1. A Letter in Reply

On 11 September 2026, twenty-five Fields medallists issued a statement about what they called a serious misalignment in AI-driven mathematics. Terence Tao republished it on his blog. Its central distinction was between solving problems and the understanding for which problem-solving serves as a means. An answer, however difficult to obtain, is not the whole purpose of mathematics.

A related distinction appears in a letter written 196 years earlier.

Niels Henrik Abel died of tuberculosis on 6 April 1829, aged twenty-six. In June the following year, the Paris Academy shared its mathematics prize between the dead Norwegian and Carl Gustav Jacob Jacobi, a twenty-five-year-old professor in Königsberg. Working independently, the two had transformed the theory of elliptic functions. Adrien-Marie Legendre, who had spent decades cultivating that subject, responded with remarkable generosity: he added supplements to his own work to accommodate what the younger men had discovered. Jacobi had already asked him, in March 1829, how Abel's substantial memoir, submitted to the Academy in 1826, could have attracted so little attention.

Jacobi's book on elliptic functions appeared in 1829. When Poisson discussed it for the Academy, he repeated a criticism attributed to Fourier: Abel and Jacobi ought to have given priority to problems concerning heat. Fourier's own great mathematical achievement had grown from heat conduction. He died on 16 May 1830, while serving as one of the Academy's permanent secretaries. A different manuscript, submitted for that year's competition by the eighteen-year-old Évariste Galois, failed to reach the final assessment and disappeared. That institutional failure should not be turned into a confidently reconstructed chain of personal responsibility.

On 2 July, Jacobi wrote to Legendre. He praised Poisson's account of two transformations and recognized the influence of Legendre's supplements. Then he objected to the revival of Fourier's remark. In Jacobi's account, Fourier had located mathematics' chief purpose in public utility and the explanation of natural phenomena. Jacobi countered that science existed for the honour of the human mind; understood that way, a question about numbers could be as valuable as a question about the system of the world.

This is Jacobi's response, not a verbatim statement by Fourier independently located for this review. The distinction matters. The letter itself is available in the correspondence published in 1875; Fourier's position reaches us here through reporting and disagreement.

Detached from its occasion, Jacobi's sentence became a portable motto. Science sometimes became mathematics, the exchange with Fourier disappeared, and the claim was assimilated to mathematics for its own sake. Jean Dieudonné used its central phrase for a book title in 1987. Yet Jacobi had named a beneficiary: the human mind. His argument was not that mathematics existed for mathematics.

Utility and human intellectual flourishing need not be enemies. They are nevertheless two different answers to the question of purpose, and this exchange brings their difference into focus.

2. For Use

The practical answer is older than Jacobi's letter by several millennia.

Many of the earliest Mesopotamian tablets record deliveries, obligations, and distributions. They were made so that an institution could remember what had happened and calculate what should happen next. Later scribal education has left literary accounts of its own routines. In the composition conventionally called Schooldays, a pupil goes to school with his tablet, recites, eats, prepares further work, and repeats his lessons before his father at home. The next school day brings inspection and punishment.

This is a school text, not a disciplinary record. Its beatings concern such matters as talking, movement, and handwriting; it would be misleading to present it as documentary evidence that a particular numerical error earned a particular blow. But the institutions into which scribes graduated depended heavily on accounts. Grain, labour, and taxes had to be recorded accurately. The child was being trained to become a dependable part of a larger apparatus.

Fourier represents the practical tradition at its most intellectually distinguished. He accompanied Napoleon's expedition to Egypt and subsequently served as prefect of Isère from 1802 to 1815. His work on heat developed during those years in Grenoble. The physical problem led him to represent functions through trigonometric series, opening questions that would transform analysis. In this instance, usefulness and depth were not rival virtues. They grew together.

The twentieth century extended this relationship into artillery tables, wartime cryptanalysis, and financial pricing. Mathematical workers entered states and corporations in increasing numbers. At Britain's Government Communications Headquarters, Clifford Cocks devised in 1973 a public-key system mathematically equivalent to the later RSA construction. It remained secret until 1997. James Ellis, whose earlier idea of non-secret encryption had helped open that route, died shortly before the work was publicly disclosed. The institution obtained the benefit decades before the researchers could receive ordinary public recognition.

The most thoughtful defence of contact with the world is not a demand that every theorem immediately serve an application. In his 1947 essay The Mathematician, John von Neumann warned that a subject could become detached from its empirical origins through successive generations of abstraction, then drift toward elaborations of diminishing significance. Renewal required movement between abstraction and renewed contact with experience. Mathematics needed the world, on this account, to remain fertile.

That is a powerful intellectual argument. It leaves another question unanswered: how are the people doing the mathematics treated?

The schoolchild, the young cryptanalyst bound by secrecy, the woman whose pay category required describing her as a linguist, and the inventor unable to publish his work for twenty-four years all supplied something useful. Their mathematical abilities became resources. Too often the person became a resource as well. An institution can correctly value a calculation while valuing its maker chiefly as an interchangeable means of obtaining it.

3. For Its Own Sake

In 1940, with another world war under way, G. H. Hardy offered a very different defence. A mathematician, he argued in A Mathematician's Apology, makes patterns as a painter or poet does; beauty is indispensable. He also found consolation in the apparent distance between his own work and the conduct of war. Parts of the number theory he cherished would later become central to cryptography, a reminder that an author's preferred use cannot determine a subject's future.

The Bourbaki treatises gave impersonal mathematics a particularly influential literary form. Definitions came first, historical notes later; examples were often displaced into exercises. Results appeared in a carefully ordered structure, with much of the uncertainty and individuality of discovery removed. This was a programme for presenting mathematics, not evidence that every modern mathematician intended to abolish the human presence. But it made a powerful image available: a subject accountable to its internal architecture.

A useful mathematics can subordinate its workers to an external objective. An impersonal mathematics can make them disappear from the finished account. These are different mechanisms, not identical histories, yet both invite us to ask what has been left outside the frame.

Turing's 1936 paper began with a human calculator. His abstraction retained the rule-governed operations that could be performed step by step and set aside the rest of the person's life. That restriction made a mathematical theory of computation possible. It was not a proof that a human being consists entirely of those operations.

Emmy Noether's career exposes a different kind of omission. Her early Göttingen teaching appeared under Hilbert's name, with her described as assisting. Habilitation eventually allowed her to lecture in her own right, but did not initially provide a salaried position. Later remuneration remained modest. On 25 April 1933 a telegram ordered her suspension. Fourteen mathematical assessments, including those of Hardy and Takagi, together with a petition from twelve students, documented her extraordinary standing. They did not reverse the outcome. Gender restrictions had first obstructed her; Nazi racial and political persecution then expelled her. Mathematical value was not what the institution was prepared to recognize.

The four-colour theorem brings less conspicuous labour into view. The 1976 announcement involved a list of 1,936 configurations; the published proof used a revised list of 1,482. Children from the Appel and Haken families helped check printed material. Dorothea Haken contributed substantial work while at school and university. In Haken's later recollection, thirteen-year-old Laurel Appel spent a month finding roughly eight hundred copying or diagram errors across some seven hundred pages. These were corrections to working materials, not eight hundred surviving counterexamples to the theorem. The children were not named as authors, although absence from a byline should not be confused with absence from every acknowledgment.

Perelman placed his papers online in 2002 and 2003, making them available without requiring the reader to acknowledge his status. Later interviews attributed to him the view that a correct proof needed no additional recognition. His withdrawal also expressed dissatisfaction with mathematical institutions. Neither the independence of a proof from its author nor the author's refusal of a prize makes the treatment of the person irrelevant.

4. Bringing the Person Back

Repeatedly, mathematicians have challenged the idea that their subject is exhausted by a stock of results.

Brouwer's 1907 dissertation located mathematics in constructive mental activity. Existence required an appropriate construction, not merely a classical argument that nonexistence would lead to contradiction. This is a technical and philosophical position, not a requirement that every infinite process physically finish. In 1908 he questioned the unrestricted use of the law of excluded middle, initiating a dispute that would eventually divide him sharply from Hilbert.

Hermann Weyl, Hilbert's former student, was for a time drawn strongly toward Brouwer. His 1921 discussion of a new foundational crisis presented intuitionism as revolutionary. Elsewhere he compared mathematizing with creative activities such as language and music: its historical choices could not all be explained away as uniquely compulsory responses to an independent object. Reflecting in 1946 on the costs of foundational controversy, he connected those costs with asking what mathematics meant within a human life of knowledge, suffering, and creation. These were philosophical commitments, not conclusions every mathematician accepted.

The computer-assisted four-colour proof brought another version of the question before a much wider audience. At a Toronto meeting in August 1976, Haken described work involving around twelve hundred hours of computer calculation. Polite applause did not extinguish unease. The obstacle was not a theorem establishing that human checking was logically impossible. It was the practical difficulty of examining an enormous computation by familiar human means, and the uncertainty about what kind of knowledge its successful execution supplied.

In 1994, William Thurston framed the question differently in On Proof and Progress in Mathematics. Instead of asking principally how mathematicians prove theorems, he asked how they advance human understanding. He read the four-colour controversy as expressing a desire for intelligibility beyond confidence in a result's truth. That was his interpretation of the controversy; some critics had also raised more direct concerns about reliability.

Thurston's account of his own work on foliations gave the argument an uncomfortable specificity. In the early 1970s he proved major theorems, but other researchers subsequently left the field. Someone told him, intending a compliment, that he was killing it. He came to see that his papers had not communicated enough of the ways of thinking that made the results accessible to him. A finished theorem could raise the entry barrier rather than open a route for other people.

What colleagues most wanted from him, he found, was often his way of seeing. The point applied even to his celebrated work on large classes of cases of geometrization, before Perelman completed the full programme. Mathematical progress depended on transmitting capacities, not simply depositing conclusions.

Brouwer, Weyl, and Thurston did not share one philosophy. Their arguments concern different questions about existence, creativity, and communication. What joins them for this series is a refusal to let the human activity disappear behind its products. Mathematics is something people do, and its purposes cannot be read directly from a list of true propositions.

5. Taking a Position

The opposing view deserves its strongest formulation.

In section 24 of the Apology, Hardy treats the primality of 317 as a fact independent of what people think or how their minds happen to be constituted. Gödel's 1964 account of mathematical realism goes further: he describes a kind of intuition of set-theoretical objects and principles, analogous in some respects to perception. Neither statement is merely a claim that mathematicians enjoy certainty. Both suggest that their thought answers to a reality it did not create.

No vote can make 317 composite under the ordinary definitions. The truth of a mathematical result does not depend on the nationality of its discoverer or the year of its publication. Any account that ignores this resistance has missed something essential.

This series nevertheless adopts a broadly constructivist interpretation of mathematical history. That is the author's position, not an established conclusion of the history surveyed here. The names negative and imaginary preserve old difficulties in admitting objects. Greek treatments of incommensurability changed what could count as equality of ratios. Non-Euclidean geometry loosened the apparent necessity of the parallel postulate. Nineteenth-century limit methods replaced many earlier uses of infinitesimals with quantified control of finite increments and errors, while still employing infinite sets and processes. Twentieth-century nonstandard analysis then supplied rigorous number systems containing infinitesimals.

These developments show that mathematicians make consequential choices about languages, definitions, and methods. They do not by themselves refute realism. A realist may interpret the choices as alternative ways of approaching an independent domain. The present series instead emphasizes the building of the frameworks through which mathematical objects become available to us.

Once ordinary divisibility and primality are fixed, 317 has no positive divisors other than 1 and itself. The conclusion is not negotiable. The philosophical claim here is that objectivity can arise within humanly articulated frameworks without requiring the further conclusion that those frameworks are wholly independent of human activity.

Throughout the series, framework names a structure built to accommodate something: a notation, an axiomatic theory, a rule of proof, and, at another level, a university, charter, or prize. Remainder names what such a structure fails to accommodate. The diagonal of a square, alternative geometries, and unfamiliar infinities have played that role in mathematical arguments. Noether and the unrepresented aspects of Turing's human calculator enter the series as social and conceptual analogies. Exclusion by a discriminatory institution is not literally the same mechanism as inexpressibility in a formal language.

Some theories do obtain completeness and decidability by restricting their expressive resources. That does not mean their decision procedures are easy. Gödel's incompleteness theorem, strengthened by Rosser, establishes a different, precisely bounded result: a consistent, effectively axiomatized theory capable of expressing sufficient arithmetic cannot decide every sentence in its language.

Within those hypotheses, a limit has been proved. Outside them, the claim that every framework produces a remainder is the series' interpretive proposal, not Gödel's theorem in disguise. Its ethical proposal is equally explicit: frameworks are means; people should not be reduced to serving them.

6. The Machines Arrive

At the 2025 International Mathematical Olympiad on Australia's Sunshine Coast, the official results record 630 contestants from 110 countries. They faced three problems in each of two four-and-a-half-hour sessions. Seventy-two students earned gold medals; the threshold was 35 out of 42.

Two AI systems also produced work assessed at that score. Google's advanced Gemini Deep Think solved five problems, with its solutions assessed by IMO coordinators. OpenAI reported five solutions from an experimental model, graded by three former IMO medallists. The second evaluation was not official IMO grading. Neither system was a national-team contestant, and neither received a student medal.

The previous year, DeepMind's AlphaProof and AlphaGeometry 2 had jointly solved four problems for 28 points, one below the gold threshold. Humans translated the problems into formal language where required, and some solutions took as long as three days. Timothy Gowers and Joseph Myers assessed the work. The progress from that experiment to natural-language solutions reported within the 2025 examination limits was substantial. The experimental conditions were not interchangeable.

Both companies said their 2025 systems observed the two examination time limits. But grading an answer does not verify the entire process that generated it. The IMO's closing-day statement expressly said it could not validate such matters as computational resources, human involvement, or reproducibility in private AI tests. Gold-medal-standard performance is consequently a more accurate description than winning an IMO gold medal.

The announcements also created a dispute about timing. OpenAI announced on the evening of the closing ceremony, saying it understood that publication should wait until the ceremony had ended. DeepMind announced on Monday after clearance through its collaboration with the organizers. Accounts of requested delays and permissions differed. It is unsafe to turn those accounts into a settled finding that one precisely documented universal embargo was broken. What was visible was a clash between the wish to preserve attention for students and the commercial importance of announcing first.

In October, a different announcement blurred another boundary. An OpenAI executive described answers to ten Erdős problems as solutions of previously unsolved problems. Thomas Bloom, who maintains a catalogue of Erdős questions, objected that the model had located existing literature unknown to him. A site's incomplete knowledge was not the same thing as an open problem throughout mathematics. The post was deleted. Recovering overlooked papers is useful; presenting retrieval as new discovery obscures their authors.

Genuine new mathematical results must be distinguished from that episode. A May 2026 paper by nine mathematicians presents a model-generated counterexample to an Erdős conjecture on unit distances. Let u(n) be the largest possible number of unit-distance pairs among n points in the plane. The conjectural bound was n raised to 1 plus a quantity tending to zero. The construction instead gives, along an unbounded sequence, at least n raised to 1 plus a fixed positive constant. What fails to shrink is the excess exponent, not the numerical excess of pairs over points.

The human authors, including Bloom, checked, refined, contextualized, and explained the construction. Their paper acknowledges earlier mathematical ingredients. Recognizing that labour does not diminish the machine's contribution. It states more accurately what the achievement comprised.

7. Who Is Responsible?

In January 2023, Nature explained why a large language model could not be listed as an author: authorship carries accountability that the tool cannot assume. Similar rules have been adopted by other major publishers. These are institutional responsibility requirements, not a theorem about every possible future intelligence.

On 4 September 2026, Anthropic announced what it described as the first complete computer-checked formalization of Fermat's Last Theorem. According to the company, dozens of agents worked for eleven days, producing roughly thirteen million lines of code and about 30,300 proved intermediate results, of which approximately 29,500 entered the final proof. The theorem itself had been proved by humans in the 1990s. The new claim concerned comprehensive formal verification, not discovery of a new theorem.

Kevin Buzzard supplied an important independent check. In his same-day account, he reported compiling the material and running the comparison check for the theorem's statement. That is stronger evidence than simply repeating a company's announcement, but it is not a claim that he manually inspected every line. Formal assurance still depends on the statement encoded, the accepted axioms, and the trusted checking infrastructure.

Buzzard's own five-year, £1 million project had begun in October 2024, initially undertaking to reduce Fermat's Last Theorem to results available in the 1980s. He congratulated the new work while distinguishing his continuing objective: a living document through which people could explore modern number theory. The Anthropic formalization followed an older exposition rather than his project's modern route. Compilation alone does not explain which ideas matter or how a reader might carry them elsewhere.

A useful comparison is the 2023 formalization of the characteristic-two polynomial Freiman–Ruzsa theorem. Following the proof by Gowers, Green, Manners, and Tao, a collaborative project organized the argument through a public blueprint linking mathematical statements to formal code. Within roughly three weeks, a group of around two dozen contributors had completed the formalization. The blueprint remains available as an invitation to understand how the argument fits together.

Another announcement followed on 8 September 2026. OpenAI said a deployment of roughly ten thousand agents had found a candidate solution to a permitted breakdown version of the Navier–Stokes Millennium Problem. Its account distinguishes approximately eighty-eight hours of discovery from a further seventeen hours of Lean formalization. The claimed construction concerns finite-time breakdown with a smooth external force, not the unforced problem in every formulation. The company said it would not claim the prize.

On 11 September, the Clay Mathematics Institute said the problem appeared to have been settled, while stressing deliberately unhurried evaluation. Its statement did mention the million-dollar prize and its rules; it did not announce an award. As of this essay's review on 1 October 2026, that provisional language must not be silently converted into completed institutional certification.

There was also a dispute about priority and provenance. Tristan Buckmaster and Levent Alpöge released related work on forced Euler and other equations shortly before OpenAI's announcement. Buckmaster's signed statement dated their results to August and described Alpöge's participation as personal research, not an Anthropic project. It raised questions about whether unpublished ideas might have reached OpenAI, including through coding-tool usage, and reported an alleged proposal concerning Alpöge's removal from a byline. Buckmaster explicitly said he was not accusing anyone.

OpenAI denied having seen their work before publication and subsequently said its internal investigation found that the relevant coding-tool material could not have influenced the result through training. Those are competing first-party accounts, not an independently adjudicated finding. This review found no public independent determination resolving the dispute by 1 October. Provenance questions must also be kept distinct from whether a mathematical argument is correct.

The Fields medallists' statement appeared against this background. It defended exposition, attribution, the extraction of reusable ideas, and the formation of future mathematicians. Its objection was not that machines should never contribute. It was that rapid production of answers could displace the human work through which answers become mathematics people can understand and develop.

8. For Whom?

Jacobi's letter and the statement of September 2026 stand 196 years apart. One speaks of the honour of the human mind; the other distinguishes conceptual understanding from the instrumental achievement of solving problems. Their circumstances and assumptions differ, but both locate something worth preserving in people.

Recent mathematical AI has made two human roles especially visible. At one end stands responsibility: authors who answer for claims, researchers who check that a formal statement captures the intended theorem, institutions that distinguish an announcement from an accepted result. At the other stands understanding: readers, teachers, collaborators, and students who want to acquire ways of thinking rather than merely receive certified conclusions.

Much more can now occur between those ends without direct human execution. A model can search, propose, code, and help verify on a scale that changes what research is possible. None of that by itself decides what mathematical institutions ought to reward, which forms of knowledge they should preserve, or how they should treat the people whose work supplies their foundations.

The 630 students on the Sunshine Coast were not simply inexpensive devices for generating six answers. A machine's reported 35 points is a meaningful technical result. The students' education, effort, and developing mathematical capacities are different goods. To treat the contest solely as a mechanism for obtaining solutions would fail to explain why a human mathematical community values holding it.

The position of this series is that mathematics is constructed within human practices, and that people should remain its ends. The first claim is philosophical and contestable; the second is an ethical commitment. Neither follows mechanically from the correctness of a proof.

The history examined here repeatedly changes the relation between those ends and means. Sometimes mathematics is elevated above its makers, and the finished account leaves out the calculator's life or the child checking diagrams. Sometimes it is enlisted for taxation, military advantage, or profit, and the mathematician becomes another input. These are not reasons to reject abstraction, application, or machines. They are reasons to examine the institutions through which their benefits and burdens are distributed.

Thirteen million lines of code, ten thousand agents, eighty-eight hours: such quantities tell us something important about scale. They do not answer the question of purpose. The farther a system can operate beyond any one person's unaided inspection, the more deliberately its users must decide what kinds of responsibility and understanding surround it.

In 1843, Ada Lovelace added extensive notes to her translation of an account of an unbuilt calculating machine. One of those notes denied that the Analytical Engine could originate anything on its own. That historical claim concerned a particular proposed machine. It cannot simply be carried forward as a verdict on every system that came after it.

The machine has changed. The question remains.

What, and whom, is all this calculating for?