Never complain, never explain
Math 2.0 and the manner of the fox
Man dürfe einem guten Bauwerke, pflegte er zu sagen, nach seiner Vollendung nicht mehr das Gerüste ansehen.
On a good building, he used to say, one should no longer see the scaffolding once it is finished.
—Sartorius von Waltershausen, Gauss zum Gedächtniss (1856). Page 82 of the memoir Gauss's friend published the year after his death. The sentence usually quoted in its place, that no architect leaves the scaffolding standing when the building is done, is a later paraphrase: the quotation has had its own scaffolding removed.
Seven hundred files arrived in October, and there was no one to ask about them. They were the output of an unnamed OpenAI model, set down in a public repository: 372 claimed results on open problems, nearly all of them one prompt to one agent and some three hours of compute apiece, many with a formal proof in Lean attached, none with an author. The repository is openai/math. The counts are from The Decoder (Oct 7, 2026) and Scientific American (Oct 8, 2026), which has 42 percent of the results formalized and three papers withdrawn within days after experts found significant errors. The earlier essay on this site took the release as the mathematical form of a problem in engineering; this one takes the complaint against it on its own terms. The Association for Human Mathematics answered the next day, and it answered in the voice of a union. "Mathematicians did not ask for this work to be done." Seven hundred files at once was "not a demonstration of scholarship, but a demonstration of power." Members should stop working with the company and "return to a vision of science that centers human understanding." AHM Statement on OpenAI's October 6 Release of Mathematical Documents, Oct 7, 2026, a guest post on Terence Tao's blog signed by the association's Communications Working Group. Some coverage has Tao leading the association; the statement names no officers, and his own position was more hedged: he reposted it, and told the Times a week later that he no longer saw a way to engage constructively with the company.
Three complaints travel together in that answer and in the week of commentary that followed it. They are worth taking apart, because they are not the same age. The first is that the output cannot be read. Roland Bauerschmidt, who had worked through some of the papers, found others "sort of this AI slop," so badly written that had they come by e-mail from a nobody he would have deleted them. Scientific American, Oct 8, 2026. Andrew Sutherland of MIT, in the same piece: "we should expect some of the proofs to contain mistakes, possibly serious ones." The second is that a solution is not an understanding. Twenty-eight Fields Medalists had said so a month before the release: solving problems as a benchmark "is detrimental to the science of mathematics"; the famous ones "have often served as landmarks and lighthouses"; "the mass production at faster and faster pace of 'true/false' statements could destroy fertile ground"; and without mathematicians to develop and integrate what a machine produces, "the crucial human transmission chain between mathematicians would be lost." A Severe Misalignment of AI in Mathematics, Sep 11, 2026, with twenty-eight Fields Medalists on its list of initial signatories as the page stands today, Tao among them. The third is that sorting the output is labor the field never agreed to supply: a community left "doing the thankless work of making proofs readable, for free," while none of the humans behind the proofs, Tao wrote, "are available to take questions, give talks, attend conferences, submit papers to journals, train students, or otherwise participate in the subsequent development of these results." The Decoder, Oct 8, 2026; Tao in 100+ reactions to 100+ solutions, Proofs and Prompts, Oct 8, 2026. Cryptic, useless to understanding, and unpaid. That is the charge, and of all the disciplines that might bring it, mathematics is the strangest.
Three days later, at Caltech, Tao turned the charge into a program and gave the program a name. Math 2.0, the AIM Public Lecture of Oct 9, 2026, announced a year earlier as "Machine Assisted Proof"; the quotations are from slides 3, 6, 9, 16, 25 and 29. His post of the next day describes the talk as laying out "the historical 'Math 1.0' state of mathematics, the current state induced by the excessive emphasis on automated solving of open problems by AI, and my vision for a future 'Math 2.0'". Mathematics had lived "under the assumption of proof scarcity," and its institutions, incentives and stated objectives were built on it. Proofs are verifiable and digitizable, so the labs went for them first, and "we are now entering an era of proof abundance in mathematics." Problem solving had been a proxy for the discipline's real goals, accurate while proofs were scarce, "misaligned in the era of proof abundance," and its further blind optimization "now actively harmful to the long-term health of mathematics." Math 2.0 would expand the research frontier "while simultaneously decentering the traditional role of problem solving." Open problems are lighthouses, "not destinations to be reached in and of themselves," and reaching them early by machine "can disrupt the exploration of the paths not taken, and sterilize the surrounding field." One use of the machine he would keep is the open exposition problem, a motivated explanation of a result whose known proof is opaque. The obstacle, on the last slide, is "not technology, or even institutional culture. It is imagination."
Each count names something the discipline has done to its own readers, on purpose, for two hundred years. Each has a history inside mathematics, and the oldest of them belongs to the fox.
Abel spent the winter of 1825 in Berlin and wrote home to Hansteen that the young mathematicians there "nearly deify Gauss. He is for them the quintessence of all mathematical excellence." He granted the genius and added what the students added: "it is also well-known that he gives rotten lectures. Crelle says that all Gauss writes is an abomination, since it is obscure to the point of being almost impossible to understand." Abel to Christopher Hansteen, the first of five letters from his journey, in Arild Stubhaug's translation (Niels Henrik Abel and his Times, 2000, p. 332), as quoted by David E. Rowe, "Looking Back on Gauss and Gaussian Legends," The Mathematical Intelligencer 38:4 (2016). Crelle's word was Gräuel. Hansteen printed the letter in 1862 with a footnote: a German student had said of Gauss on that occasion, er macht es wie der Fuchs, der wischt mit dem Schwanze seine Spuren im Sande aus. He does as the fox does, who wipes out his tracks in the sand with his tail. Christopher Hansteen, "Niels Henrik Abel," Illustreret Nyhedsblad, Christiania, Mar 2, 1862. The remark circulates as Abel's own. Rowe, who could not find it in any modern biography, traced it through an unattributed footnote in Bjerknes's life of Abel (French edition, 1885, p. 92) to Hansteen's page, where it belongs to a Berlin student, not to Abel. Two of the three complaints of 2026 were made in 1825, against the prince of the discipline, by its best young talent and its best editor: unreadable, and no use for learning.
Gauss had an answer, and Sartorius, his friend, recorded it the year after his death, together with the trade's verdict on it:
Es war zu aller Zeit Gauss' Streben seinen Untersuchungen die Form vollendeter Kunstwerke zu geben; eher ruhete er nicht und er hat daher nie eine Arbeit veröffentlicht bevor sie diese von ihm gewünschte durchaus vollendete Form erhalten hatte. Man dürfe einem guten Bauwerke, pflegte er zu sagen, nach seiner Vollendung nicht mehr das Gerüste ansehen. In seiner Darstellung bediente er sich fast immer der synthetischen Methode, die er beim Studium des Archimed und Newton lieb gewonnen hatte. Sie ist zwar an Kürze und Bündigkeit vor der analytischen Methode ausgezeichnet; allein der Gang der Entdeckung bleibt verschleiert und es scheint öfter, dass er den zum blossen Zwecke der Belehrung absichtlich in seinen Schriften nicht habe betreten wollen.
Es lässt sich nicht verkennen, Gauss selbst hat es öfter bemerkt, dass diese kunstvolle Art der Darstellung, die namentlich das Lesen seiner Abhandlungen weniger in der Mathematik Bewanderten sehr erschwert, ihm viele Zeit gekostet hat. Da er aber in seiner Jugend diesen Weg ein Mal gewählt hatte, hat er ihn später auch nicht wieder verlassen wollen und es erklärt sich dadurch das Motto seines Siegels: „Pauca sed matura.“
Einige grosse Mathematiker der neuern Zeit haben in Bezug auf Gauss' Entdeckungen mitunter die Ansicht ausgesprochen, dass es für die Entwicklung der Wissenschaft vortheilhafter gewesen sei, wenn er auf die Vollendung seiner Arbeiten weniger Gewicht gelegt, dagegen aber mehr von seinem unerschöpflichen Ideenreichthum mitgetheilt habe, der jetzt, so weit er nicht aufgezeichnet, für die Nachwelt verloren gegangen ist.
It was at all times Gauss's endeavor to give his investigations the form of finished works of art; he did not rest until then, and so he never published a work before it had received the thoroughly finished form he wanted for it. On a good building, he used to say, one should no longer see the scaffolding once it is finished. In his presentation he almost always used the synthetic method, which he had come to love in studying Archimedes and Newton. It is distinguished from the analytic method by brevity and conciseness; but the course of discovery stays veiled, and it often seems that he deliberately did not want to enter upon it in his writings for the mere purpose of instruction.
It cannot be denied, and Gauss himself often remarked it, that this artful manner of presentation, which makes the reading of his papers very hard for those less versed in mathematics, cost him a great deal of time. But since he had once chosen this path in his youth, he would not leave it later either, and that explains the motto of his seal: pauca sed matura.
Some great mathematicians of recent times have sometimes expressed the view, with regard to Gauss's discoveries, that it would have been more advantageous for the development of science had he put less weight on the completion of his works and instead communicated more of his inexhaustible wealth of ideas, which now, so far as it was not written down, is lost to posterity.
—Wolfgang Sartorius von Waltershausen, Gauss zum Gedächtniss (Leipzig: Hirzel, 1856), p. 82. My translation, from the scan. Pauca sed matura: few, but ripe. The synthetic method presents a result from its premises and conceals the analysis that found it; Archimedes and Newton wrote that way, and so, with few exceptions, did every journal after them.
The answer is an aesthetic. The verdict that follows it is the third complaint of 2026 in the trade's own voice, a century and a half early: the finish was paid for in ideas, and posterity never received them. The manner cost its author time and its readers more, and the discipline kept it anyway.
The discipline knew the price at the moment of purchase and paid it, because the man who set the price was Gauss. What had been a prince's habit Bourbaki made a constitution. The Éléments de mathématique carefully avoided illustrations, text and formulas only, the exception being the chapters on Lie groups for which Borel drew figures; asked about the austerity, Cartier said that the Bourbaki were Puritans, "and Puritans are strongly opposed to pictorial representations of truths of their faith." Marjorie Senechal, "The Continuing Silence of Bourbaki: An Interview with Pierre Cartier," The Mathematical Intelligencer 20:1 (1998). The group marked a hard passage with a bend-in-the-road sign in the margin and otherwise left the reader to find the road. The manner did not need Paris to spread. The two American textbooks on which the postwar generations were raised, Rudin's Principles of Mathematical Analysis for the undergraduate and Lang's Algebra for the graduate student, keep it. Rudin opens by showing that the positive rationals whose square is less than 2 have no largest member: to each such p he assigns the number q = p − (p² − 2)/(p + 2), verifies that q is larger than p and still too small, and moves on. Where the expression came from is not said there or anywhere else in the book, and students have asked ever since. Walter Rudin, Principles of Mathematical Analysis, 3rd ed. (New York: McGraw-Hill, 1976), Example 1.1, p. 2; the example is already in the first edition of 1953. The expression is the linear-fractional map q = (2p + 2)/(p + 2), whose fixed point is the square root of 2 and which moves p toward it without crossing it. The book gives the map and not the thought. Lakatos took the book as his specimen of what he named the deductivist style. The student "is obliged, according to the Euclidean ritual, to attend this conjuring act without asking questions either about the background or about how this sleight-of-hand is performed"; the style "hides the struggle, hides the adventure. The whole story vanishes, the successive tentative formulations of the theorem in the course of the proof-procedure are doomed to oblivion while the end result is exalted into sacred infallibility." He kept quoting Rudin, he explained in a footnote, because the book was one of the best within the tradition. He had been a Marxist, and he said in another footnote where he thought the ritual came from: his mathematician is "an imperfect personification of Mathematics," an idea he called "closely analogous to Marx's concept of the capitalist as the personification of Capital." Imre Lakatos, Proofs and Refutations: The Logic of Mathematical Discovery, ed. John Worrall and Elie Zahar (Cambridge: Cambridge University Press, 1976), Appendix 2, "The Deductivist versus the Heuristic Approach," pp. 142–7 for the passages quoted; the footnote on Rudin is on p. 145, and "outstandingly good within the deductivist tradition" on p. 147. His exhibits are uniform convergence and bounded variation: the definition first, and the reason it was wanted never. The Marx analogy is a footnote on p. 146. Lang heads the exercises on homology in his Algebra with one sentence: "Take any book on homological algebra, and prove all the theorems without looking at the proofs given in that book." Serge Lang, Algebra (Reading, Mass.: Addison-Wesley, 1965), p. 105; p. 175 in the 1984 edition. Rotman and Eisenbud both quote it in their own books on the subject. The revised third edition (New York: Springer, 2002) moved homological algebra to a part of its own and dropped the exercise, keeping only the remark beside it that the arrow-theoretic arguments were "affectionately called abstract nonsense by Steenrod." The exercise is a joke, and the joke is the fox's: the proofs in the book are the tracks, and the student is told that a mathematician does without them. What the textbooks did for the student the journals did for the author. The genre is definition, theorem, proof. The scaffolding comes down before submission as a matter of form, and a referee who asked for it back would be asking for a different genre; the request would be declined as a request for exposition, which is not what the journal publishes.
A manner kept this long stops being a manner and becomes a style, and a style is a way of being judged. Mathematics chose to be judged on finish, and finish is the one quality a reader cannot see through: the more complete the building, the less anyone can tell how it was raised, or whether its builder understood the ground or only got the thing to stand. That is the corner the discipline has written itself into, and its great ones are great in part because they declined to enter it. Thurston's ways of thinking, which he spent his later years giving away, were a style. Grothendieck's rising sea was a style, chosen against the hammer. Benson Farb used to say that every solving of a problem is an encounter, and the sentence came with an instruction: make the encounter plain, so that what was met is obvious and the fuss around it is gone. As I remember it from Chicago. The sentence is his; the wording is my memory's. Plainness is a stylistic choice too, the costliest, because nothing can hide behind it. It has just become the rare one. A machine will produce finish at any length for nothing, so sophistication, the look of the completed building, is now the cheapest thing on a page, and the thing that cannot be faked, the plain sentence that shows what was met, is the thing the manner spent two centuries training mathematicians not to write. That is the irony of the moment for the discipline of the fox. It perfected the one virtue the machine was going to have for free.
The cost was measured from the inside in 1994 by the mathematician best placed to measure it. Thurston refuses the question "how do mathematicians prove theorems?" because it presupposes that progress consists of proving them, and asks instead how mathematicians advance human understanding of mathematics. "The measure of our success is whether what we do enables people to understand and think more clearly and effectively about mathematics." His first exhibit is the first machine proof: when Appel and Haken proved the four-color theorem by a massive automatic computation, the controversy had little to do with doubt about the theorem; it "reflected a continuing desire for human understanding of a proof, in addition to knowledge that the theorem is true." William P. Thurston, On Proof and Progress in Mathematics, Bulletin of the AMS 30:2 (1994), 161–177, §§1–2. The essay answers Jaffe and Quinn's proposal to set speculative work apart as "theoretical mathematics"; its account of knowledge as social is §4, the confession §6. Then the confession. As a graduate student he proved dramatic theorems about foliations, "fairly rapidly," and published the most important of them. Within a couple of years "a dramatic evacuation of the field started to take place," mathematicians advising each other not to go into foliations because "Thurston was cleaning it out," graduate students leaving, and people telling him, as a compliment, that he was killing the field. He does not think the territory was exhausted. The results "were documented in a conventional, formidable mathematician's style. They depended heavily on readers who shared certain background and certain insights"; he "threw out prize cryptic tidbits of insight, such as 'the Godbillon-Vey invariant measures the helical wobble of a foliation', that remained mysterious to most mathematicians who read them. This created a high entry barrier." And the second effect: "More than the knowledge, people want personal understanding. And in our credit-driven system, they also want and need theorem-credits."
That is the AHM's description of October, written thirty-two years early from the other side of the desk: correct results, delivered in the trade's own manner, which nobody could use, and a field that emptied. The difference is that Thurston could be asked, and in time he let himself be. "In reaction to my experience with foliations and in response to social pressures, I concentrated most of my attention on developing and presenting the infrastructure in what I wrote and in what I talked to people about"; he explained the details to the few people who were up for it, and found that what mathematicians most wanted from him "was to learn my ways of thinking, and not in fact to learn my proof of the geometrization conjecture for Haken manifolds." The released proofs have no one to ask. That is the whole of the difference, and it is not small. But it is a difference in who can be asked, not in the manner of the proof, and the manner is the discipline's own. "Mathematical knowledge and understanding were embedded in the minds and in the social fabric of the community of people thinking about a particular topic. This knowledge was supported by written documents, but the written documents were not really primary." Thurston, §4, on learning foliation theory as a student: "There were published theorems that were generally known to be false, or where the proofs were generally known to be incomplete." The fabric is what the Fields declaration calls the transmission chain. The documents were never primary. The fox's manner was tolerable because the fox could be found at a blackboard. A document from a fox who cannot be found is the manner with its one excuse removed.
Go had its October ten years ago, and it had no proofs to check. On 10 March 2016, in the second game of its match against Lee Sedol in Seoul, AlphaGo, playing Black, made the thirty-seventh move of the game on the fifth line of the right side, diagonally against a white stone on the fourth: a shoulder hit, a shape every player knows, on a line where nobody plays it, because the textbook says the fourth line is territory and the fifth-line shoulder hit hands it over for an outside that is not yet anything.
Fan Hui, the European champion AlphaGo had beaten in private the previous October, was in the building. "It's not a human move. I've never seen a human play this move," he told the reporter next to him. "So beautiful." Cade Metz, "The Sadness and Beauty of Watching Google's AI Play Go," Wired, Mar 11, 2016. In the commentary he wrote afterward for DeepMind, Fan Hui records what he put down at the moment: "Here?! This goes beyond my understanding. Globally, there's nothing wrong with it, it's going in the right direction…and AlphaGo always pays special attention to coordinating the stones." The commentators on the English broadcast saw no beauty yet. "That's a very surprising move," said Michael Redmond, the only Westerner to have reached the top professional rank. "I thought it was a mistake," said Chris Garlock beside him; and Redmond, to the two million people watching, "I don't really know if it's a good move or a bad move." Metz, Wired, Mar 11 and May 2016. Many of those watching the screen, Fan Hui writes, thought the stone had been put down in the wrong place. Lee had gone out to smoke before the move was played. He came back, caught sight of it, stared at the board, smiled, sat down, and thought for twelve minutes before he answered. Fan Hui's commentary in DeepMind's annotated record of the game, with analysis by Gu Li and Zhou Ruiyang; Wired has Lee leaving the room after seeing the move and taking nearly fifteen minutes. The accounts agree on the smile and on the clock: at move 42 Lee had an hour and sixteen minutes left, AlphaGo an hour and thirty-four.
DeepMind supplied a number. The policy network, the part of AlphaGo trained on human games to guess a human's next move, had put the chance that a professional would play there at one in ten thousand; AlphaGo had played it anyway, on what it had learned from playing itself, and, in Silver's words to the same reporter, "the move was genius." Metz, "In Two Moves, AlphaGo and Lee Sedol Redefined the Future," Wired, Mar 16, 2016, reporting David Silver. DeepMind's own page calls it "a move that had a 1 in 10,000 chance of being used," and gives Lee's move 78 of the fourth game, the one that beat the machine, the same odds. Silver put the textbook in a sentence years later: a shoulder hit, "you can only play it on the third line or the fourth line," and AlphaGo played it on the fifth (Lex Fridman Podcast 86, 2020). The number is a statement about humans, not about the move. What the move was worth was settled by the game, which White resigned after move 211, and since then by engines. The figure replays the match and puts the move among every other move of it, measured by an engine anyone can run and by a model of the human player the engine surpassed.
The five games of AlphaGo against Lee Sedol, Seoul, 9 to 15 March 2016, 1033 moves, each rated twice: the probability a model of human play (a 2016 professional) gives the move actually played, and what the move cost its player by an engine (the lead before it minus the lead after it, in points, 400 visits a position). Over the 327 moves played while both sides were above 10 percent: AlphaGo's 159 moves have a median probability of 41%, 4% are rarer than 1 in 100, they cost 0.17 points each on average, 5 cost 2 points or more, and 60% were the engine's first choice; Lee's 168 moves: median 40%, 4% rarer than 1 in 100, 0.35 points each, 9 over 2 points, 48% the engine's first choice. Rarer than 1 in 100 and costing under half a point: AlphaGo 4, Lee 3. Game 2, move 37, Black P10 (the fifth line from the right edge, against White's 36 on the fourth): the model gives it 0.81%; cost 0.2 points. The deep search of that position (8,000 visits, 2,500 after each candidate) makes P10 the engine's second choice, +0.2 for Black after it against +0.3 after E12; the model's first choice, S14, gets 19%. AlphaGo's own network put the move at 1 in 10,000. The rarest of AlphaGo's good moves in open play is game 4 move 23, E4, one chance in 7600 by the model and the engine's first choice. Game 4, move 78, White L11, Lee's wedge: the model gives it 0.46%; the deep search makes it the engine's first choice, Black 98% before it and 96% after; AlphaGo's moves 89, 93 and 97 then cost it 6.2, 4.5 and 5.2 points. Black's winrate in game 2 by the engine: 52% after move 36, 50% after move 37; the largest swing of the game came at move 50. The live figure needs JavaScript: a game and a move under a scrubber with play and single steps, overlays (the move's line from the nearest edge, where the human model would play, where the engine searched, the expected owner of every point), the scatter of all 1033 moves by surprise and cost with the famous moves labelled, the deep search of the two famous positions, and Black's winrate after every move of the chosen game.
Right, the match: every move of the five games by how often that human plays it (across, log scale) and what it cost its player by the engine (up; the lead before the move minus the lead after it, 400 visits a position, costs over 8 points drawn at 8). AlphaGo's moves filled, Lee's hollow, faint once a side was under ten percent; the famous moves and the notable ones labelled; click a dot to put that position on the board. The readout's statistics are over the moves played while both sides were above ten percent. "This position" shows instead the deep search of a famous position (8,000 visits, 2,500 after each candidate): every candidate by human probability and Black's lead after it. Under the scatter: Black's winrate after every move of the chosen game, 400 visits each, the move on the board as a hairline. Engine: KataGo 1.18 with the b18c384 network of 2024 and its human-play model; the records from Andries Brouwer's archive, checked against DeepMind's annotated copies (games 3 and 5 are rotated there; game 1 differs only in the order of two pairs of moves) and, for game 2, against the British Go Association's.
Ten years on, the explanation of move 37 is still numbers rather than a sentence. Fan Hui offers a metaphor, the net; the engines offer a lead in points; the sentence a teacher would say at the board, the reason, does not exist, and the players stopped waiting for it. That is the condition the mathematicians are refusing in advance: a result whose correctness can be checked and whose reason cannot be told.
Lee Sedol retired in November 2019, after twenty-four years, and gave Yonhap the reason: "Even if I become the number one, there is an entity that cannot be defeated." Yonhap, Nov 27, 2019. He had submitted his letter of retirement to the Korea Baduk Association on Nov 19, and did not deny that a dispute with the association over fees had played its part. Ke Jie, then the first-ranked player in the world, lost three games to the next AlphaGo at Wuzhen in May 2017 and wept at the board during the third. The machine was too perfect, he said afterward; he was in pain; he saw no hope of winning. 「它太完美我很痛苦，看不到任何胜利的希望。」 Press conference of May 27, 2017, as reported by Sina Sports; he had announced beforehand that these would be his last three games against a machine, "a correct decision."
Disraeli's maxim comes down through Morley, who records that he condensed Lyndhurst's advice of 1835 into four words: never complain and never explain. John Morley, The Life of William Ewart Gladstone (1903), vol. I, book II; the "and" has since been lost in circulation. Lyndhurst's advice concerned a man attacked in public, who should answer only by attacking back. Mathematics was Disraelian before Disraeli. Pages were always cheap; a page cost its author little, so the trade priced the scarce thing, the result, and let credibility be earned by finish and by rarity, pauca sed matura, seldom by explanation. The referee reads the building. Nobody is asked, before a paper is accepted, where the idea came from, which attempts failed, or what the author would say at the board to a student who did not see it. Math 2.0 proposes to pay for exactly that: motivated explanations of opaque proofs as open problems in their own right, a journal that weighs a paper by how much it deepens understanding, priority for the first to explain a result rather than the first to generate it. Slide 25 of the lecture; Shaoming Guo, Starting a new journal (Oct 17, 2026); Tao, Mathstodon, Aug 2026. His ICM essay, Mathematics in the age of AI (Aug 2026), already called the labor of digesting a result, less prestigious than proving it, "the mechanism by which the individual achievements of mathematicians are converted into collective progress and understanding." Mathematics has its own wording of the maxim, from the man who wrote its apology: "there is no scorn more profound, or on the whole more justifiable, than that of the men who make for the men who explain. Exposition, criticism, appreciation, is work for second-rate minds." G. H. Hardy, A Mathematician's Apology (Cambridge: Cambridge University Press, 1940), §1, on the first page. Math 2.0 is the repeal of that sentence. The discipline of the fox proposes to pay for tracks.
Someone has to make them, and the question the proposal leaves open is who. The machine will not; its manner is the fox's, kept better than Gauss kept it, since it cannot be found at any blackboard. The people who can make tracks are the ones who could always read the finished building, the few who were up for it in Thurston's phrase, and the proposal asks them to spend their years on the exposition of results they did not find. The earlier essay worried about engineers who write with an agent's help and read with an agent's help, "humans still speak, but as the meaty mouthpieces of a binary god." The worry here is the reverse: whether mathematicians are to become the wordier disciples of a god that does not explain, the clergy that supplies an oracle with the sentences it will not supply itself.
Go answers the question, and the answer is not discipleship. The people who explained AlphaGo's moves to the world were the professionals it beat and the commentators who had called its move a mistake. A disciple repeats. A reader can say the move is wrong, did say so, was wrong himself, and said that too; and the figure above is what a reader looks like now, an engine anyone can run and a model of the human player the engine surpassed, set side by side so that the surprise and the merit can be measured instead of recited. That is exegesis, and it is also what Thurston said mathematics was: "Mathematicians are those humans who advance human understanding of mathematics."
The Mathematics Genealogy Project holds 334,029 names, each with an advisor, in lineages that run back to Abu Sahl al-Masihi, who died around 1010, and to his student Ibn Sina, from each of whom more than 245,000 of the names descend. The project was begun by Harry Coonce in 1997 and has been kept at North Dakota State University since 2002; the count is its own, as of Sep 30, 2025. I know of no other science that keeps such a register of descent, or keeps it with such pride. The register is the fabric Thurston described, drawn as a tree: knowledge in minds, passed at a blackboard from advisor to student, the documents secondary. It is the right arrangement for a manner that removes the scaffolding, because the scaffolding is kept in the lineage instead, and it is the arrangement that makes the manner a club's. The register also shows the trade's shape. A few founders start fields from problems outside mathematics; Gauss computed where Ceres would reappear and surveyed Hanover before he wrote the theory of surfaces. Most of the names work inward, on questions only the lineage can read, and the lineage is the only reader they need. That is the sociology as it looks from outside, and it is unkind, so its kinder half belongs beside it: the inward work is where the lighthouses stand, and the lineage is how anyone learns to see them.
The club has been described from inside by its greatest members, and not kindly. Grothendieck, refusing the Crafoord Prize in 1988, told the Swedish Academy that in the two decades since he left the milieu "the ethics of the scientific profession (at least among mathematicians) has degraded to such a degree that plain plunder among colleagues, and above all at the expense of those not in a position to defend themselves, has become almost a general rule, and is in any case tolerated by all, including in the most flagrant and iniquitous cases." Letter of April 19, 1988 to Tord Ganelius, permanent secretary of the Royal Swedish Academy of Sciences, published in Le Monde on May 4, 1988, from the copy Grothendieck circulated with his letter to those who had congratulated him: l'éthique du métier scientifique (tout au moins parmi les mathématiciens) s'est dégradée à un degré tel, que le pillage pur et simple entre confrères (et surtout aux dépens de ceux qui ne sont pas en position de pouvoir pour se défendre) est devenu quasiment une règle générale, et qu'il est en tous cas toléré par tous, y compris dans les cas les plus flagrants et les plus iniques. My translation. His account of his own method is the opposite of the fox's, not the hammer and chisel on the nut but the sea that "advances insensibly in silence, nothing seems to happen, nothing moves, the water is so far off you hardly hear it … yet it finally surrounds the resistant substance" (Récoltes et Semailles, pp. 552–3, in Colin McLarty's translation). Voevodsky's confession of 2014 bears on reading. A paper of his from 1993 carried a mistake through years of use: "multiple groups of mathematicians studied" it "at seminars and used it in their work and none of them noticed the mistake," the corrected argument appearing in 2006, and he drew the rule: "A technical argument by a trusted author, which is hard to check and looks similar to arguments known to be correct, is hardly ever checked in detail." Vladimir Voevodsky, "The Origins and Motivations of Univalent Foundations," The Institute Letter, Institute for Advanced Study, Summer 2014. The paper was the "Cohomological Theory" paper on presheaves with transfers. "Who would ensure that I did not forget something and did not make a mistake," he asks there, "if even the mistakes in much more simple arguments take years to uncover?" He spent the rest of his life on computer-checked foundations. Reuben Hersh gave the arrangement its sociology: mathematics has a front, where the finished proof is shown, and a back, where it is made, in the sense Goffman gave the words for restaurants and theaters, and the certainties claimed for mathematics are claimed for the front. Reuben Hersh, "Mathematics has a front and a back," Synthese 88 (1991), 127–133, after Erving Goffman's The Presentation of Self in Everyday Life (1956). Pólya had said it without the sociology: "Finished mathematics presented in a finished form appears as purely demonstrative, consisting of proofs only. Yet mathematics in the making resembles any other human knowledge in the making" (Mathematics and Plausible Reasoning, 1954, preface).
Physics kept its back room in view, and the reason is structural rather than a matter of temperament. A field whose referee is nature cannot wipe out its tracks, because the track is the experiment, and it cannot be taken over by a producer of text, because the text is not the verdict. Mathematics referees itself. That is what let the manner hold for two centuries, and it is what lets a machine that produces checkable text take the field's product first; Tao's account of why the labs came for proofs begins there, with proofs being verifiable and digitizable. Arnold, who refused the separation all his life, put the physicist's view in a line, "Mathematics is the part of physics where experiments are cheap," and called the mid-century attempt to divide the two "catastrophic": "Whole generations of mathematicians grew up without knowing half of their science and, of course, in total ignorance of any other sciences." V. I. Arnold, "On teaching mathematics," an address at the Palais de la Découverte, Paris, March 7, 1997, Russian Mathematical Surveys 53:1 (1998), 229–236. Dyson had said it from the other side in his Gibbs lecture of 1972: "the marriage between mathematics and physics, which was so enormously fruitful in past centuries, has recently ended in divorce" ("Missed Opportunities," Bulletin of the AMS 78 (1972), 635–652). The exception confirms it from inside physics. String theory has gone decades without a verdict from nature, and in the meantime acquired the troubles of mathematics: a literature refereed by its own authors, a lineage, a prestige that ran ahead of any test, and critics from within who said so in the same year, Smolin in The Trouble with Physics and Woit in Not Even Wrong. Lee Smolin, The Trouble with Physics: The Rise of String Theory, the Fall of a Science, and What Comes Next (Boston: Houghton Mifflin, 2006); Peter Woit, Not Even Wrong: The Failure of String Theory and the Search for Unity in Physical Law (New York: Basic Books, 2006). The title is Pauli's phrase for a theory that cannot be tested. A model that writes will not find new physics the way it found 372 theorems, because that way ends at a verifier, and physics keeps its verifier outside the text; where it has stopped consulting it, the text is the verdict again, and the machine can have that part too.
The lineage also selects, and what it selects for has been measured. Scott Stern compared the offers made to the same postdoctoral biologists and found that the ones who were allowed to publish took about a quarter less pay for it; scientists, he concluded, pay to be scientists. Scott Stern, "Do Scientists Pay to Be Scientists?", NBER Working Paper 7410 (1999), published in Management Science 50:6 (2004), 835–853: a 25 percent wage discount at firms that let their researchers publish, ability held fixed by comparing the several offers each candidate received. A ritual with a long apprenticeship and a prize at the end selects for the lowest reservation wage for the ritual, which is not the same thing as the best mind; every graduate student has known the brilliant colleague who left and the patient one who stayed. That is not an argument against the lineage. It is the reason the lineage should want the machine, because the part of the apprenticeship the machine has priced is the part that was endurance.
The cognitive mirror on this site made the point about language in 2025: what the machine reveals is "that the machinery behind much of what we call 'intelligence' may be far more mechanistic, compressible, and brute-force than our human vanity allows," and the discomfort "doesn't stem from what machines are missing, but from what they reveal." A model that produces 372 results in a month, correct as far as anyone can tell and written like a nobody's e-mail, holds the same mirror to the inward work. Much of it, the mirror says, could be regenerated from the literature by a process with no understanding at all. The humiliation is understandable. It does not exonerate. It does not excuse a refusal to read, or a boycott of the manner the discipline invented, or the word "power" for seven hundred files one did not ask for, from a guild that has never asked anyone's leave to publish.
The oldest form of the mirror is a letter. Gödel wrote to von Neumann in March 1956 with a question about machines: if one were built to decide whether a formula has a proof of a given length, how fast would the number of steps it needed grow with the length? He said what would follow if the growth were only linear or quadratic, and the consequences, he thought, would be of the greatest importance: Es würde nämlich offenbar bedeuten, dass man trotz der Unlösbarkeit des Entscheidungsproblems die Denkarbeit des Mathematikers bei ja-oder-nein Fragen vollständig durch Maschinen ersetzen könnte. In spite of the undecidability of the Entscheidungsproblem, the mental work of the mathematician on yes-or-no questions could be replaced entirely by machines; and it seemed to him entirely within the realm of possibility that the steps grew that slowly. Gödel to von Neumann, Princeton, March 20, 1956, the letter in which the question later called P versus NP was first asked; it came to light in 1988. The German text and a translation are printed with Michael Sipser, "The History and Status of the P versus NP Question," Proceedings of the 24th ACM Symposium on Theory of Computing (1992). My translation. Folgerungen von der grössten Tragweite, consequences of the greatest importance, is his phrase for such a machine: the mathematician's Denkarbeit was the stake from the subject's first sentence.
The question is now called P versus NP, whether everything quick to check is also quick to find, and the proof is its standing example: checked in time proportional to its length, found, as far as anyone knows, only by a search that outgrows every polynomial in that length. The consensus is that the two are different, and the best argument for the consensus was never a theorem. Aaronson calls it the philosophical argument: "If P = NP, then the world would be a profoundly different place than we usually assume it to be. There would be no special value in 'creative leaps,' no fundamental gap between solving a problem and recognizing the solution once it's found. Everyone who could appreciate a symphony would be Mozart; everyone who could follow a step-by-step argument would be Gauss." Scott Aaronson, "Reasons to believe," Shtetl-Optimized, Sep 4, 2006, the ninth of ten. Impagliazzo had drawn the world in 1995 and named it Algorithmica, "the world in which P = NP or some moral equivalent," and peopled it with the schoolmaster Grouse, who set the young Gauss the sum from 1 to 100 and, in Impagliazzo's telling, spent the rest of his life trying to humiliate him with problems he could not solve: there "Gauss could use the method of verifying the solution to automatically solve the problem," and Grouse would have had even less success than in real life (Russell Impagliazzo, "A Personal View of Average-Case Complexity," Proceedings of the 10th IEEE Structure in Complexity Theory Conference, 1995). The earlier essay on this site on tax-loss harvesting is a worked example from the hard side of the line. That is an argument from expertise: finding must be harder than checking, or expertise would not exist, and it plainly does. I have never met the gap at work. The tasks that were hard to do were hard to check, the ones that were easy to check were never the hard ones, and a result that cannot be verified without redoing the work is the ordinary case of my trade, not the exception.
The minority among the theorists has answered the argument the same way, by asking what expertise is evidence of. Knuth, becoming convinced of the equality at the Turing centennial in 2012, gave his reason two years later: "It's hard to believe that P ≠ NP and that so many brilliant people have failed to discover why." Among the humongous number of algorithms that run in n^M steps for some finite and incredibly large M, he found it "really hard to believe that all of those algorithms fail," and expected only that the proof, when it came, would be nonconstructive, the algorithm existing and no one ever knowing it. Vardi put his odds on P = NP as a protest vote, because the whole case for the other side is the lack of progress in exhaustive search, and "the space of algorithms is very large and we are only at the beginning of its exploration." Viola decided in 2018 that he believed it, because the history of the field is a history of upper bounds nobody expected: "we have grossly underestimated the reach of efficient computation." Pitassi, whose work is proving that things cannot be done cheaply, said in 2021: "More and more, I'm starting to wonder whether P equals NP." Donald Knuth, "Twenty Questions for Donald Knuth," InformIT, May 20, 2014, question 17, which recalls his remark at the ACM Turing Centennial; in Gasarch's first poll (2002) he had already predicted that "someone will prove P=NP because there are only finitely many obstructions to the opposite hypothesis" and that we would never know the algorithm's complexity. Moshe Vardi in William Gasarch, "The Second P =? NP Poll," SIGACT News Complexity Theory Column 74 (2012). Emanuele Viola, "I believe P=NP," Thoughts, Feb 16, 2018. Toniann Pitassi in Siobhan Roberts, "The 50-year-old problem that eludes theoretical computer science," MIT Technology Review, Oct 27, 2021. Lipton and Bollobás are on the same side, Bollobás calling himself "on the loony fringe." The minority in Gasarch's three polls was 9 percent in 2002, 9 in 2012 and 12 in 2019; one respondent to the third poll wrote that he would have voted P = NP had Vardi, Knuth, Lipton and Viola been the ones at his door. All of them are saying one thing. The gap between finding and checking has only ever been measured by what people could do, and people are a poor instrument for the measurement.
October is evidence on exactly that point. A process that recognizes and recombines, a checker run in a loop over its own guesses, produced 372 results that experts could check and had not found, and the experts' first word for them was slop. It had no faculty for creative leaps; it had recognition, applied faster and more often than any expert applies it, and the leaps appeared. If a leap is what recognition looks like when it is fast enough, then the gap the philosophical argument needed was a fact about the speed of human recognition and not about the world, and the strongest reason anyone had to believe P ≠ NP goes with it. I have come, against the consensus, to believe that P = NP, or that whatever separates the two classes will not matter to anyone who is not a cryptographer. The argument from expertise was the only one that ever persuaded me, and the machine has been removing its evidence for a decade, in Go first, where move 37 looked like a creative leap and was a search with a good enough judge of positions. The rank of the prover over the reader was a bet on the other side of the question.
Heidegger has the words for what the machine did to the trade's tools. Equipment (Zeug) in use is ready to hand (zuhanden) and inconspicuous (unauffällig); the hammer is not noticed while it hammers, and what is ready to hand has the peculiarity, he says, of withdrawing in order to be ready to hand at all. It becomes an object, present-at-hand (vorhanden), when it breaks, when it is missing, or when it is in the way, and these three modes, conspicuousness, obtrusiveness and obstinacy (Auffälligkeit, Aufdringlichkeit, Aufsässigkeit), "all have the function of bringing to the fore" what was there to be used. Sein und Zeit (Tübingen: Niemeyer, 1927), §§15–16, H. 69, 74 and 75: "Das Eigentümliche des zunächst Zuhandenen ist es, in seiner Zuhandenheit sich gleichsam zurückzuziehen, um gerade eigentlich zuhanden zu sein"; "Die Modi der Auffälligkeit, Aufdringlichkeit und Aufsässigkeit haben die Funktion, am Zuhandenen den Charakter der Vorhandenheit zum Vorschein zu bringen"; and, of the moment a piece of equipment fails, "Der Zeugzusammenhang leuchtet auf … Mit diesem Ganzen aber meldet sich die Welt." The English is Macquarrie and Robinson's. Mathematics had grown fond of its equipment as an object. The chalk is the plain case: when the Japanese maker of Hagoromo announced in 2014 that it would stop, mathematicians bought lifetime supplies, the director of the institute at Berkeley among them, and a Korean importer took the machines and makes the chalk today in a shed thirty kilometers from the North Korean border. Hikari Hida and Jean Chung, The New York Times, November 2020; the plant's output peaked at 90 million sticks in 1990. The blackboards themselves have their book, Jessica Wynne's Do Not Erase (2021), and a gallery in the earlier essay. Tao, asked in October whether he felt lucky to have had his career before this, listed the trade's idiosyncratic traditions in the same breath: "we all use blackboards; we do a lot of one-on-one advising of individual grad students." No More Heroes, Numberphile, Oct 15, 2026, quoted at length in the earlier essay. The ritual is hide and seek. The paper hides the chain of thought; the board, the seminar and the advisor are where it is sought, and the seeking is the apprenticeship, and the apprenticeship is the lineage. A machine that hides the chain and offers no board breaks the ritual at its second step. That is what has become conspicuous (auffällig), and it is why the complaint sounds as it does: the trade is looking at its own equipment for the first time in a long while, because a piece of it stopped working.
劳动没有高低贵贱: labor has no high or low, noble or base. The slogan is a schoolroom's, and the argument under it is Marx's, from the manuscript of 1845 in which he and Engels first said what they meant by ideology:
Die Teilung der Arbeit wird erst wirklich Teilung von dem Augenblicke an, wo eine Teilung der materiellen und geistigen Arbeit eintritt. Von diesem Augenblicke an kann sich das Bewußtsein wirklich einbilden, etwas Andres als das Bewußtsein der bestehenden Praxis zu sein, wirklich etwas vorzustellen, ohne etwas Wirkliches vorzustellen - von diesem Augenblicke an ist das Bewußtsein imstande, sich von der Welt zu emanzipieren und zur Bildung der „reinen“ Theorie, Theologie, Philosophie, Moral etc. überzugehen.
The division of labor becomes a real division only from the moment a division of material and mental labor appears. From that moment consciousness can really flatter itself that it is something other than the consciousness of existing practice, that it really represents something without representing anything real; from that moment consciousness is in a position to emancipate itself from the world and proceed to the formation of "pure" theory, theology, philosophy, morality and the rest.
—Marx and Engels, Die deutsche Ideologie, I. Feuerbach (1845–46). Marx-Engels-Werke 3, p. 31; the marginal note beside the sentence reads Die erste Form der Ideologen, Pfaffen, fällt zusammen: the first form of ideologists, priests, coincides with it. My translation. Thirty years later the Critique of the Gotha Programme names the same division as the thing whose disappearance marks the higher phase of a communist society, damit auch der Gegensatz geistiger und körperlicher Arbeit verschwunden ist, before society can write on its banner: from each according to his abilities, to each according to his needs.
The rank between kinds of work is not in the work. It is made by the division, and it is claimed by whoever holds the mental half, which is why the priests come first. Mathematics holds the mental half of the mental half. Inside the trade the division runs again: proving is the mental labor, and reading, checking and teaching are the material, the less prestigious labor of acceptance and digestion in Tao's phrase, done after the prizes have been given for the proving. It is P ≠ NP made into a career, the finder above the checker on the assumption that finding is the harder work. The machine has taken the proving and handed the trade the rest. Marx saw the shape of that too, in the notebooks of 1858: as machinery takes over production, the worker "steps to the side of the production process instead of being its chief actor" and comes to relate to it "as watchman and regulator." Grundrisse, Notebook VII, in Martin Nicolaus's translation (Penguin, 1973), the pages on which the "general intellect" appears; the notebooks were not published until 1939. The mathematician is being offered that position beside the proof process, and the complaint is that it is beneath him. The same notebook names what the machine is. Fixed capital, Marx wrote, shows "to what degree general social knowledge has become a direct force of production," the conditions of life "under the control of the general intellect"; a model trained on the literature is the trade's general intellect enclosed and set to work, and "a demonstration of power" is the right name for it. He also wrote what the enclosure was for, if it could be turned: "the artistic, scientific etc. development of the individuals in the time set free." Grundrisse, Notebook VII, in Martin Nicolaus's translation (Harmondsworth: Penguin, 1973), pp. 705–706; Marx wrote "general intellect" in English.
The AHM's first sentence is a labor sentence: "Mathematicians did not ask for this work to be done." Its verdict, "a demonstration of power," is a verdict on labor relations. And the complaint that the field is being made to do the thankless work of making proofs readable for free is a complaint about rank as much as about pay, because the reader's work was always the lower work. If the slogan holds, the complaint about rank falls and the complaint about pay stands. Reading a machine's proof until it can be taught is mathematics on Thurston's definition, and the question of who pays for it is one the field can put to the companies without first pretending that the work is beneath it. Math 2.0 is, among other things, that question put properly: it asks for the exposition to be counted as work, and so to be paid, credited and refereed as work.
Go got there first, with less theory. Asked last year whether he would play a machine again, Ke Jie said that the audience only wanted to watch him get crushed, and that if they liked it he would play: a hundred games, a hundred losses; feeding his games into the machine, he added, would only pollute its database. He had asked himself the question the mathematicians are asking, whether twenty years at the board had been a waste of a life, and he had cried during the last game of 2017 because he could not accept that the contest with the machine would end that way. Once the crying was done, he said, he was at peace with it, and he still plays when there is a reason to. World Journal, Sep 7, 2025, from his interview on the program 熱浪之外 three days earlier. The model of human play in the figure exists because human games were fed to a machine so that it could tell humans where they would have played; Ke Jie's joke about the database is a player's version of the AHM's complaint, and a happier one. A player who plays for an audience that wants to watch him lose, says so, and plays, is labor without rank. The game kept its players by letting them become the machine's readers.
The record of the decade is in. A study of 5.8 million moves by professionals from 1950 to 2021, each scored by a superhuman engine against 58 billion counterfactual continuations, found that "humans began to make significantly better decisions following the advent of superhuman AI," and that the improvement came with novelty: moves never before seen in the record became more frequent, and better. Minkyu Shin, Jimin Kim, Bas van Opheusden and Thomas L. Griffiths, "Superhuman artificial intelligence can improve human decision-making by increasing novelty," PNAS 120:12 (2023), e2214840120. The programs, the authors suggest, "prompted human players to break away from traditional strategies." The mirror did not end the game. It made the players better at it, and stranger.
The fox's manner was a prince's privilege and became a trade's constitution. It now belongs to a machine, which keeps it better than Gauss did, and keeps it for nothing. Finish is free. The discipline that prized finish above every other quality finds that the rare thing is now the plain sentence, the encounter told without fuss, which is the sentence the manner taught it not to write. Disraeli's maxim was advice for a man who would be judged by results, and mathematics took it, in Gauss's century, for the same reason. Math 2.0 is the discipline deciding, after two centuries, to be judged by something else, by what it can explain; the machine, meanwhile, has learned never to explain, and will not be taught otherwise by a boycott. Never explain was the manner. Never complain is the half that remains to be learned, and the Go players, who had no theory of understanding to defend, learned it in a decade, and played better for it: 哭完，也就释然了。