Mathematics and Refusal

A declared limit

At CERN, Alexander Grothendieck described his position with a severity that has shaped every later account of his life. He had reached, he said, a decision to abandon all scientific research. He distinguished paid mathematical work from research pursued for its own sake and noted that mathematics was his profession. Later in the same talk he said that his interest in disinterested research had vanished, practically from one day to the next, after the group Survivre began.11Grothendieck, “Allons-nous continuer la recherche scientifique ?” 3, 7.

These declarations should not be weakened into a metaphor. He meant that the activity which had organized his adult life no longer possessed an overriding claim on his time. Problems that once exerted fascination no longer summoned him in the same way. He judged it an indefensible use of effort to devote his best forces to pure research while questions of survival were urgent. The change involved desire as well as duty: he insisted that he had not regretted it.22Grothendieck, “Allons-nous continuer la recherche scientifique ?” 2–7.

The declarations should not be converted into a false chronology either. During the years in which he said research had stopped, Grothendieck taught advanced mathematics, lectured on group schemes and Riemann–Roch, maintained contact with students, and carried a last mathematical program on Barsotti–Tate groups and Dieudonné crystals into courses and publications. Volumes from earlier seminars appeared under his name. In 1972–73 he taught future mathematics teachers at Orsay. Together, these records indicate that teaching, advising contact, and exposition continued.

The record supports a narrower distinction than the apparent contradiction: mathematics and research were not synonyms. Grothendieck was drawing a hierarchy inside mathematical labor. New problems pursued for their own sake stood at one level. Teaching, livelihood, responsibility to students, exposition of existing work, and the momentum of collaborative publication stood at others. His refusal displaced the first from the center; it did not instantly cancel every relation built around it.

What continued

The simplest evidence is institutional. At Queen’s University early in 1971 he lectured on group schemes. At the Collège de France he gave a course on Barsotti–Tate groups and proposed another under a Dieudonné title. In December 1971 he lectured in Germany on Riemann–Roch. He told CERN that mathematical invitations still arrived and that he accepted some on condition of a public debate. The lecture circuit depended on mathematics even as he redirected it.33Grothendieck, “The Responsibility of the Scientist Today”; Grothendieck, “Remous au Collège de France”; Grothendieck, “Allons-nous continuer la recherche scientifique ?”; Scharlau, Spirituality.

Students also remained inside the story. Luc Illusie completed a major work on the cotangent complex and deformation theory in 1971, with a second volume in 1972. William Messing’s book on crystals associated to Barsotti–Tate groups appeared in 1972. The chronology and later accounts indicate that both projects were connected to ideas and lectures in Grothendieck’s orbit, but neither was a passive transcription of a master. Their authors had to create proofs, organize machinery, and carry questions into publishable form.

Publication continued through another kind of delay. SGA 6 appeared in 1971, SGA 4 in three volumes across 1972 and 1973, and SGA 7 in 1972 and 1973. These dates fall after the break but record seminars held from 1963 to 1969. They made the earlier program newly material at the very moment its director was denouncing the priorities of research. Publication date is not research date; the books nevertheless required editors, authors, publishers, corrections, and the ongoing authority of a name.

Thus the mathematical life did not stop like a clock. It continued through obligation, collaboration, students, texts already in motion, and the need to earn a salary. Grothendieck’s own interest could vanish while the social world made by that interest continued around him. Refusal entered a field with memory.44Grothendieck, “The Responsibility of the Scientist Today”; Grothendieck, “Remous au Collège de France”; Scharlau, Spirituality, ch. 7.

A last active thread

The period sources use groupes de Barsotti–Tate; modern English more often uses p-divisible groups. Their relation to crystalline Dieudonné theory was the mathematical subject crossing the rupture. Grothendieck presented the topic in Montreal in 1970 and in a paper for the Nice congress published in 1971. His Collège course returned to it. Scharlau, using testimony from participants, places the main interest from roughly 1969 to 1972.55Grothendieck, “Groupes de Barsotti-Tate et cristaux”; Alexandre Grothendieck, Groupes de Barsotti-Tate et cristaux de Dieudonné, Séminaire de mathématiques supérieures 45 (Presses de l’Université de Montréal, 1974); Scharlau, Spirituality, ch. 7.

The terminology therefore carries two dates. The 1970–1971 sources anchor the historical program, while the precise English definition below uses a later standard formulation checked against Illusie’s retrospective account. The chapter does not imply that every modern axiom or notation appears in the period paper in the same form.66Grothendieck, “Groupes de Barsotti-Tate et cristaux”; Luc Illusie, “Déformations de groupes de Barsotti-Tate (d’après A. Grothendieck),” Astérisque 127 (1985): 151–98, https://www.numdam.org/item/AST_1985__127__151_0/.

This was not a minor exercise performed after the great architecture had ended. The subject joined deformation, group schemes, abelian varieties, characteristic p, and crystalline methods. It asked how to understand compatible p-power torsion when the base itself varied infinitesimally. The program carried Grothendieck’s characteristic concern with families and functorial structure into a domain where classical linear algebraic classification had worked over much simpler bases.77Illusie, “Déformations de groupes de Barsotti-Tate (d’après A. Grothendieck)”.

It would be misleading to call the work a final theorem and assign it to one author. Grothendieck sketched programs, announced results, lectured, and communicated methods. Illusie, Messing, Pierre Berthelot, Lawrence Breen, and others developed parts of crystalline and deformation theory over subsequent years. Luc Illusie’s 1985 account explicitly presented deformation of Barsotti–Tate groups after A. Grothendieck, while documenting a mathematical history larger than a single source.88Illusie, “Déformations de groupes de Barsotti-Tate (d’après A. Grothendieck)”.

The thread matters biographically because its content and its mode diverged. The content opened durable arithmetic geometry. The mode—letters, courses, students, unpublished proofs, later reconstructions—showed a program no longer driven toward Grothendieck’s own finished publication. Mathematics survived the refusal by passing through other hands.99Illusie, “Déformations de groupes de Barsotti-Tate (d’après A. Grothendieck)”; Scharlau, Spirituality, ch. 7.

Why characteristic p changes the question

Over the complex numbers, many geometric objects can be studied through smooth shapes, differential forms, and analytic topology. In characteristic p, the equation $p=0$ changes elementary operations. The derivative of $x^p$ is zero. The Frobenius map, sending a function to its p-th power, becomes a fundamental geometric operation. Infinitesimal and inseparable phenomena appear that a picture made from complex points does not detect.

Abelian varieties illustrate the challenge. Over the complex numbers, the torsion points of an abelian variety have a relatively uniform description. In characteristic p, the p-power torsion can have connected and étale parts and can vary in families in ways that record subtle geometry. Looking only at ordinary points loses scheme-theoretic information. A p-divisible group keeps the compatible system of p-power torsion as an object capable of seeing that information.

Classical Dieudonné theory had shown that certain formal groups over a perfect field of characteristic p could be translated into modules equipped with operators related to Frobenius and Verschiebung. The achievement was a form of linearization: nonlinear group data could be studied by algebra. Grothendieck’s question was how far such a translation could survive when the base was not one perfect field but a scheme with infinitesimal variation.

This relative move was continuous with his earlier geometry. A mathematical object should not be understood only over one fixed ground field. It should be studied in families and under base change. Properties that seem absolute may become fibers of a more general relation. In the Barsotti–Tate program, the relative viewpoint met the arithmetic difficulty of characteristic p and the nascent language of crystalline cohomology.1010Grothendieck, “Groupes de Barsotti-Tate et cristaux”; Illusie, “Déformations de groupes de Barsotti-Tate (d’après A. Grothendieck)”.

The mathematical transition is important because it occurred beside a political rejection of abstraction. Nothing in the subject was less abstract than schemes or topoi. Grothendieck did not move in 1969 toward elementary calculation and then leave because depth had ended. He was extending the same relative style into technically demanding territory. The political turn cannot be explained as an intellectual inability to continue.1111Grothendieck, “Groupes de Barsotti-Tate et cristaux”; Scharlau, Spirituality, ch. 7.

Nor should the word crystalline tempt a biography into symbolic readings. The compatibility of data across infinitesimal thickenings has no direct ethical message about rigidity, transparency, or life. Mathematical language often borrows ordinary words and strips them into definitions. Reattaching political meaning because the same person studied crystals and survival would be decorative rather than explanatory.

What does connect the two activities is a habit of questioning the given base. In mathematics Grothendieck asked how an object behaved when the base changed. In politics he asked what institutional base made research possible. This resemblance concerns a form of questioning, not an identity of subject. The standards of proof, agency, and consequence remain different.

The durability of the p-divisible-group thread also marks a limit to his own value judgment. Work pursued at the edge of refusal became useful to questions other mathematicians considered central. He could decide not to spend his life on it. He could not decide in advance what mathematical lives it would have.1212Illusie, “Déformations de groupes de Barsotti-Tate (d’après A. Grothendieck)”.

P-power layers

For a fixed prime number $p$, the $p$-power torsion of an abelian variety or a suitable commutative group scheme in characteristic $p$ occurs in finite subgroup schemes killed by $p$, $p^2$, $p^3$, and so on. A p-divisible group packages these compatible layers as an inductive system

$$G_1 \hookrightarrow G_2 \hookrightarrow G_3 \hookrightarrow \cdots$$

For height $h$, each $G_n$ is a finite, locally free commutative group scheme of rank $p^{nh}$, and $G_n$ identifies with the kernel of multiplication by $p^n$ on $G_{n+1}$, in the standard equivalent formulation of the exactness axioms. The display suppresses the full base-scheme hypotheses but preserves the central idea: the object remembers all p-power torsion coherently, not one finite layer in isolation.

Tower is a useful picture only up to a point. It suggests levels and compatibility. It can wrongly suggest that the p-divisible group is a large set formed by stacking finite groups. Scheme structure matters: nilpotent functions and behavior under base change carry information that sets of points miss. Exact maps among layers matter. The base over which the object lives matters.

An elliptic curve $E/S$ supplies a true example: its compatible subgroup schemes $E[p^n]$ form the p-divisible group $E[p^\infty]$, of height $2$ under the standard hypotheses. A single finite subgroup scheme such as $E[p]$ is not a p-divisible group; it lacks the full compatible system.

Organizing all levels supports deformation questions. If an abelian scheme or p-divisible group is known over a base that is thickened by infinitesimal nilpotent directions, the theory asks whether the object lifts, how lifts differ, and which class obstructs a lift. A coherent p-power object can retain local information sensitive to those questions.

The human intuition is archival rather than political. If one wants to understand how a structure changes, preserving compatible records at every resolution can reveal constraints invisible in one snapshot. The analogy stops there. A mathematical inductive system has exact axioms; a life or movement does not. This chapter does not turn p-divisibility into a metaphor for survival.

Crystals

Classical Dieudonné theory associates linear-algebraic data to certain commutative formal or p-divisible groups over a perfect field of characteristic $p$. Linear data can be easier to calculate and deform than the original nonlinear object. Grothendieck sought an extension over more general bases on which $p$ is nilpotent. The language of crystals was designed to track data compatibly across infinitesimal thickenings.

A mathematical crystal is neither a mineral nor an arbitrary vector bundle with a poetic name. Very roughly, one considers a site whose objects include divided-power infinitesimal thickenings and requires data that pull back compatibly among them. The rigidity suggested by the word crystal concerns this compatibility: the value over one thickening is not chosen independently from the value over another.

The phrase linear shadow helps only if its limit is stated. A shadow usually loses information; in a good classification or deformation theory, the linear object can retain exactly the information needed under specified hypotheses. Those hypotheses are the mathematics. One cannot infer a general equivalence from the intuition alone.

Under the theory’s hypotheses, the Dieudonné crystal associated with $E[p^\infty]$ for an elliptic curve in characteristic $p$ is a concrete example. An arbitrary vector bundle on the base is not: it lacks compatible values and pullback isomorphisms over the required divided-power thickenings.

The historical use was that infinitesimal changes to abelian schemes and their p-power torsion could be approached through modules, filtrations, operators, and compatible crystalline data. This linked a local deformation problem to structures that could be calculated and compared. It also required the cotangent and obstruction machinery being developed around Illusie’s work.

A deformation in slow motion

Suppose a base scheme $S_0$ sits inside a slightly thicker scheme $S$, where the extra directions are nilpotent: they cannot be seen as new ordinary points, but functions remember them to first order or beyond. Suppose a group scheme or abelian scheme is known over $S_0$. A deformation problem asks whether an object over $S$ restricts to the given one, how different lifts are related, and what prevents a lift from existing.

The word slightly is mathematical, not visual. Nilpotent ideals can encode an infinitesimal neighborhood with no counterpart in a drawing. One studies the extension in stages because small thickenings make obstructions tractable. A successful theory locates choices and failures in algebraic groups or cohomology classes rather than only announcing that a lift exists.

For a p-divisible group, compatibility across every p-power level makes lifting delicate. Choosing a deformation of each finite layer independently will not generally produce one coherent system. The layers must remain compatible with inclusion and multiplication by p. Crystalline Dieudonné data offers a linear setting in which this compatibility can be organized.

Illusie’s cotangent complex enters at the level of general deformation machinery. For a morphism, it records differential information robust enough for singular and derived situations where an ordinary cotangent bundle is insufficient. Maps out of the complex can classify first-order changes; higher Ext groups can carry obstructions. Specific theorems determine which groups and which degrees apply.

The layered explanation has a biographical use. Grothendieck’s late course did not repeat a known classification. It joined a new cohomological language to active deformation problems. A complete proof could depend on machinery a student was constructing. The relation between adviser and student was therefore reciprocal in mathematical substance even if unequal in status.1313Scharlau, Spirituality, ch. 7.

It also clarifies why courses and letters mattered. A research program at this stage could consist of a framework and a route through lemmas rather than a polished article. A lecture transmitted choices about definitions and proof strategy that a six-page congress announcement could not contain. When the lecturer’s priority shifted, notes and memories became part of the mathematical archive.1414Grothendieck, “Groupes de Barsotti-Tate et cristaux”; Illusie, “Déformations de groupes de Barsotti-Tate (d’après A. Grothendieck)”; Scharlau, Spirituality, ch. 7.

Later expositions reconstructed arguments from these materials. Reconstruction is not simple recovery. A later author knows subsequent results, repairs gaps, and chooses notation. Describing a 1985 proof as what Grothendieck proved in 1971 can hide that labor. The safest attribution names the originating program and the authors who made its proofs available.

For general readers, the deformation question illuminates the humanity of the work without turning it into metaphor. Mathematicians were trying to know how a structure could persist through an infinitesimal change. They required exact compatibility, not inspiration alone. The political life around them was also changing, but it offered no comparable obstruction group and no unique lift.

Before and after December 1969

The chronology begins before Grothendieck left IHÉS. In December 1969 he sketched principles of the theory in a long letter to Illusie; Scharlau also refers to a January 1970 letter to David Mumford. These letters are not directly reproduced for this account and therefore support only cautious chronology, not quotation. The program was active while the funding dispute unfolded.1515Scharlau, Spirituality, ch. 7.

That overlap guards against a simple causal tale. The military-funding discovery did not arrive after mathematics had already become empty. A serious research direction existed. Leaving IHÉS and turning toward Survivre meant redirecting energy away from work still capable of development, not escaping a field with nothing left to ask.

At CERN, Grothendieck recalled initially imagining one bridge rather than total cessation: he might spend several years learning basic biology and apply or develop mathematical techniques for biological problems. He said that the plan would preserve research while moving it toward life and that, soon after the movement formed, even that interest disappeared.1616Grothendieck, “Allons-nous continuer la recherche scientifique ?” 7.

The abandoned bridge is revealing. It assumed that mathematics could serve an urgent field once redirected. The later new science critique questioned the assumption that an outsider’s sophisticated technique should organize biological knowledge. A mathematics of life could reproduce the hierarchy between expert and object. His refusal deepened from choosing better applications to questioning the social and epistemic form of application.

Montreal and Nice

The Montreal summer school joined the two trajectories in one visit. Grothendieck lectured on crystalline cohomology and Barsotti–Tate groups while requiring equal space for the military address. Mathematicians who responded to the second talk included people engaged with the first. William Messing, working in the same mathematical domain, joined Survivre’s provisional administration. Gordon Edwards’s dissertation would concern group-scheme questions even as activism redirected his career. Mathematics and refusal were socially entangled from the movement’s formation.1717Grothendieck, “Groupes de Barsotti-Tate et cristaux”; Grothendieck, Groupes de Barsotti-Tate et cristaux de Dieudonné; Scharlau, Spirituality, ch. 6.

At Nice the mathematical paper states a program compactly for the international congress. The activist report distributed at the same congress challenged military applications and professional complicity. A reader could encounter the same name in the proceedings and on a tract. Later mythology separates these as the last mathematician and the first ecological rebel. The congress placed them on the same dates.

The paper’s six pages and the bulletin’s many pages had different regimes of authority. The mathematical text addressed readers able to supply definitions and verify arguments. The political text appealed to moral experience, institutional evidence, and common danger. Grothendieck’s public crisis included dissatisfaction with the social elevation of the first regime over the second. It did not make their standards interchangeable.

His intervention during a congress talk about possible military application tested that boundary badly and powerfully. A mathematical speaker might mention an application descriptively; Grothendieck demanded an ethical judgment in the technical session. The interruption said that consequences belonged inside content. It could also prevent the audience from hearing or evaluating the actual mathematics. The cost was not accidental to his method.

The course as an unpublished medium

Grothendieck’s mathematics had always traveled orally. The Séminaire de Géométrie Algébrique was a collective writing system built around talks: problems assigned, exposés delivered, notes circulated, corrections made, and volumes assembled later. A course at Montreal or the Collège therefore did more than popularize results already complete. It could be where definitions stabilized and proof responsibilities were distributed.

The lecture medium was productive precisely because an audience interrupted. A listener could expose a missing hypothesis, ask whether a construction commuted with base change, or volunteer to write the argument. The authority of the lecturer structured the program, but the text that emerged could contain many hands. This collective practice sits uneasily beside Grothendieck’s CERN portrayal of the lecture as a humiliating ritual. Both forms existed: the active research seminar and the star performance before an audience unable to follow.

When he brought public criticism into mathematical visits, he altered both. Time used for a survival debate was time not used for an additional theorem. Critical remarks could help participants question professional ritual and could break the concentration needed for a technical sequence. The decision was not costless because oral mathematics depends on sustained shared attention.

The Collège proposal made that trade explicit. At least eighteen hours would remain for Dieudonné theory; the number of preceding nontechnical sessions would depend on the audience. The mathematical program no longer possessed the calendar automatically. It had to wait while the people in the room considered whether its continuation was justified.1818Grothendieck, “Remous au Collège de France,” 1–6.

No surviving record in the available record gives a complete set of course notes. This absence limits claims about which theorems were fully proved there. Scharlau reports that the course contained complete proofs partly using obstruction theory related to Illusie’s thesis. Illusie’s later article confirms Grothendieck’s foundational role in the deformation theorem. The chapter therefore names the program and interaction without pretending to reconstruct every lecture.

Unpublished teaching is especially vulnerable to biography. Admirers may credit the lecturer with all later developments; critics may count nothing not printed under his name. The proper unit is a chain of transmission: question, talk, letter, student proof, book, later exposition. Different links carry different kinds of authorship.

This chain also resists the dramatic date of abandonment. A person can stop seeking new questions while still explaining a program already alive in a seminar. A student can ask a question that draws new thought from an adviser who says research has ended. There is no administrative test that separates the last original idea from the first act of teaching.

The course carried salary and responsibility without proving that research had regained its former priority. Participants could use a mathematical insight offered in a lecture whatever ethical category the speaker assigned to it.

Illusie and the cotangent complex

Luc Illusie’s Complexe cotangent et déformations appeared in two volumes in 1971 and 1972. In the later mathematical literature, the cotangent complex provides a derived object that organizes differentials and controls deformation questions more faithfully than an ordinary module of differentials in singular or non-smooth settings. Deformations, their automorphisms, and their obstructions can be expressed through associated Ext groups under appropriate hypotheses.

The language controls must not become magic. A cotangent complex does not automatically lift every object or erase geometric work. It provides a framework in which the existence and failure of lifts can be located. Specific group schemes and p-divisible groups still require structure, calculation, and arguments particular to them.

Illusie had participated in the SGA enterprise for years and completed a doctoral work under Grothendieck’s influence. Scharlau reports his admiration for Grothendieck as an adviser. That testimony belongs to Illusie’s experience, not a universal verdict on the relationship. The books themselves establish his authorship. To describe them only as Grothendieck’s ideas would repeat the credit hierarchy that later concerned both men.

The interaction illustrates how mathematical influence can continue after direct research slows. A program can reside in questions given, letters written, seminar talks, and a student’s machinery. The result need not belong wholly to the person who initiated it. Refusal by one mathematician may create space or burden for another; the archive rarely measures the division cleanly.

Messing and the work of proof

William Messing’s The Crystals Associated to Barsotti–Tate Groups: With Applications to Abelian Schemes appeared as Lecture Notes in Mathematics 264 in 1972. Its title records the bridge from Grothendieck’s program to a systematic treatment. Later accounts connect the work to a Montreal lecture, a strategy for a Serre–Tate result, and Illusie’s deformation machinery.1919William Messing, The Crystals Associated to Barsotti–Tate Groups: With Applications to Abelian Schemes, Lecture Notes in Mathematics 264 (Springer, 1972), https://doi.org/10.1007/BFb0058301; Illusie, “Déformations de groupes de Barsotti-Tate (d’après A. Grothendieck)”.

Strategy is not proof. Messing had to develop the argument, make definitions work, and produce a text other mathematicians could use. Scharlau reports that Messing did much of this independently and later remembered Grothendieck as an excellent adviser. The same source notes that Grothendieck omitted Messing from a later list of students. The discrepancy warns against making formal academic status the sole measure of influence or memory the sole measure of contribution.

The book’s applications to abelian schemes show why the theory mattered. Abelian schemes vary in families; their p-divisible groups encode infinitesimal and p-adic information relevant to deformation and moduli. Crystalline realisation gave a way to study that information when ordinary characteristic zero intuitions failed. Later arithmetic geometry would continue to develop the theory far beyond the 1970 lectures.

Biographically, Messing’s work is evidence against two opposite errors. It contradicts the claim that nothing mathematical remained near Grothendieck after IHÉS. It also contradicts the claim that continued influence meant he quietly resumed the old program. The mathematics advanced through another author while his own daily priority had shifted.

Influence without ownership

Mathematical communities need compressed attributions. A theorem, program, or style receives a name so that people can refer to it. Biography expands the compression and discovers unequal forms of labor: posing a problem, finding the concept, proving a special case, writing details, correcting errors, teaching notation, and recognizing consequence. The Barsotti–Tate story contains all of them.

Grothendieck’s gift often lay in seeing which concept and statement organized a field. That gift could make later proofs seem like execution of an idea already complete. In practice a program can be wrong, underdetermined, or dependent on machinery not yet built. The person who makes it work changes what the original idea was. Illusie’s complex and Messing’s crystals were not clerical services.

The social relation made recognition difficult. A famous adviser could mention a direction and later remember having given the essential insight. A student could experience months of independent work and remember the adviser chiefly as catalyst. Both accounts could be sincere. Publication provides one anchor: the named author assumes responsibility for the text. Letters and testimony provide other anchors without dissolving that one.

Grothendieck’s political critique sharpened the ethical stakes. Survivre attacked hierarchy, competition, and the conversion of technical prestige into power. The mathematical program around him remained structured by his authority. Younger mathematicians benefited from extraordinary attention and could also labor inside an architecture publicly associated with his name. The contradiction did not make the collaboration fraudulent; it made credit a question his later writings would revisit intensely.

Refusal complicated ownership further. His rejection of research as a governing vocation did not settle his claim on a program developed by others, or whether others’ use of orally released ideas fulfilled the gift. The 1970–73 record does not show him trying to stop this mathematical development. It shows transmission continuing while his public values changed.

The omission of Messing from a later student list is a small but telling archival problem. It may reflect formal supervision, memory, classification, or grievance; the available record cannot decide. A cautious narrative does not correct the omission by making Messing a passive Grothendieck student. It names the mathematical relation and preserves the author of the book.

Illusie’s later phrase after A. Grothendieck offers a useful attributional form. It acknowledges an originating theorem or program and places the present expositor in the title. The preposition contains history without erasing the writer. Similar care is needed throughout SGA, where a director’s vision and exposé authors’ proofs coexist.

The wider lesson belongs to refusal itself. One can refuse to continue producing and still owe recognition to people who carry shared work. One can receive an idea from a person who has left and still make something one’s own. Ethical departure is not the same as solitary ownership of the past.

The books arriving after the departure

SGA publication makes chronological language treacherous. SGA 6, on intersection theory and Riemann–Roch, appeared in 1971 but came from the 1966–67 seminar. The three SGA 4 volumes on topoi and étale cohomology appeared in 1972, 1972, and 1973 but recorded the 1963–64 seminar. SGA 7 on monodromy appeared in 1972 and 1973 from work in 1967–69.

To a reader scanning a bibliography, the years suggest sustained output by Grothendieck in the period of activism. To a reader of his speeches, research ended. Both impressions omit the lag between oral seminar, written exposé, collective editing, and print. The correct statement is more interesting: the publication machine of the previous decade continued delivering its volumes after the person at its center had questioned the machine’s purpose.

These were not vanity publications. SGA 4 and SGA 7 remained foundational sources. Their arrival extended the mathematical transformation associated with IHÉS into new generations. Students could be formed by books whose director was telling public audiences that much contemporary research might deservedly disappear.

The temporal disjunction makes refusal a conflict with one’s own durable effects. Grothendieck could stop initiating work but could not recall concepts, withdraw students’ labor, or prevent readers from using the volumes. He did not attempt to do so in these years. Later he would make more explicit claims over unpublished writings and publication, but the early 1970s were not an erasure campaign.

The books also embody collective intelligence. Artin, Verdier, Deligne, Berthelot, Illusie, Katz, and many exposé authors cannot be folded into one name. Grothendieck directed questions and architecture; the series existed because others wrote and corrected. His political criticism of hierarchy can be read against a mathematical practice that was collective and still asymmetrical.

Berlin and Bielefeld

On December 6, 1971, Grothendieck gave a Riemann–Roch colloquium at the University of Bielefeld. A handwritten entry titled Hexenküche appears in the upper half of digital PDF page 13 of the Faculty of Mathematics Kolloquiumsbuch 1971; the item is rotated 180 degrees in the full scan. Scharlau reports that the technical lecture lasted roughly two hours and was followed by discussion of ecology and science. The verified object establishes the event, date, title, author, and locator, but no wording from it is quoted here.

The entry is often read as a repudiation of the Grothendieck–Riemann–Roch theorem, one of the works that made him famous. Yet he had just taken the time to give the lecture. The observable sequence joined mathematical exposition to public criticism of the value attached to specialized work.

The mathematical content had a history of its own. Grothendieck’s version recast Riemann–Roch functorially through K-theory and characteristic classes, turning a formula into a relation stable across broad geometric settings. Its generality later enabled applications not reducible to the motives of its authors. Calling it socially remote in 1971 did not determine its future use or intellectual worth.

Scharlau preserves colorful recollections around the German visits. Some are second-hand and theatrical; this chapter does not use them as scene detail. The institutional register, primary scan, and documented itinerary are enough to establish the bounded sequence.

Production as an imperative

At CERN, Grothendieck said that scientific production had become an imperative in itself, like production elsewhere in industrial civilization. Publications were necessary for jobs, promotion, and recognition; in extreme cases output was counted by pages. The critique was not that researchers never loved their subjects. It was that institutions converted love into compulsory measurable work.2020Grothendieck, “Allons-nous continuer la recherche scientifique ?” 5.

At CERN, he included his former self among the privileged and described research as a source of intense fulfillment under exceptional conditions. He contrasted that position with students and colleagues subject to rank and career pressure. This retrospective self-criticism gave the argument more force than an accusation directed solely at administrators.2121Grothendieck, “Allons-nous continuer la recherche scientifique ?” 4–7.

The generalization to most scientists exceeded his evidence. Conversations across campuses provided testimony, not a representative survey. Some researchers accepted career pressure and still found substantial freedom. Others worked in collaborative or applied settings unlike his mathematical world. The claim is best read as a diagnosis emerging from a formerly protected position, not a measured sociology of every laboratory.

The diagnosis nevertheless anticipated a durable problem. When publication functions simultaneously as communication, proof of employment, prestige, and institutional metric, quantity can displace judgment. Competition can narrow questions while appearing to reward novelty. Refusal can then mean declining not knowledge but the demand to produce it on command.

Pure, applied, and useful

Grothendieck had been formed inside a strong distinction between pure and applied mathematics. Pure work claimed autonomy from immediate use; its value lay in structure, depth, and internal truth. That autonomy protected inquiry from state or industrial direction. It could also protect mathematicians from asking how their institutions were financed and how mathematical authority served social selection.

At Nice, applied mathematics was gaining official recognition in France. Control theory, computation, modeling, and links with engineering answered state demands for modernization. Some critics of Bourbaki treated this turn as an opening toward the real world. Grothendieck distrusted the conclusion. A mathematics directed toward industry or military planning might be socially engaged in the least emancipatory sense.

His position therefore crossed the standard camps. He attacked the ivory tower without endorsing usefulness as defined by government and firms. He defended no return to pure research insulated from consequences. He also rejected the idea that every mathematical result should acquire an application. The relevant question was who defined the need and what social relation the work produced.

This made useful unstable. A theorem without present application might become fruitful later. A model with an immediate use might optimize an unjust system. Elementary teaching might have greater human value than celebrated research without producing novelty. A public discussion might be useful even if it yielded no paper. Grothendieck’s refusal reordered these forms rather than supplying one metric.

The military-funding dispute demonstrated the inadequacy of content alone. His algebraic geometry was not a weapons calculation. He still considered the funding relation disqualifying. Use therefore included institutional legitimation alongside downstream application. Critics could respond that every public institution is entangled in objectionable power, making the rule impossible to satisfy. He increasingly accepted the entanglement and shifted from clean separation toward transformation of life.

Mathematics itself complicates prediction of use. Abstract number theory, logic, and geometry have acquired applications generations later; applied models have vanished with the systems that commissioned them. Responsibility cannot rely on omniscience about consequences. It can examine patrons, purposes, secrecy, foreseeable uses, labor relations, and the openness of decision. Grothendieck’s categorical language dramatized this examination without producing a general ethics for uncertain futures.

His new science proposal placed essential material needs first, but even those needs require theoretical work whose use may not be immediate. Soil ecology, epidemiology, energy storage, and medicine depend on long investigations. A society that allowed inquiry only after a direct need was certified could create a new bureaucracy of usefulness. The danger mirrors the publication imperative: an external metric replaces curiosity.

The unresolved achievement of his position was to deny both easy refuges. Purity did not absolve institutional responsibility. Application did not prove social good. Mathematics had to be judged through a plural account of truth, freedom, patronage, consequence, teaching, and need. Refusal named the moment when he judged the existing balance intolerable.

Pleasure under judgment

At CERN, Grothendieck described the pleasure of discovery as real and asked whether it justified social investment or rested on pressure below highly placed researchers. The question resembled his critique of energy demand: an experienced desire was real but not automatically self-justifying.

At CERN an audience member compared research with music and art. The response forced him to distinguish esoteric mathematics from forms of creation widely accessible without long training. He estimated that only a handful of people could understand his recent work. The comparison was polemical. Advanced music also has histories of training; simple mathematical play can be widely shared. Audience size does not decide artistic value.

The deeper issue was social hierarchy. A culture could honor one difficult activity as the summit of intellect and treat people unable to perform it as less knowledgeable. Grothendieck’s argument sought to strip mathematical creation of that ranking power. It did not offer a full account of how a society might support rare, demanding work without turning rarity into domination.

This gap matters because his own work depended on long apprenticeship, libraries, international exchange, and concentrated time. A new science organized only around immediate communal needs might have no place for it. He publicly accepted that possibility. The position was radical, and it asked future people to surrender possibilities whose value no contemporary speaker could determine.2222Grothendieck, “Allons-nous continuer la recherche scientifique ?” 35–40.

Biology and the abandoned redirection

At CERN, Grothendieck recalled a plan to learn biology and bring mathematical methods to it. The plan fits a common account of responsible science: redirect powerful knowledge toward health, agriculture, or ecology. He soon considered that insufficient. The relation between mathematical expert and biological problem could reproduce the separation of intellectual and practical work.2323Grothendieck, “Allons-nous continuer la recherche scientifique ?” 7.

His critique of analytic method went further. Science divided reality into specialities, isolated rational faculties from sensation and ethics, and then assembled experts unable to recover the whole. He speculated that biology’s complexity required a transformed relation to living processes and that mathematics itself would need renewal if it were to model life faithfully.

This speculation was not a mathematical result. It should not be credited with later systems biology, complexity theory, or ecological modeling. Nor did Grothendieck acquire the biological knowledge needed to judge the field. Its historical significance lies in the point where he refused the easy solution of applying his established mathematical methods to a socially approved object.

There was humility and presumption in that refusal. Humility: mathematics could not arrive as universal master. Presumption: from limited acquaintance he pronounced large parts of existing knowledge doomed. His public program joined both claims in a proposed practice where knowledge would not dominate its object.

Science of the people

The Grothendieck–Guedj theses proposed science of the people rather than science for the people. The distinction shifted agency. A benevolent expert could produce for people while retaining control over questions and methods. Science of the people would arise from ordinary work, bodily experience, and collective need.2424Grothendieck, “Allons-nous continuer la recherche scientifique ?” 6.

The program did not reject rational thought. It rejected the monopoly of one experimental-deductive form and its separation from intuition, ethics, sensation, and beauty. It imagined farmers, gardeners, healers, technicians, and builders as researchers in their principal activity. Mental and manual work would rejoin.

Such integration can romanticize experience. Traditional or local knowledge can contain hierarchy and error; formal methods can protect against wish and authority. The history of medicine shows both the violence of professional monopoly and the value of controlled evidence. A democratic science needs ways to test claims without converting testing into caste power. Survivre identified the problem more fully than it designed the method.

Grothendieck’s refusal therefore should not be called a simple turn against reason. He continued to argue, distinguish, and demand coherence. He used scientific studies in the nuclear campaign. He taught mathematics. The refusal targeted the social sovereignty of research and the claim that its current methods alone defined knowledge.

Truth is not the same as value

One distinction remained easier in mathematics than in the political debate. A theorem can be true under stated definitions and hypotheses even if the institution producing it is unjust. Military funding does not change a valid proof into an invalid one. Grothendieck never argued otherwise. His refusal concerned the value assigned to producing the theorem and the relations that supported production.

This separation protects mathematical integrity and can become moral evasion. A researcher may say that only truth belongs to the work while patronage, labor, application, and exclusion belong to someone else. Grothendieck’s institutional argument rejoined them without confusing their standards. The proof remains true; accepting the grant remains an action open to judgment.

Social value also cannot be deduced from truth. A correct result may serve a harmful purpose, a benign purpose, remote curiosity, or several at once. A false ecological claim may mobilize people toward a worthwhile institution and still require correction. Good intention does not change truth value, and truth value does not settle political good.

Survivre’s attack on scientism sometimes blurred this careful distinction by treating scientific method, ideology, and social use as one system. The blur captured their interaction and risked making criticism immune to evidence: if a scientific objection was automatically an expression of scientism, no correction from science could be heard. Samuel’s reservations and the CERN audience resisted that closure.

Mathematical practice offers general critical rules: state assumptions, distinguish implication from analogy, preserve counterexamples, and admit where a conclusion exceeds the data. In documented public answers, Grothendieck sometimes followed these rules under challenge; his catastrophic forecasts often did not. The record does not divide reason and politics into separate periods.

The social value of teaching demonstrates another difference. A classroom explanation may contain no new theorem and have considerable value. A paper may contain a new theorem and deepen a destructive competition. Publication metrics invert these judgments because novelty is countable and another person’s understanding is not. Refusal made room to value the second without denying the first.

Mathematical truth also outlived his judgment about civilization. Later researchers could use p-divisible groups under institutions and purposes he did not foresee. Their work did not refute his right to leave. It demonstrated that no creator holds final authority over the value of a creation.

The phrase mathematics and refusal therefore joins two independent questions: what is mathematically the case, and what one should do with a life capable of discovering it. His demonstrated ability in the first domain did not make him infallible about the second. His decision about the second did not diminish the truth of the first.

What refusal gave and withheld

Refusing new research freed immense time for correspondence, travel, publication, and public discussion. It allowed Grothendieck to question institutions without treating activism as a weekend supplement. It signalled that military and ecological responsibility could outweigh the rewards of the highest scientific career. The act’s credibility came partly from the value of what he left.

The same act withheld future mathematics from collaborators and students. No one can know what he might have developed, and counterfactual mourning easily becomes a claim that society owned his talent. It did not. A gift for mathematics does not impose a lifelong duty to produce. Yet collaborative programs create expectations and dependencies, and an abrupt redirection has consequences for people carrying shared work.

Grothendieck’s later writings revisited credit and abandonment. In 1970–73 the immediate evidence shows younger mathematicians completing books and theories while he turned elsewhere. Later testimony must remain attached to each named speaker; this chapter does not infer a common relationship outcome.

It also produced a public simplification. Newspapers and later profiles could say that a prominent mathematician had abandoned mathematics, a dramatic story easier to tell than the mixed calendar of teaching, old publications, student work, and livelihood. Grothendieck’s sharp public phrasing emphasized the value reversal. Historical precision must restore the mixture without undoing the reversal.

Freedom to leave

A society that celebrates creation can begin to regard a creator as public property. The stronger the gift, the stronger the claim that it must continue. Later profiles converted Grothendieck’s departure into a loss for mathematics, listing hypothetical theorems, absent advising, and diminished institutional prestige. Those texts record reception, not the private states of all colleagues or students.

No person owes indefinite production because earlier work was exceptional. Freedom in research includes freedom to stop, change subject, teach, or refuse the institution. This principle is not sentimental. Without it, the ideal of free inquiry becomes a demand that gifted people serve a community’s desire under the name of vocation.

Freedom to leave does not cancel obligations already undertaken. Students may need supervision; co-authors need decisions; manuscripts and seminars contain shared labor. A departure can be exercised responsibly or carelessly. The documentary record in this period shows both continued teaching and unfinished collective expectation. It is too incomplete to render a single moral verdict.

The distinction between right and consequence helps. Grothendieck had the right to stop research. The consequence included disruption. Naming the consequence is not asserting ownership of his talent; naming the right is not denying other people’s loss. Biography often collapses these because it seeks a hero or a culprit.

His material freedom was also unequal. The Collège and Orsay provided work; international reputation made invitations available; he could reclassify mathematics as livelihood without immediately becoming unemployable. A young researcher who refused publication might lose a career before finding another post. Grothendieck discussed career constraint at CERN, but his prescription sometimes underestimated how privilege made refusal possible.

This does not invalidate exemplary action. Privilege can be spent to protect itself or risked to contest the source of privilege. Grothendieck left a concentrated research environment and redirected working time toward Survivre, public meetings, and teaching. The records used here do not quantify further personal losses. The relevant caution is against turning his act into a universal test of courage for people with different dependence.2525Grothendieck, “Comment je suis devenu militant ?”; Grothendieck, “Remous au Collège de France”; Grothendieck, “Allons-nous continuer la recherche scientifique ?”

Refusal also became a new source of prestige. The famous mathematician who left science was more newsworthy than an unknown scientist who quietly chose another life. Moral renunciation can accumulate symbolic capital even when the person does not seek it. Grothendieck used that visibility for Survivre and could not fully control the legend it produced.

The most defensible meaning of his act is therefore neither sacrifice nor desertion. It was an exercise of freedom made as a political claim: research was not the highest good by default. Its effects on mathematics, students, and his own later life remained open. The claim asks every institution to justify the labor it demands; it does not require every mathematician to give the same answer.

A theory that did not vanish

The Barsotti–Tate and crystalline thread continued long after the activist years. Illusie’s later exposition, Messing’s book, and subsequent crystalline Dieudonné theory made the program part of arithmetic geometry. P-divisible groups became central in deformation of abelian varieties, moduli, p-adic geometry, and the study of reduction in characteristic p.2626Messing, The Crystals Associated to Barsotti–Tate Groups: With Applications to Abelian Schemes; Illusie, “Déformations de groupes de Barsotti-Tate (d’après A. Grothendieck)”.

Later importance does not refute Grothendieck’s 1972 judgment, because his judgment concerned priority and social form rather than a prediction that the theory contained no true results. It does show the limit of any person’s ability to know what a mathematical idea will become. Work that appeared distant from life acquired uses inside a broad mathematical culture; whether those uses justified the institutions remained a separate question.

The afterlife also redistributed authorship. Later mathematicians proved, generalized, corrected, and taught. Grothendieck’s theory became shorthand for a genealogy. This is the normal life of mathematics and a challenge to possessive biography. His refusal did not freeze the idea at the moment he left it.

The result is not an ironic victory of mathematics over its deserter. It is evidence that creation exceeds intention. The same principle supports his political demand that scientists cannot control all uses of their work. It also weakens a total condemnation based on present use. Unpredictability makes responsibility harder, not impossible.

Mathematics after the center

By the 1972–73 academic year, Grothendieck’s principal teaching base was Orsay. The available account places him before research specialists and in preparation for the national agrégation, working with students headed toward ordinary teaching. Mathematics remained his livelihood and a relation to people who did not inhabit the summit he had left.

This setting altered the question. At IHÉS, he had built a program among advanced collaborators with exceptional freedom. At the Collège, he placed survival before a specialist course. At Orsay, he would comment on elementary and classical mathematics presented by students. Refusal moved from public declaration into pedagogy.

The change should not be sentimentalized. The agrégation is a competitive national examination, part of the educational hierarchy he criticized. Preparing students for it was salaried work inside that hierarchy. Yet within that constraint, the reported course emphasized concepts, motivation, questions, and patience rather than tricks. The next chapter begins there.

Mathematics and refusal were not two lives separated by October 1, 1970. They were two judgments occupying the same person and often the same day. He could explain a p-divisible group and believe the civilization supporting the explanation was destructive. He could reject research’s claim on his life and offer a student an illuminating account of a group. He could be wrong in his forecasts and exact in a proof. The years after IHÉS are intelligible only if those statements are allowed to remain together.