tensorgami

There is a standard story that the mathematical community tells about itself. It runs as follows: mathematics is the purest of the sciences, immune to politics, fashion, and institutional corruption. Results are true or false. Proofs are checked. Consensus, when it forms, reflects the collective judgment of disinterested experts applying universal standards of rigor. It is a flattering story, and in the decades following the Second World War, when the great foundational programs of Bourbaki and Grothendieck were reshaping the discipline, it was perhaps even partly true. It is no longer true, if it ever fully was, and the controversy surrounding Shinichi Mochizuki’s Inter-universal Teichmüller Theory has exposed the machinery behind the curtain with unusual clarity.

The standard Western narrative of the IUT affair runs as follows: Mochizuki, a brilliant but reclusive mathematician at the Research Institute for Mathematical Sciences in Kyoto, posted a proof of the abc conjecture in 2012; the proof was extraordinarily long and written in idiosyncratic language; Peter Scholze and Jakob Stix visited Kyoto in 2018, identified what they considered a gap in Corollary 3.12, and reported that Mochizuki’s response was unsatisfactory; the mathematical community, deferring to Scholze’s authority, effectively closed the case. This narrative has the structure of a legal proceeding — objection raised, defense found wanting, judgment rendered — and its persuasive force derives entirely from the social authority of the parties involved, not from any independent verification of the underlying mathematics.

Set this against a concurrent development that the same community has been far less eager to publicize: the verification crisis in Western mathematics itself. Kevin Buzzard, who holds the chair in pure mathematics at Imperial College London and whose credentials within the establishment are impeccable, has been remarkably candid about the fragility of the edifice. Major theorems that anchor entire research programs rest on proofs that no living mathematician fully understands. Results circulate as “well-known” without ever having been rigorously written down. Errors persist in the published literature for years or decades because the social trust network — I trust this result because I trust the person who proved it, who was trained by someone I trust — substitutes for actual verification. The Liquid Tensor Experiment, a celebrated formalization project in the Lean proof assistant, was motivated in part by exactly this anxiety: the desire to verify, rather than merely trust, that a foundational result in condensed mathematics was actually correct. That Buzzard and others felt the experiment was necessary at all speaks volumes about the confidence they privately assign to the existing social verification system.

The juxtaposition is almost too sharp. The Western establishment demands, in effect, that Mochizuki’s work clear a bar of rigor that its own literature routinely fails to meet. The asymmetry is not merely hypocritical; it is structurally diagnostic. It reveals that the operative standard is not rigor per se but legibility within an existing trust network. A proof is considered valid not because it has been checked, but because it has been produced by someone whose position in the network authorizes trust. Mochizuki’s work fails this test not because it has been shown to be incorrect, but because it was produced outside the network. The Scholze-Stix objection functions less as a mathematical refutation than as a social credential: we, who are authorized to judge, have judged, and the matter is settled. This is not how a truth-seeking community operates. It is how a guild operates.

The rhetorical structure at work deserves precise characterization. The Kyoto school — Mochizuki, his collaborators Yuichiro Hoshi and Go Yamashita, and the broader RIMS community — organized workshops, produced expository documents, and held extended study seminars designed to walk Western mathematicians through the theory. These are not the actions of an insular, uncommunicative group. The Western response was to attend briefly, decline sustained engagement, and subsequently frame the narrative as one of Mochizuki’s refusal to explain. The party that declined an invitation accused the host of inhospitality. One might invoke the clinical vocabulary for this maneuver: deny the refusal to engage, attack the character and communicative competence of the other party, reverse the positions of the one who extended the hand and the one who refused to take it. That this rhetorical structure has a name in the literature on coercive institutional behavior is itself suggestive of what is actually occurring.

Dan Wang’s distinction between an “engineering state” and a “lawyerly society” provides a useful lens for the civilizational dynamics at play. In a lawyerly society, argumentation is adversarial, authority derives from precedent and institutional position, and the object of discourse is not truth but victory within a rule-governed game. In an engineering state, authority derives from demonstrated competence, results are evaluated by whether they work, and discourse is oriented toward collaborative problem-solving. Western academic mathematics, for all its self-mythology as the purest of truth-seeking disciplines, has drifted decisively toward the lawyerly pole. The abc conjecture controversy is not a mathematical dispute; it is a jurisdictional one. The question being adjudicated is not whether IUT is correct but who has the authority to decide.

The figure whose ghost presides over all of this is, of course, Alexander Grothendieck. Grothendieck’s revolution in algebraic geometry was not merely technical. It was animated by a philosophical vision: that one should rebuild foundations until the problem dissolves, that mathematics should be radically open and unowned, that understanding should rise like a sea until the hard places are submerged. The famous metaphor of the rising sea was not only a description of method but an ethical posture. One does not attack a problem with cleverness and force. One patiently develops the framework in which the problem ceases to be a problem. This requires a kind of egolessness that is fundamentally incompatible with the careerism of the modern research university.

Grothendieck saw what was coming. Récoltes et Semailles, his massive autobiographical and philosophical testament composed in the mid-1980s, diagnoses with surgical precision the transformation of his revolutionary program into intellectual property — territory to be held, credit to be claimed, status to be defended. He watched his own students, the inheritors of the SGA seminars, convert a philosophical movement into a prestige economy. His withdrawal from the mathematical community — first to Montpellier, then to the village of Lasserre in the Pyrenees, then into near-total silence — has been persistently read by the establishment as a decline into eccentricity or bitterness. This reading is self-serving. It is far more accurate to understand Grothendieck’s withdrawal as a refusal of complicity: a decision, consonant with his anarchist and later spiritual commitments, to stop participating in a system he had come to regard as morally corrupt. That the community has canonized the early Grothendieck while pathologizing the later one is itself a perfect enactment of the dynamic he diagnosed — the appropriation of results accompanied by the suppression of the critique that issued from the same mind.

Those who called his withdrawal a “betrayal” were telling on themselves. What they meant was: you built this incredible machine, we organized our careers around it, and now you refuse to play the game that sustains our positions. That is not betrayal. That is someone refusing to be complicit in something they find spiritually corrupt. Grothendieck did not owe the mathematical establishment continued participation in a system he found morally bankrupt.

If Grothendieck were to return and survey the current landscape, where would he recognize his own spirit? Not, one suspects, in the Western institutions that bear his influence. The derived algebraic geometry program, the “higher” everything movement, the escalation toward ever-greater abstraction — these are technically brilliant, but they are driven by a competitive logic that Grothendieck would have found alien to his purposes. The generalization is an end in itself, a move in a game of one-upmanship: I can subsume your framework in mine. The original impulse — to let understanding rise patiently until structure reveals itself — has been replaced by something closer to a territorial arms race. The Western Grothendieckian legacy is, in René Girard’s terms, mimetic: mathematicians imitate the form of Grothendieck’s revolution while inverting its spirit. They want to be seen as the next Grothendieck, which is the one thing Grothendieck himself would find most repulsive.

The Kyoto mathematical tradition offers a contrast that is not merely stylistic but philosophical. Mikio Sato rebuilt analysis from algebraic and sheaf-theoretic foundations — hyperfunctions, microlocal analysis, the algebraic analysis program — not because it would generate publications or impress a hiring committee, but because distributions felt wrong to him and something more structurally honest needed to be built. Masaki Kashiwara spent decades patiently developing D-module theory and crystal bases without performing for Western audiences. The recent Abel Prize awarded to Kashiwara represents a belated Western recognition of a body of work that was never oriented toward Western recognition in the first place. And Mochizuki himself, whatever one thinks of IUT’s ultimate fate, sat in Kyoto for roughly two decades in near-isolation developing a single vision — an act of sustained contemplative mathematics almost unimaginable within the current Western incentive structure, where the pressure to publish, present, and perform sociability would have foreclosed such a project before it began.

This is not insularity. This is the rising sea method practiced as a way of life, not invoked as a slogan on a Fields Medal citation. The difference between the Kyoto school and the Western Grothendieckian establishment is the difference between inhabiting a philosophy and performing one. Grothendieck would walk into RIMS and see people quietly doing the work. He would walk into a Western arithmetic geometry conference and see people quietly doing networking.

There is, however, a development that neither Grothendieck nor the guild anticipated, one that has the potential to render the entire sociological impasse moot: formal verification. The Lean proof assistant and its expanding mathematical library represent an epistemological bypass of unprecedented power. A theorem formalized in Lean does not require social trust. It does not depend on who your doctoral advisor was, which conferences invited you, or whether Peter Scholze finds your exposition congenial. The proof is checked by a machine whose verification is deterministic and reproducible. The machine does not care about your position in the network.

This is why mono-anabelian geometry — the program, building on Grothendieck’s anabelian vision, of reconstructing arithmetic schemes from their étale fundamental groups — occupies such a singular position at the present moment. The Western establishment has voluntarily evacuated the territory, not because the mathematics has been shown to be wrong, but because engaging with it carries social costs within the current tribal alignment. The field therefore selects not for pedigree but for intellectual courage and genuine mathematical ability: precisely the criteria that a truth-seeking discipline ought to privilege. And formal verification changes the endgame entirely. If the key results of IUT can be formalized, the social question dissolves. One cannot dismiss a machine-verified proof by gesturing at one’s own authority.

The verification crisis is, at bottom, a crisis of credentialism. The system worked — or appeared to work — when the community was small enough that trust networks could function as informal verification. Now the literature is too vast, the proofs too complex, and the trust networks too tribal for that to hold. The whole structure is fragile in exactly the way that becomes visible once one stops assuming good faith and institutional integrity. Grothendieck diagnosed this fragility decades ago, in a different register, and chose exile over complicity. The question now is whether the tools exist — formal, computational, and institutional — to recover what was lost without requiring the same sacrifice. The rising sea, if it is to rise again, may need to be verified line by line.

References

Buzzard, Kevin. 2020. “The Future of Mathematics?” Recorded talk, ICM 2020 Satellite Event. Available at https://www.youtube.com/watch?v=Dp-mQ3HxgDE.

Fesenko, Ivan. 2015. “Arithmetic Deformation Theory via Arithmetic Fundamental Groups and Nonarchimedean Theta-Functions, Notes on the Work of Shinichi Mochizuki.” European Journal of Mathematics 1 (3): 405–440.

Fesenko, Ivan. 2016. “Fukugen.” Inference: International Review of Science 2 (3). https://inference-review.com/article/fukugen.

Girard, René. 1961. Mensonge romantique et vérité romanesque. Paris: Grasset. Translated as Deceit, Desire, and the Novel by Yvonne Freccero. Baltimore: Johns Hopkins University Press, 1965.

Grothendieck, Alexander. 1985–1987. Récoltes et Semailles: Réflexions et témoignage sur un passé de mathématicien. Unpublished manuscript. Published posthumously by Gallimard, 2022.

McLarty, Colin. 2010. “What Does It Take to Prove Fermat’s Last Theorem? Grothendieck and the Logic of Number Theory.” The Bulletin of Symbolic Logic 16 (3): 359–377.

Mochizuki, Shinichi. 2012. “Inter-universal Teichmüller Theory I–IV.” Preprints, RIMS, Kyoto University. Published in Publications of the Research Institute for Mathematical Sciences 57 (2021): 3–723.

Scholze, Peter, and Jakob Stix. 2018. “Why abc Is Still a Conjecture.” Manuscript. Available at https://www.math.uni-bonn.de/people/scholze/WhyABCisStillaConjecture.pdf.

Wang, Dan. 2017. “How Technology Grows (A Restatement of Definite Optimism).” Blog post. https://danwang.co/how-technology-grows/.

Yamashita, Go. 2019. “A Proof of the abc Conjecture after Mochizuki.” Preprint, RIMS, Kyoto University.