extensively explaining obvious parts.

More subtly, proofs might also be “over-polished.”

In human writing, places where the author deliberated extensively usually retain some natural friction, prompting the reader to slow down. Over-smooth AI text might present difficult and routine steps with equal ease, causing readers to lose clues to gauge importance.

This is not romanticizing errors, nor is it suggesting that ambiguity is better. It reminds us: good explanation does not mean superficial frictionlessness.

Truly excellent exposition should help readers establish the correct cognitive rhythm: where to skim quickly, where to pause and re-derive, and where the core ideas of the entire work reside.

As AI-assisted writing becomes more common, “linguistic fluency” will become less scarce; the ability to accurately present the difficulty structure, origin of ideas, and logical flow of reasoning will become a more important writing skill.

Lecture and Leiden Declaration Practical Recommendations #

Tao does not attempt to exhaust all institutional issues in one lecture. The following cautious and specific practical recommendations come partly from his lecture and partly from the Leiden Declaration, which he explicitly recommended.

1. Public Disclosure of AI Use #

Researchers should frankly state in which stages AI was used, what tools were employed, and how it influenced the research process. Responsible disclosure should become an academic norm, and researchers should not conceal their AI use due to concerns about peer evaluation.

2. Human Authors Retain Responsibility #

The Leiden Declaration recommended by the lecture emphasizes that even with the use of automated tools, the correctness of the results, the sufficiency of the arguments, and the completeness of the citations remain entirely the responsibility of the human authors. AI can participate in proof generation, formalization, writing, and literature review, but it cannot become a black hole of responsibility.

3. Actively Support Review and Verification #

Authors should not only pursue generation speed but also strive to reduce the verification cost for peers, including providing complete citations, clearly explaining AI involvement, and, when appropriate, submitting formal proofs or reproducible materials.

4. Reduce Excessive Rewards for “Being the First to Solve a Problem” #

When proof generation becomes increasingly inexpensive, simply rewarding “the first answer” will amplify rushing, concealment, and low-quality output. Academic evaluation should elevate the status of explanation, validation, editing, reproduction, and theoretical consolidation.

5. Consider “Ability to Explain Clearly” as a Publication Threshold #

Tao proposed a simple yet weighty judgment criterion: if an author cannot explain their results with expert-level clarity, correctness, and appropriate attribution, then the work is not yet suitable for formal publication.

6. Retain Necessary Human Training in Education #

In educational and research training stages, AI use needs to be strictly limited in some scenarios. Learners still need to retain the ability to independently derive, identify errors, and perceive difficulty; otherwise, even with powerful tools, they will be unable to judge when the tools deviate from the goal.

7. Build New Workflows and Infrastructure #

Traditional journals, peer review, and textbook systems struggle to directly absorb large-scale machine-generated results. The mathematical community needs to proactively design new mechanisms for verification, discussion, formalization, knowledge consolidation, and public communication, rather than completely entrusting the rules to commercial model providers.

The common principle behind these recommendations is:

AI can expand the capabilities of mathematicians, but it cannot replace transparency, responsibility, judgment, and community collaboration.

Why I Admire This Lecture #

In my view, there are three points that make this lecture most admirable.

First, it maintained a rare restraint regarding short-term capability predictions, elevating the problem to a more enduring institutional level. Model performance will change rapidly, but understanding, responsibility, education, and community will not automatically be resolved by the next version update.

Second, it accurately identified the bottleneck shift. AI lowers the cost of generation, but it might transfer costs to verification, review, maintenance, and integration. In software engineering, we already see similar cost shifts: after code became easier to generate, what became truly expensive was requirements judgment, architectural constraints, testing, deployment, and long-term maintenance.

Third, it re-evaluated the “invisible labor” within the academic system. Reviewers, editors, teachers, textbook authors, and knowledge maintainers were often seen as supporting roles after innovation; in an era of proof abundance, they might become the core force sustaining knowledge quality.

Precisely because I take this lecture seriously, I am willing to continue pursuing several directions still to be explored, following the problem awareness it initiated.

First, the lecture intentionally bundled differences across fields, tasks, and supervision methods into a broad working hypothesis for clarity of condition analysis. However, as the discussion further delves into policy and resource allocation, these differences still need to be unpacked: problems with high formalizability and clear verification feedback will not be affected at the same rate as problems that rely on conceptual creation, long-term theoretical development, and research taste.

Second, while “community recognition” is irreplaceable, it is also not inherently fair. Academic communities can also tend to be conservative, form hierarchies, and be influenced by resources, prestige, and personal networks. The AI era requires both valuing human expert judgment and continuously improving human-made institutions, rather than idealizing existing peer review.

Tao also explicitly stated at the end that similar analyses should be conducted for teaching, mentoring, hiring, grant applications, and public communication. Building on this, I believe academic authorship, concentration of computational power, closed commercial models, and participation opportunities among different countries and research institutions also need to be incorporated into the same framework.

Raising these extended questions is not to demand that one lecture answer everything; on the contrary, that these questions naturally arise from the lecture demonstrates Tao’s success in opening up a space for long-term discussion within the mathematical community. For me, this is the most valuable aspect of the lecture: it is not a self-proclaimed complete, definitive answer, but a research agenda solemnly proposed by a top scholar, inviting the entire community to continue advancing it.

Insights for AI Engineering and Academic Work #

As someone involved in both AI engineering and academic review, I prefer to interpret this lecture as a thinking framework transferable to many knowledge-based tasks.

Generation capability does not equal responsibility transfer. Models can draft code, proofs, reports, and review comments, but ultimate responsibility must still rest with individuals who can explain, verify, and bear the consequences.

Investing in generation capabilities must be accompanied by investments in validation and digestion capabilities. Stronger models, without corresponding testing, review, knowledge management, provenance tracking, and quality thresholds, will only push the bottleneck downstream. Reviewing, editing, teaching, and maintenance should also receive evaluation commensurate with their importance.

Educational stages require retaining independent judgment. The goal of training is not just to get answers faster, but also to develop a sense of problems, errors, and judgment. Some difficulties cannot be entirely outsourced, as they are part of the process of capability formation itself.

Conclusion: Answers Become Cheaper, Judgment Becomes More Expensive #

AI will not necessarily end mathematics, nor will it render mathematicians meaningless. It is more likely to prompt the mathematical community to re-answer a question long obscured by technological progress: Is mathematics merely about producing more theorems, or is it about helping humanity understand and think more clearly and effectively?

When proofs were still scarce, these two goals seemed almost identical; when proofs began to become abundant, they truly diverged.

The Chinese term ‘xué wèn’ (学问), meaning ’learning’ or ‘knowledge,’ inherently implies more than just answers: it’s about both ’learning’ and ‘inquiring’ (学 and 问); it’s about seeking truth, but also about articulating principles clearly and transmitting them onwards. From this perspective, the understanding, responsibility, and community that Tao cherishes are not mere add-ons to mathematical discovery, but the very foundation upon which learning can grow and be passed down.

The most valuable capabilities in the future may no longer be just producing answers faster, but judging which questions are worth asking, which results are worth believing, which ideas are worth transmitting, and who is willing to take responsibility for these judgments.

In this sense, AI brings not merely an upgrade to mathematical tools, but a re-pricing of knowledge, value, and community.

When answers become cheaper, understanding, judgment, and responsibility become more expensive.

References #

Mathematics in the Age of AI Infographic