+ 4 +

Mathematics as an Addiction:
Following Logic to the End

The question is sometimes asked, To be a great mathematician, does being crazy help? The simple and straightforward answer is, No, of course not. Working in a university math department, or attending the meetings of the American Mathematical Society, one cannot help observing the pervasive normality. Still, there is something different about mathematicians compared to, say, chemists or geologists or even English professors. It is possible to be “crazy”—that is, conspicuously eccentric, very odd, even antisocial—and still hold a job as a math professor. Even, perhaps, as an industrial mathematician in certain organizations. If you are really good at solving hard problems and can communicate with other human beings well enough to convince them of what you have done, then in many universities no one will care too much if you work all night and sleep till noon or are careless about keeping your hair combed and your shoelaces tied. There is a certain casualness or “sloppiness” detectable at math meetings that would not be present at a convention of brain surgeons or chemical engineers. Standards of conformity and conventionality are more liberal and tolerant in the mathematical community than in many other academic or professional communities.

Of course, it can still happen that a mathematician succumbs to real mental illness. The case of John Nash became famous through Sylvia Nasar’s book, A Beautiful Mind, which even became a Hollywood movie. After a brilliant early career, Nash was disabled by paranoid schizophrenia. He ultimately made a good recovery, in time to receive a Nobel prize. It would be very complicated to disentangle the relation, if any, between his mathematical genius and his madness.

As we mentioned in the last chapter, some mathematicians under extreme stress have found that their discipline provided solace and relief. Others, lacking a sense of balance, have put themselves at risk by making mathematics the only focus of their lives. For some mathematicians, it seems, mathematics can be a destructive addiction. In some cases, there was a specifically mathematical style or flavor to their delusions.

Our first example is by far the most important and takes up about half of this chapter. Alexandre Grothendieck was one of the preeminent mathematicians of his time, most famous for completely re-creating and transforming algebraic geometry. His life story is truly fascinating and amazing, raising questions not only about the potentially addictive nature of mathematics but also about the psychological destruction resulting from the cataclysmic wars and persecutions in Europe in the 20th century. In contrast with most mathematicians, Grothendieck wrote at length about his emotions. This apparent need to share his feelings and experience, and to describe the sources and direction of his discoveries, may be rooted in his lonely childhood.

After describing his achievements and his descent into solitary delusions, we present five cases of actual criminal or suicidal insanity in other mathematicians, sometimes with claims that their violent actions were “logical.” Of course, none of the other mathematicians we discuss in this chapter are comparable to Grothendieck as creators of great mathematics. The lesson is that mathematics can indeed be dangerous for vulnerable minds in unfavorable circumstances.

We now turn to the amazing life story of Alexandre Grothendieck, from his heritage of revolutionary rebellion to his tragic final alienation from the mathematical community. Today Alexandre Grothendieck is a pacifist hermit, living at a secret address in a remote village in the Pyrenees Mountains. From 1950 to 1970, Grothendieck reshaped functional analysis and algebraic geometry. In 1970 he started his withdrawal from the famous institutions of the mathematical world, declaring them corrupt and vicious, although he continued to create some mathematics for another 17 years.¹

Understanding Grothendieck’s creations requires serious preparation in algebraic geometry and category theory. We can only offer partial and incomplete renderings. We get help from his famous or infamous 1000-page unpublished work Récoltes et Semailles (Reaping and Sowing), as partially translated by Roy Lisker,² in which he combines personal and mathematical reflections. As Allyn Jackson says, it is “a dense, multi-layered work that reveals a great and sometimes terrifying mind carrying out the difficult work of trying to understand itself and the world. . . . He often succeeds at describing things that at first glance would seem quite ineffable.”³ Jackson’s brilliant biography and tribute has contributed to our own presentation, together with materials from Leila Schneps and the Grothendieck circle.

Early Years

Alexandre Grothendieck’s father, Alexander (Sascha) Shapiro, was born around 1890 to Hasidic Jewish parents in Novozybkov, a little town near the place where Russia, Ukraine, and Belarus now meet. At age 17 he was arrested for joining in the unsuccessful 1905 revolution in Russia, but his youth saved him from a death sentence. He spent 10 years in prison in Siberia after running away and being recaptured a few times. He lost an arm, attempting suicide to avoid capture by the police. Upon release in 1917 he quickly became a leader of the Socialist-Revolutionaries of the Left, a party that was soon outlawed by Lenin. He then took part in several other failed European revolutions. During the 1920s he participated in the armed clashes of the leftist parties opposing Hitler and the Nazis in Germany. He supported himself as a street photographer. In Berlin he met Hanka (Johanna Grothendieck). She had become part of the radical avant garde after escaping from her Lutheran bourgeois family in Hamburg. On March 28, 1928, Hanka had a son, Alexandre.

When Hitler took power in 1933, Sascha fled to Paris, and Hanka soon followed him. She left 5-year-old Alexandre hidden near Hamburg in a libertarian private school run by a Christian idealist, Wilhelm Heydorn. Dagmar, Heydorn’s wife, later recalled young Alexandre as very free, completely honest, and lacking in inhibitions. As he recalled in Récoltes et Semailles, “When I was a child I loved going to school. I don’t recall ever being bored at school. There was the magic of numbers and the magic of words, signs and sounds. And the magic of rhyme. In rhyming there appeared to be a mystery that went beyond the words. . . . For a while everything I said was in rhyme. . . . I wasn’t what would be considered ‘brilliant.’ I became thoroughly absorbed in whatever interested me, to the detriment of all else, without concerning myself with winning the appreciation of the teacher.”

The sudden separation from his parents was very traumatic for him. They were apart for 6 years. In 1939 war was imminent and political pressure was increasing. The Heydorns could no longer keep their foster children. Grothendieck was an especially difficult case—he looked Jewish. Through the French consulate in Hamburg, Dagmar managed to reach Shapiro in Paris and Hanka in Nimes. In May 1939, 11-year-old Alexandre was put on a train traveling from Hamburg to Paris.

Sascha and Hanka had gone to Spain in 1936. There Sascha fought with the anarchist militia in the civil war against Franco’s Fascists. After their defeat, Sascha and other loyalist fighters fled to France. In 1939 Alexandre spent a brief time with both his parents before his father was taken away and imprisoned in Le Vernet, the worst of the French detention camps. In October 1940, after France capitulated to Hitler, the Vichy government shipped Shapiro and others off to be killed at Auschwitz. Alexandre and his mother were left to survive as best they could.

In his recollections Grothendieck wrote:

For my first year of schooling in France, 1940, I was interned with my mother in a concentration camp, at Rieucros, near Mende. [While in this camp, Hanka contracted tuberculosis, from which she continued to suffer until she perished of it in 1957.] It was wartime and we were foreigners—“undesirables” as they put it. But the camp administration looked the other way when it came to the children in the camp, undesirable or not. . . . I was the oldest and the only one enrolled in school. It was 4 or 5 kilometers away, and I went in rain, wind and snow, in shoes if I was lucky to find them, that filled up with water. . . . During the final years of the war, during which my mother remained interned, I was placed in an orphanage run by the “Secours Suisse,” at Chambon sur Lignon. Most of us were Jews, and when we were warned (by the local police) that the Gestapo was doing a round-up, we all went into the woods to hide for one or two nights. . . . This region of the Cevennes abounded with Jews in hiding. That so many survived is due to the solidarity of the local population.⁴

At the College Cevenol, Alexandre was remembered as being very intelligent and always immersed in thought, reading, and writing. He was a fierce chess player, loud, nervous, and brusque. In his recollections he recalled that he devoured his textbooks as soon as he got them, hoping that in the coming year he would really learn something interesting. He writes, “. . . I can still recall the first ‘mathematics essay.’ The teacher gave it a bad mark. It was to be a proof of ‘three cases in which triangles were congruent.’ My proof wasn’t the official one in the textbook he followed religiously. All the same, I already knew that my proof was neither more nor less convincing than the one in the book, and was in accord with the traditional spirit of ‘sliding this figure over that one.’ It was evident that this man was unable or unwilling to think for himself in judging the worth of a train of reasoning.”⁵

Once the war ended in May 1945, he and his mother lived outside Montpellier. There he attended the university, which offered him little but time, during which he developed some of his own ideas and unknowingly rediscovered some classical mathematics. In the autumn of 1948 he went to Paris, carrying a letter of recommendation to Henri Cartan from his calculus teacher at Montpellier. Cartan’s seminar attracted the most brilliant and aggressive young mathematicians, graduates of the elite École Normale Superieure. Although Grothendieck was an unknown foreigner and a newcomer from the provinces, he was accepted among them, and his talent was recognized. In fact, several of the young men he met then would later become his collaborators. Nevertheless, in view of the deficiencies in his preparation at Montpellier, it seemed that he would do better in a less pressured environment; so he was advised to go to Nancy to study under Laurent Schwartz. To Laurent Schwartz, working in a somewhat isolated academic environment, “the most fantastic gift to Nancy came in the person of Alexandre Grothendieck.”⁶

Schwartz and his wife, Helene, gave their new colleague a sense of belonging seldom experienced by a young man who had often lived as an outsider. They were caring people whose interests went beyond mathematics. Here we insert, parenthetically, Schwartz’s political history. In Schwartz’s younger days he was an active Trotskyist. In 1943, in hiding both as a Jew and as a Trotskyist, “life had become a constant series of dangers. You had to keep your eyes open and stay lucid. For this reason I decided to stop mathematical research during this period. I kept myself from the distraction of research which might have led me to relax my vigilance.” For the rest of his life Schwartz was a leading activist for human rights. He opposed both of France’s colonial wars, in Algeria and Indo-China, as well as Stalinist and fascist repression. In revenge for opposing the war in Algeria, his apartment building was plastic-bombed by an underground pro-war group, the Secret Army Organization (OAS).⁷

As Schwartz’s student, Grothendieck wrote a dissertation of over 300 pages containing a vast generalization and abstraction of Schwartz’s famous theory of distributions. Schwartz called it “a masterpiece of immense value.” He wrote, “It was difficult. I spent six months full time on it. What a job, but what a pleasure!”⁸ At schools dominated by Bourbaki, it produced a major transformation of functional analysis, although specialists in more application-oriented places like the Courant Institute continued to concentrate on the more traditional Banach space and Hilbert space. Nevertheless, because Grothendieck had no citizenship and as a matter of principle refused to apply for citizenship he couldn’t be hired at a French university. He spent a few years in Paris as a postdoctoral fellow. Then he taught in São Paulo, Brazil. He spent part of 1955 in the United States in Kansas and at Chicago. Then he returned to France, supported by fellowships and without a regular job. That was when he changed his mathematical interests from analysis to geometry. He wrote, “It was as if I’d fled the harsh arid steppes to find myself suddenly transported to a kind of ‘promised land’ of superabundant richness, multiplying out to infinity wherever I placed my hand on it, either to search or to gather. . . .”⁹

In 1957 there was established, first in Paris and then on a hill in the countryside southwest of Paris, the French counterpart of Princeton’s Institute for Advanced Study, the Institut des Hautes Études Scientifiques (IHES). Grothendieck and Jean Dieudonné were chosen as the two math professors. Since the IHES is not part of the French government, Grothendieck’s statelessness was not a problem.

His mother Hanka had been nearly bedridden for several years, suffering from tuberculosis and severe depression. She and Alexandre had grown inseparable. In his personal writings the image of the mother appears repeatedly as the source of life and creativity. In her last months she was so bitter that his life became extremely difficult. A close friend of hers named Mireille, several years older than Alexandre, helped him care for Hanka during her last months. Mireille was fascinated and overwhelmed by Alexandre’s powerful personality and fell in love with him. Hanka died in December 1957. Her death so shocked Grothendieck that he left mathematics for a few months. But then he returned to mathematical research and married Mireille.

In 1958 Oscar Zariski invited Grothendieck to visit him at Harvard. But in order to get a U.S. visa, he had to sign an oath pledging not to work to overthrow the U.S. government. This Grothendieck refused to do. Zariski wrote to him, warning that Grothendieck’s views might land him in prison. Grothendieck responded that that would be fine, as long as he could have books and visits from students. In February 1959 Mireille bore their first child, Johanna.

In March 1959 Grothendieck started giving seminars on algebraic geometry at the IHES. In collaboration with Jean-Paul Serre, with whom he shared many of his emergent ideas in correspondence as well as in person, he was building on ideas put forward by Jean Leray and André Weil. It is said that he worked 12 hours a day, 365 days a year, for 10 years. In his office at the IHES hung a portrait of his father, Sascha Shapiro. It was the only decoration. In 1961 he did visit Harvard, with Mireille as his wife. In July a son named Alexander was born, called Sasha after Alexandre’s father.

During these years of deep absorption in mathematics, Grothendieck paid little attention to the larger world and politics. He seldom read the newspapers. But after the outbreak of the Algerian uprising against France, he changed. On October 5, 1961, there was a curfew on French Muslims from Algeria. On October 17 thousands of Algerians took to the streets of Paris to protest. There was a police massacre that day that left dozens of bloody bodies piled in the streets or floating down the Seine.

During the war against Algeria, Grothendieck refused to demand special exemption from military service for mathematics students. Rather, he wrote to Serre, “The more people there are who, by whatever means, be it conscientious objection, desertion, fraud or even knowing the right people, manage to extricate themselves from this idiocy, the better.”¹⁰

In 1965 Mireille and Alexandre’s third child was born: a son, Mathieu. In this period, Mireille described him as working all night by the light of a desk lamp. She slept on a sofa in the study near him. Occasionally she would wake to see him slapping his head with his hand, trying to get ideas out faster.

Figure 4-1. Alexandre Grothendieck (center) among his colleagues. Courtesy of Archives of the Mathematisches Forschungsinstitut Oberwolfach.

Grothendieck poured out new ideas, and Dieudonné kept up as his scribe. Grothendieck was author or coauthor of around 30 volumes in the IHES blue series, most of which comprise over 150 pages. Éléments de Géométrie Algébrique (EGA), which he created in collaboration with Jean Dieudonné, and Séminaire de Géométrie Algébrique (SGA), consist of about 10,000 pages. His other works add a couple of thousand more pages. It was too much for Grothendieck to write, so he depended on a large group of more or less willing and able students and colleagues. There was a sense that a revolution was underway, as his ideas transformed algebraic geometry into one of the most abstract and technical fields in mathematics.

Grothendieck’s Theories

In the next few paragraphs we will try to give an idea of this work accessible to the curious nonmathematician. We quote extensively from Récoltes et Semailles. Some of these quotes are vague and hard to comprehend precisely. But they are Grothendieck’s own words! Nowhere else in the literature have we seen any account of these ideas written for the nonprofessional. Some readers may wish to skip over these paragraphs.

To begin savoring a slight taste of the flavor of this mathematics, consider first, Why did mathematicians need negative numbers? Because we wanted to be able to subtract any integer from any other regardless of which is bigger. Then we needed complex numbers because the solutions of quadratic and cubic equations cannot be fully understood within the domain of real numbers. And next we needed things called quaternions (or vectors), in order to use algebraic methods in three-dimensional space. At each stage, new kinds of numbers were created to solve more difficult problems.

Similarly, three daring conjectures by André Weil demanded the creation of new kinds of geometries. Grothendieck wrote:

These utterly astounding conjectures allowed one to envisage, for the new discrete varieties (or “spaces”), the possibility for certain kinds of constructions and arguments which to that moment did not appear to be conceivable outside of the framework of the only “spaces” considered worthy of attention by analysts—that is to say the so-called “topological” spaces (in which the notion of continuous variation is applicable). One can say that the new geometry is, above all else, a synthesis between these two worlds, which, though next-door neighbors and in close solidarity, were deemed separate: the arithmetical world, wherein one finds the so-called “spaces” without continuity, and the world of continuous magnitudes, “space” in the conventional meaning of the word. In this new vision these two worlds, formerly separate, comprise but a single unit. The embryonic vision of this Arithmetical Geometry (as I propose to designate the new geometry) is to be found in the Weil conjectures. In the development of some of my principal ideas, these conjectures were my primary source of inspiration, all through the years between 1958 and 1969.¹¹

Weil’s conjectures were very precise and very specific, and Weil could prove them in certain important special cases. But in full generality they were fantastically difficult because their very meaning could not be made precise without the development of a whole new theory that did not exist. One of Weil’s conjectures proposed that the number of solutions of a diophantine equation—an equation whose solutions are required to be integers—could be found by a kind of algebra that was invented for the study of continuous functions. On the one hand, the desired information, the number of solutions of diophantine equations, pertained to the realm of the discrete—whole numbers. On the other hand, the claimed solution, “Betti numbers,” are part of homology theory, which made sense only in the context of continuous manifolds (spheres, toruses, and their higher-dimensional analogs). Something very big was missing—a general theory that could bring together the discrete and the continuous, that could make the machinery of homology and cohomology, which was efficient and powerful in topology, valid in the remote realm of whole numbers.

The missing theory required generalizing the geometric notion of a space. Many spaces are considered in mathematics—Euclidean and non-Euclidean, the projective plane where a line at infinity is adjoined, curved manifolds both smooth and rough, Einstein-Minkowski space-time, knots and braids, pretzels and multipretzels, even the infinite-dimensional spaces used in quantum mechanics. They all can be thought of as sets of points, subject to conditions and constraints of one kind or another. But Grothendieck went further, to spaces without points. What sense does it make to talk about spaces without points?

The trick is to “algebrize” everything. Here is our own example (much simpler, of course, than what Grothendieck was concerned with). Out of pure algebra, one can create a familiar geometric space, the complex plane (the two-dimensional set of points represented by complex numbers of the form a + bi). Start with the set of all quadratic polynomials, ax² + bx + c, where a, b, and c are arbitrary real numbers. This set of polynomials is closed under addition and under multiplication by real numbers. Such an algebraic structure is called a “ring.” Then we make the following agreement: if two of those polynomials differ by a multiple of the special polynomial (x² + 1), we will regard them as “equivalent.” In other words, we decide to treat (x² + 1) as zero! The set of all scalar multiples of (x² + 1) is called an “ideal,” for some reason. By this equivalence relation, the ring of all polynomials in x is broken down into subsets. Each subset consists of polynomials equivalent to each other, meaning that the polynomials in each subset differ by a multiple of (x² + 1). It then appears that there is a natural and convenient way to add and multiply these equivalence classes to one another. In fact, this algebraic structure of equivalence classes is isomorphic to the complex numbers!

Where does i, the square root of –1, come from? Well, a moment’s thought will show that if (x² + 1) is equivalent to 0, then x² is equivalent to –1, and so the equivalence class of {x} works like i, the square root of –1. Thus, instead of defining complex numbers as points in the plane endowed with certain additive and multiplicative properties, we start with an algebra of polynomials and succeed at the end in constructing the space of complex numbers from the polynomial algebra. Thus a ring of polynomials, with a certain equivalence relation defined on it, turns out to be a space!

Here we are working with a very specific ring, the real polynomials in x. In more general contexts, including problems in number theory associated with diophantine equations, one again finds the algebraic structure, with operations of addition and multiplication, that is called a ring. Grothendieck’s idea, roughly speaking, was to impose on any ring at all a superstructure constructed to have the axiomatic properties of what we usually call a space.

A standard, principal tool in algebraic geometry and topology was the “sheaf” invented by Jean Leray. A circle with a tangent line attached at every point is an elementary example of a sheaf. The next example is any smooth surface in 3-space with the variable tangent plane attached at each point. A third example is the same smooth surface, but with the line perpendicular to the tangent plane attached at each point. In fact, we can take any manifold (such as a higher-dimensional sphere or torus or multitorus) and at every point attach a plane, a hyperplane, or some other algebraic structure.

Grothendieck had the boldness to consider, with each geometric object such as a manifold, roughly speaking, the set of all possible attached algebraic objects, including lines or planes. He defined these huge structures axiomatically and called them “schemes” or “toposes.” Then he showed how to do mathematics—prove theorems—about them. This was possible by using methods from a new, superabstract branch of mathematics called “category theory,” which had recently been developed by Samuel Eilenberg, Saunders MacLane, and Henri Cartan, among others. In category theory, one doesn’t start with sets, one starts with mappings (“arrows”) connecting undefined objects. Grothendieck’s toposes, then, are defined axiomatically to be like spaces with all possible sheaf structures. They don’t have points. What they have is “cohomology”—a certain algebraic structure, that works to classify topological spaces.

For Grothendieck, a theorem had to be exactly right in every detail. Here is how he described his own way of working: “The first analogy that came to my mind is of immersing the nut in some softening liquid, and why not simply water? From time to time you rub so the liquid penetrates better, and otherwise you let time pass. The shell becomes more flexible through weeks and months—when the time is ripe, hand pressure is enough; the shell opens like a perfectly ripened avocado!”

Grothendieck wrote, “The two powerful ideas that had the most to contribute to the initiation and development of the new geometry are schemes and toposes. The very notion of a scheme has a childlike simplicity—so simple, so humble in fact that no one before me had the audacity to take it seriously.”¹²

Grothendieck thinks of sheafs as

various “weights and measures” [which] have been devised to serve a general function, good or bad, of attaching “measures” (called “topological invariants”) to those sprawled-out spaces which appear to resist, like fleeting mists, any sort of metrizability. . . . One of the oldest and most crucial of these invariants, introduced in the last century (by the Italian mathematician Betti) is formed from the various “groups” (or “spaces”) called the “cohomology” associated with this space. . . . It was the Betti numbers that figure (“between the lines” naturally) in the Weil conjectures, which are their fundamental “reason for existence” and which give them meaning. Yet the possibility of associating these invariants with the “abstract” algebraic varieties that enter into these conjectures . . . that was something only to be hoped for.

He further wrote:

The new perspective and language introduced by the use of Leray’s concepts of sheaves has led us to consider every kind of “space” and “variety” in a new light. The “new principle” that needed to be found was the idea of the topos. This idea encapsulates, in a single topological intuition both the traditional topological spaces, incarnation of the world of the continuous quantity, and the so-called “spaces” (or “varieties”) of the unrepentant abstract algebraic geometers and a huge number of other sorts of structures which until that moment had appeared to belong irrevocably to the “arithmetical world” of discontinuous or “discrete” aggregates. Consider the set formed by all sheaves over a given topological space, or, if you like, the formidable arsenal of all the “rulers” that can be used in taking measurements of it. . . . We will treat the “ensemble” or “arsenal” as one equipped with a structure that may be considered “self-evident,” one that crops up in front of one’s nose, that is to say, a categorical structure. It functions as a kind of “superstructure of measurement,” called the “category of sheaves” (over the given space). It turns out that one can “reconstitute” in all respects, the topological space by means of the associated “category of sheaves” (or arsenal of measuring instruments) . . . the idea of the topos had everything one could hope to cause a disturbance, primarily through its “self-evident” naturalness, through its simplicity . . . through that special quality which so often makes us cry out: “Oh, that’s all there is to it” in a tone mixing betrayal with envy, that innuendo of the “extravagant,” the “frivolous,” that one reserves for all things that are unsettling by their unforeseen simplicity, causing us to recall, perhaps, the long buried days of our infancy.¹³

Grothendieck’s Later Years

At this point we return to Grothendieck’s chronological life history. In 1966 the International Congress of Mathematicians, meeting in Moscow, awarded him the Fields Medal, the highest international award for mathematical research. In protest against the repressive policies of the Soviet regime, he refused to attend to receive the prize.

In 1970, at the age of 42, still at the height of his creative powers and international fame, it is reported that Grothendieck learned that 5 percent of his institute’s budget came from the French military. He demanded that the military money be refused. The administration of the IHES at first promised to do so but then accepted the money anyway. Grothendieck then left the institute, never to return.

The so-called “productive period” of my mathematical activity, which is to say the part that can be described by virtue of its properly vetted publications, covers the period from 1950 to 1979, that is to say 30 years. And, over a period of 25 years, between 1945 (when I was 17) and 1969 (approaching my 42nd year) I devoted virtually all of my energy to research in mathematics. An exorbitant investment, I would agree. It was paid for through a long period of spiritual stagnation, by what one may call a burdensome oppression which I evoke more than once in the pages of Récoltes et Semailles . . . the greater part of my energy was consecrated to what one might call detail work: the scrupulous work of shaping, assembling, getting things to work, all that was essential for the construction of all the rooms of the houses, which some interior voice (a demon perhaps?) exhorted me to build. . . .

In evaluating his contribution to the mathematics of the 20th century, Grothendieck describes the feverish pace at which he worked and his belief, shared by many, that he had created an extraordinary new structure of ideas, particularly in algebraic geometry:

I rarely had the time to write down in black and white, save in sketching the barest outlines, the invisible master-plan that except (as it became abundantly clear later) to myself underlined everything, and which, over the course of days, months and years guided my hand with the certainty of a somnambulist . . . And by hundreds, if not thousands of original concepts which have become part of the common patrimony of mathematics, even to the very names which I gave them when they were propounded. . . . In the history of mathematics I believe myself to be the person who has introduced the greatest number of new ideas into our science. . . . The theme of schemas, their prolongations and their ramifications, that I’d completed at the time of my departure, represents all by itself the greatest work on the foundations of mathematics ever done in the whole history of mathematics and undoubtedly one of the greatest achievements in the whole history of Science.¹⁴

In 1975 Weil’s last conjecture was finally proved by Grothendieck’s student Pierre Deligne, after Grothendieck had departed from the mathematical world. His proof was a continuation of Grothendieck’s work but relied on a result from classical mathematics that Grothendieck would not have known about. Grothendieck expressed bitter disapproval. Deligne’s method for finishing the proof did not follow Grothendieck’s grander and more difficult plan.

For a few years Grothendieck devoted himself ardently to the cause of the environment. With the cooperation of fellow mathematicians Pierre Samuel and Claude Chevalley, he founded an ecology-saving organization, Vivre et Survivre. The Bourbaki member Pierre Cartier shared Grothendieck’s environmental concerns but complained that Grothendieck talked politics whenever he was invited to lecture on mathematics. This irritated everyone who came to hear him—even if they agreed with his politics. At the International Congress of Mathematicians in Nice in 1970, Grothendieck handed out leaflets and tried to set up a table for Vivre et Survivre. His old friend and collaborator, Jean Dieudonné, now a Dean at the University of Nice and one of the organizers of the Congress, prohibited Grothendieck’s table. So Grothendieck put up a table on the street outside the meeting. The chief of police showed up and asked Grothendieck to move his table back a few yards off the sidewalk. But he refused. “He wanted to be put in jail,” Cartier recalled. “He really wanted to be put in jail!” Finally the table was moved back far enough to satisfy the police.

During that time he had a position at the Collège de France in Paris, and then he had one at the University at Montpellier, where he had once been a student. David Ruelle, a leading mathematical physicist who was Grothendieck’s colleague at the IHES starting in 1964, has recently written that “Grothendieck’s program was of daunting generality, magnitude, and difficulty. In hindsight we know how successful the enterprise has been, but it is humbling to think of the intellectual courage and force needed to get the project started and moving. We know that some of the greatest mathematical achievements of the late twentieth century are based on Grothendieck’s vision. . . . Our great loss is that we don’t know what other new avenues of knowledge he might have opened if he had not abandoned mathematics, or been abandoned by it.”¹⁵

He continues, “It may be hard to believe that a mathematician of Grothendieck’s caliber could not find an adequate academic position in France after he left the IHES. I am convinced that if Grothendieck had been a former student of the École Normale and if he had been part of the system, a position commensurate with his mathematical achievements would have been found for him. . . . Something shameful has taken place. And the disposal of Grothendieck will remain a disgrace in the history of twentieth-century mathematics.”¹⁶

In May 1972, while visiting Rutgers University in New Jersey, Grothendieck met a young graduate student in mathematics named Justine (now named Justine Bumby). When he returned to France, she went with him. They lived together for 2 years and had a son John, who is now a mathematician. Once, while she was with Grothendieck at a peaceful demonstration in Avignon, the police came and harassed the demonstrators. Grothendieck got angry when they started pestering him. Justine recalled in an interview with Allyn Jackson: “The next thing we know, the two policemen are on the ground.” Grothendieck, who once practiced boxing, had single-handedly decked two police officers. At the police station, the chief of police expressed his desire to avoid trouble between police and professors, and Grothendieck and Justine were released.

For a long time Grothendieck was hospitable to all sorts of marginal “hippies.” In 1977 he was indicted and tried under a 1945 regulation that made it a misdemeanor to meet with a foreigner. He was given a suspended sentence of 6 months in prison and a fine of 20,000 francs (about $40).

He continued writing about mathematics, and circulating ideas among his acquaintances. The 1000-page Récoltes et Semailles was composed between 1983 and 1988. It’s a remarkable document, full of beautiful images and inspired rhetoric. Much of it is devoted to attacks on his pupils and followers, both for failing to carry out the projects he left for them and for shamelessly utilizing morsels, bits, and pieces of his work for their own ends.

In 1988 Sweden awarded him the $160,000 Crafoord Prize, to be shared with Pierre Deligne. Grothendieck refused the prize and wrote:

The work that brought me to the kind attention of the Academy was done twenty-five years ago at a time when I was part of the scientific community and essentially shared its spirit and its values. I left that environment in 1970, and, while keeping my passion for scientific research, inwardly I have retreated more and more from the scientific milieu. Meanwhile, the ethics of the scientific community (at least among mathematicians) have declined to the point that outright theft among colleagues (especially at the expense of those who are in no position to defend themselves) has nearly become the general rule, and is in any case tolerated by all, even in the most obvious and iniquitous cases. . . . I do not doubt that before the end of the century totally unforeseen events will completely change our notions about science and its goals and the spirit in which scientific work is done. No doubt the Royal Academy will then be among the institutions and the people who will have an important role to play in this unprecedented renovation, after an equally unprecedented civilization collapse.¹⁷

People who had been close to him were disturbed by this cryptic announcement of an apocalypse, even more than by his harsh condemnation of the mathematical world. Then, in 1992, he disappeared, cutting off all known connections with family and friends.

Grothendieck’s resentment toward his colleagues and former students is surprising in a man who wrote lovingly in earlier times about his collaborative work. The fact that many individuals worked hard to write up his discoveries also speaks of a personality that was appreciated and admired by his peers. But the course of his later life, including his growing dissatisfaction with an existence limited to research in mathematics, invites reflection. One way to think about some of his paranoid accusations, and his increasing withdrawal from his family and friends, is to speculate that his personal pain and struggles had been kept under control while he was devoting every waking hour to mathematics. That was his joy and his solace, as well as an escape from the tragic events of his childhood. While engaged in mathematics, he participated in a world that had order and beauty and was more predictable than the uprooting and losses during his early years. It was a world in which he could share his thoughts and his discoveries and put to work his incredible energy and creativity.

But however closely he immersed himself in the mathematical structures he created, it is likely that even during his most productive years he was haunted by the war years. Thinking of the many reasons he could have had to reject ordinary life, one can understand his seeking uninterrupted absorption in mathematics. But that total absorption would have excluded a more sustainable balance between his diverse interests, loves, and connection to his work. It is possible that witnessing the police brutalities against the Algerians, together with his fear of an environmental disaster, resulted in a fracture from the bounded abstract world in which he had lived productively for 20 years. His yearning for such a balance runs through many passages of Récoltes et Semailles:

This work of discovery, the concentrated attention involved and its ardent solicitude, constituted a primeval force, analogous to the sun’s heat and the germination and gestation of seeds sown in the nourishing earth and for their miraculous bursting forth into the light of day. . . . I’ve made use of the images of the builder and of the pioneer or explorer . . . in this “male builder’s” drive which, would seem to push me relentlessly to engineer new constructions, I have, at the same time, discerned in me something of the homebody, someone with a profound attachment to “the home.” Above all else, it is “his” home, that of persons “closest” to him—the site of an intimate living entity of which he feels himself a part. . . . And, in this drive to “make” houses (as one “makes” love . . .) there is above all, tenderness. There is furthermore the urge for contact with those materials that one shapes a bit at a time, with loving care, and which one only knows through that loving contact. . . . Because the home, above all and secretly in all of us, is the Mother—that which surrounds and shelters us, source at once of refuge, and comfort. . . .¹⁸

In September 1995 it was announced (by Roy Lisker) that Jean Malgoire of the University of Montpellier had visited Grothendieck, in a village in the Pyrenees Mountains. There Grothendieck meditates and sustains himself in a manner totally harmless to the environment. In Paris there is a Grothendieck circle, collecting, translating, and publishing his writings, while Grothendieck maintains his total distance from his previous mathematical life. Recently he demanded that they stop all work on his publications.

Into Madness

Mathematics involves working just with pencil and paper, or perhaps with chalk or a computer keyboard. And yet it has been wisely said that mathematics can be a very dangerous profession—dangerous especially to some of its more vulnerable practitioners. As a graduate student, I (R.H.) heard a lecture on A-contractive mappings by a Rutgers professor named Wolodymyr V. Petryshyn. Professor Petryshyn was later incarcerated. He actually murdered with a hammer his beloved wife Arcadia Olenska-Petryshyn, who was an artist of some reputation.¹⁹ According to those who knew him, Petryshyn had discovered a mistake in his estimable book Generalized Topological Degree and Semilinear Equations. His publishers assured him that it was not such a serious problem. He, however, felt that his omission of a necessary hypothesis meant total disgrace. Felix Browder, who had known Petryshyn for decades, was quoted: “It sounds as if his perfectionism drove him to insanity.”

When I (R.H.) arrived at Stanford in 1962 for the start of a 2-year instructorship, I met Professor Karel de Leeuw. Karel sat watching as I unpacked my books. He was well known in the department for his exceptional interest in students as individual human beings. Every year he ended his graduate course with a party at his house. But one of the Stanford math graduate students, Ted Streleski, decided that in retribution for his 19 years of suffering there without progressing to a degree, it was “completely logical” to murder Karel de Leeuw. He explained to a newspaper, “Stanford University took 19 years of my life with impunity, and I decided I would not let that pass.”

Maybe it’s morbid to recount such sad stories. We are telling them because total immersion in mathematical life, with the expectation of perfection, can contribute to madness. Petryshyn’s eruption was mad, but to a fellow mathematician it is in some degree understandable. We all know that our results, our publications, are supposed to be completely correct, logically irrefutable. We also know that in very many cases, even though everything looks right, there remains an aching uncertainty. Is it really absolutely correct? Haven’t I overlooked something? What if it turns out to be all wrong? With this persisting anxiety goes an underlying feeling that if that should actually be the case, then disaster and disgrace will be utter and final. All this is in fact delusion, yet it is somehow part of the ethos that we absorb somewhere along the line in our training and indoctrination. “It’s either right or wrong. You’re supposed to know the difference.”

So a particularly rigid, inflexible personality could experience a sense of total destruction if his support in perfectly rigorous mathematics turned out to be cracked and broken. And then—what then? What then?

Although mathematicians increasingly work in pairs and in groups, traditionally we mostly worked alone. To some, working alone is a matter of pride, of personal identity. But working alone makes you especially vulnerable. A partner may catch your mistake. On your own, you can kid yourself and go on and on in error. If you believe you know what is right, what is a proof, you may follow yourself into the flames. Something that is less likely to happen in a kind of work that is inherently cooperative and social.

Ted Kaczynski

In 1996 and 1997, the most famous mathematician in America was Ted Kaczynski. A Ph.D. from the University of Michigan and an ex–faculty member at Berkeley, he had not published or taught in the preceding years. He certainly was not welcomed by the mathematics profession as a representative. But he was trained and qualified as a mathematician, and his particular kind of insanity would not be found in a salesperson or an auto mechanic.

For those who don’t remember the story, the Unabomber was a mystery man for years. From 1978 to 1995, someone was mailing bombs to American scientists and doing serious damage. David Gelernter at Yale was permanently disfigured when he opened one of these nasty surprise packages. Gilbert Murray, president of the California Forestry Association, was killed. In 1995 the secret Unabomber demanded that his Manifesto be published in the New York Times.

We try to summarize his reasoning, which could even be called a theorem. We present this, not only for its intrinsic interest but also to show the mathematical nature of his obsession.

1. Industrialization is destroying the natural world, which gave rise to the human race, and on which the human race depends for its existence. Already very severe, even irreparable damage has been done to the natural world: extinction of species, depletion of resources, universal pollution by many different pollutants.

Figure 4-2. Ted Kaczynski, before he became the Unabomber. Courtesy of Archives of the Mathematisches Forschungsinstitut Oberwolfach.

2. Increasing population and the demand of capital for constant growth generate more destruction of the natural world, at an ever accelerating rate. We can see catastrophe within decades.

3. This oncoming catastrophe far outweighs any other considerations of humanity, morality, or tradition. It is the duty of every thinking person to understand it, and to act on it, immediately, and as effectively as possible.

4. But the short term self-interests of government, communications media, political parties, induce them all to close their eyes, and turn away from this overwhelmingly threatening reality.

5. Any measures to wake people up, to force them to pay attention, are justified, even demanded.

6. Threatening to kill or actually killing people who are implicated in this dreadful process is one way, maybe the only way, to get the needed attention to this cause.

7. Since I, Ted Kaczynski, do understand this, it is my absolute duty to proceed with threats and actual murders for this all-important end, to save humanity and the earth.

Reading the manifesto, the mathematician John Allen Paulos guessed from the tone, content, and structure that its author was a mathematician. He wrote an op-ed piece saying so for the New York Times. That angered some mathematicians who thought it was bad for their image, and the Wall Street Journal published a long article on the fracas. In his essay Paulos had opined that Kaczynski’s Ph.D. in mathematics was “perhaps not quite as anomalous as it seemed (despite the fact that mathematicians are for the most part humorous sorts, not asocial loners, and that the only time most of us use the phrase ‘blow up’ is when we consider division by zero)”.

Doron Zeilberger, a mathematician at Rutgers University, was prompted to look up Ted Kaczynski’s papers. He reported: “They are paragons of precision! They are written in a no-nonsense terse, yet complete step-by-step style, that makes most math papers look like long-winded sociology. The math is also beautiful! What a shame that he quit math for bombing.” Note that while it’s already a shame that Ted Kaczynski became a bomber, it will become a still greater shame when we realize that his publications showed real promise!

Kaczynski’s brother, David, recognized the mind behind the manifesto and the bombs and did the painful but necessary task of turning his brother in to the Federal Bureau of Investigation. Ted Kaczynski was found in his hermit cabin in Montana, where for years he had been thinking hard how to prevent humanity from destroying the planet. He is quoted in the New York Times, January 23, 1998: “My occupation is an open question. I was once an assistant professor of mathematics. Since then, I have spent time living in the woods of Montana.”

He is now serving a life sentence in a high-security federal prison. There he continues to correspond with like-minded anti-industrialization fanatics.

Since the conclusion of his manifesto is murder, surely it can be carried out only if there is absolute certainty that it is correct. Such absolute certainty is the special characteristic of mathematical reasoning. If Kaczynksi had not been a mathematician, he might still have believed in an imminent environmental disaster. He might still have committed murder, following his beliefs. But surely his case would not have been argued so rationally.

The answer to his argument, I believe, is simply the age-old cry of Oliver Cromwell: “I beseech you in the bowels of Christ to consider that you may be mistaken!”

André Bloch

The French mathematician André Bloch lived from 1893 to 1948 and won the Becquerel Prize for discovering Bloch’s constant which is important in the theory of univalent analytic functions of a complex variable. He was a patient of Henri Baruk, who was, according to Cartan and Ferrand, “one of the greatest French psychiatrists of the mid-twentieth century.” In Baruk’s autobiography, Patients Are People Like Us, he writes of Bloch:

Every day for forty years this man sat at a table in a little corridor leading to the room he occupied, never budging from his position, except to take his meals, until evening. He passed his time scribbling algebraic or mathematical signs on bits of paper, or else plunged into reading and annotating books on mathematics whose intellectual level was that of the great specialists in the field. . . . At six-thirty he would close his notebooks and books, dine, then immediately return to his room, fall on his bed and sleep through until the next morning. While other patients constantly requested that they be given their freedom, he was perfectly happy to study his equations and keep his correspondence up to date.²⁰

Unlike other patients, Bloch refused to go out of the building onto the grounds, saying “Mathematics is enough for me.” Bloch completed a substantial body of work: four papers on holomorphic and meromorophic functions and short articles on function theory, number theory, geometry, algebraic equations, and kinematics. He was self-taught, for his studies had been brutally and prematurely interrupted by World War I. His last paper was a collaboration with another mathematician who had been hospitalized for a short while with him in the Maison de Charenton.

A few months after Bloch’s death in 1948, his story was told by Professor Georges Valiron at the annual meeting of the Societé pour l’Avancement des Sciences. “In 1910 I had both (André and his brother Georges) in my class (at the École Polytechnique.) They left in 1914 because of the war.” Professor Valiron’s listeners needed no reminders of that war—its trench warfare and suicide attacks, its misery, degradation, and horror. He went directly to the outcome. “On November 17, 1917, at the end of a leave and three days before his 24th birthday, in the course of a crisis of madness he killed his brother Georges, his uncle, and his aunt. Declared insane, he was confined to the mental hospital at Saint Maurice, where he remained until his death on October 11, 1948.”

After months at the front, André Bloch had fallen off an observation post during a bombardment, and the shock made him unfit for active service. His brother Georges, whom he killed, had been wounded in the head and lost an eye. The murders took place at a meal in the family apartment on the Boulevard de Courcelles in Paris. Afterward André ran screaming into the street and let himself be arrested.

When Dr. Baruk read his patient’s history, he found it difficult to imagine that such a charming, cultivated, polite man could have committed such an act. “He was the model inmate, whom everyone in the hospital loved.”²¹ One day a younger brother of André appeared at the hospital. He had been living in Mexico, was passing through Paris, and wanted to see André. André gave no sign of affection or welcome to his brother. His manner was extremely cold. . . . The next day he explained to Dr. Baruk: “It’s a matter of mathematical logic. There were mentally ill people in my family, on the maternal side, to be exact. The destruction of the whole branch had to follow as a matter of course. I started my job at the time of the famous meal, but never got a chance to finish.” Dr. Baruk told Bloch that his ideas were horrifying. “You are using emotional language,” Bloch answered. “Above all there is mathematics and its laws. You know very well that my philosophy is based on pragmatism and absolute rationality.” Dr. Baruk diagnosed “morbid rationalism . . . a crime of logic, performed in the name of absolute rationalism, as dangerous as any spontaneous passion.”²²

André Bloch was an extreme example. He worked on mathematics problems for decades, for the same hours every day, sitting in the same corner in a corridor in the Charenton asylum. He was able to do important mathematical work. But his social and ethical judgment was totally impaired, akin to that of a patient suffering from prefrontal cortical injury. His was an extreme and rare case. Nevertheless, it is a useful reminder of the harm that can come from isolating abstract knowledge from real-life uses, or from constructing a life totally devoted to strenuous mental efforts without social connections or diverse interests.

Kurt Gödel

The greatest logician of the 20th, or any other century, also could have been diagnosed with morbid rationalism, although he certainly never killed any one or even raised his hand in anger. When Gödel went to a U.S. judge to attain his citizenship, accompanied by his friend Oskar Morgenstern, he insisted on correcting the judge when the judge said that events like those in Germany, resulting in Hitler’s becoming dictator, could never happen under the U.S. constitution. Gödel had noticed a way that this could happen under the constitution! Morgenstern managed to deflect his incipient lecture, so that the citizenship proceedings could continue. In his last years, after Morgenstern and his other friend at the Institute for Advanced Study, Albert Einstein, had died, Gödel simply didn’t talk to anybody—that is, talk about his real interests and concerns, mathematics and philosophy. He wrote papers for his desk drawer. His wife, Adele, of course prepared his meals. Unfortunately, she became ill and had to go to a hospital. Without her protection, he had no proof that any food he was offered was safe. He cleverly prevented himself from being poisoned by refusing to eat anything at all. Adele returned home from the hospital at the end of December and persuaded Gödel to enter Princeton Hospital. Kurt Gödel died at age 72, in the fetal position, at 1:00 in the afternoon, on Saturday, January 14, 1978. He weighed 65 pounds. According to the death certificate, on file in the Mercer County Courthouse in Trenton, he died of malnutrition and inanition caused by personality disturbance.

Thus perished the greatest logician of all time! And what about Adele? How long did she survive? How did she endure decades of serving her genius husband in total isolation from other human contact? What did she do? What did she think?

In looking back over these disturbing stories, we can’t help but remember Streleski’s and Bloch’s explanations: their crimes were completely logical! Indeed, greater crimes are committed, by politicians and generals, also claiming that their actions are “completely logical.”

It is really claimed by some philosophers that the propositional and predicate calculi—modern formal logic—are infallible (e.g., John Worrall and Elie Zahar in editing Lakatos’ Proofs and Refutations). “From true premises, true conclusions follow, infallibly.” How dangerous this dogma can be! Logic can never be anything but a tool, an action, or a procedure carried out by a human being (or by a machine created and programmed by a human being). Logic, such an essential tool of science and philosophy, sometimes becomes a sort of false god, outranking the most fundamental human impulses, such as “Thou shalt not kill.” Or even, “Eat to stay alive.”

Figure 4-3. Kurt Gödel and Adele Gödel. Courtesy of the Kurt Gödel Papers, The Shelby White and Leon Levy Archives Center, Institute for Advanced Study, Princeton, NJ, USA, on deposit at Princeton University.

In answer to our question, “Can mathematics become a dangerous addiction?” we must answer, “Yes, to a susceptible mind, under unfavorable conditions, that has indeed happened.”

Bibliography

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