“Can you give us a specific example?” he asked.
“An example of what?” asked Grothendieck. “You mean give an actual prime number?”
“Yes,” replied the questioner.
Grothendieck, in a rush to continue his main argument, said, “Well, take 57.”
Of course we know that 57 is not a prime number—it is the product of 19 and 3—but Grothendieck couldn’t care less about details such as whether a number was prime. He was concerned with much grander things—in this case, an abstraction of the idea of all prime numbers. The number 57 is now affectionately dubbed “Grothendieck’s prime.”
Grothendieck’s many major discoveries in mathematics earned him the prestigious Fields Medal—the mathematician’s Nobel Prize—as well as many other awards and distinctions. Before long, his vision of mathematics brought him into conflict with Bourbaki, which he saw as a leaderless group that lacked a clear mission. At the IHES, he instituted a lecture series that rivaled that of Bourbaki in Paris. Grothendieck’s School, as it was called, was becoming the place to be for any young mathematician seeking to make a name and exert an influence on the discipline.
Grothendieck nurtured a number of bright students and seemed to have a gift for determining which research topic was best suited for each student’s personality and interests. He has been described as a sensitive and kind person whose home was always open to everyone. Having grown up in great deprivation, he was always concerned with the fate of the downtrodden, the poor, and the persecuted. According to Cartier, for much of his life Grothendieck has practiced “dietary restrictions that could be ascribed to choice, Judaism, or Buddhism.”12
AS HIS FAME GREW, Grothendieck traveled throughout the world, lecturing about mathematics in such places as Brazil and the United States, where he toured for several years. At Rutgers University he met a student named Justine, who then accompanied him back to France and lived with him for two years. They had a son together, who is now a professor of statistics at an American university.
Like his father, Grothendieck was technically stateless. His birth certificate disappeared in Berlin during the war, and when he was being shuttled with his mother to and from various camps, they were considered “displaced persons.” It is surprising that Grothendieck, the famous French mathematician, never asked for or received French citizenship. Throughout his life, he has been traveling on a UN passport. (Although not technically a French citizen, Grothendieck nonetheless officially changed the spelling of his name from Alexander to the French version, Alexandre.)
At the Bourbaki Congress in 1957, Grothendieck cryptically told Pierre Cartier that he was considering pursuing “activities other than mathematics.” Cartier surmised that he was thinking of writing poetry or fiction, as his mother had done in the camps.13 But apparently Grothendieck had other things in mind.
Nine years later, in 1968, the big change took place. At the very height of his career, upon turning forty, Grothendieck decided to abandon mathematics altogether. He became involved in the movement against the Vietnam War, and took part in antiwar demonstrations on the streets of Paris; then he traveled with Cartier to Vietnam in order to protest against the actions of the U.S. government. This was also the year of the Prague Spring, the Czech revolt against Soviet rule, and in France stu-dents and workers were mobilized to act against the establishment under the leadership of Danny Cohn-Bendit, known as Danny the Red.
Cartier claims that what triggered Grothendieck’s strong political activism was a discovery he had made about the IHES. Apparently it was being funded in part by grants from the French Ministry of Defense. Throughout his life, Grothendieck has harbored strong antimilitary feelings, and the discovery of a connection between “his” institute and the military angered him no end.
In September 1970 Grothendieck participated in a riot in the French city of Nice and hit two policemen. He was arrested, but after the police found out that he was a prominent professor, he was quietly released. This did not diminish his political fervor. Grothendieck then started signing up to give public talks about mathematical topics—but when his turn to speak came, he would talk instead about social issues, the environment, and his rage against war and the military. Even people who agreed with him on these subjects were greatly irritated by such maneuvers. Thus the soft-spoken, kind, generally beloved mathematician began to lose friends and supporters.
That year, Grothendieck left the IHES in disgust after failing to convince the administration to reject all military-based funding. Through the help of Jean-Pierre Serre, a member of the Bourbaki group and one of the greatest mathematicians in France, he received an appointment to the Collège de France—perhaps the most prestigious of the French institutions of research and learning. But at the Collège de France, which hired him as a professor of mathematics, Grothendieck refused to lecture on the subject and, instead, talked only about political, social, and environmental issues, including his rage against nuclear arms. When his appointment expired in 1973, it was not renewed.
Grothendieck then founded a survivalist group called Survivre et Vivre (Survive and Live). He retreated into the Pyrenees, the mountainous region closest to the location of the first camp in which he was interned, Rieucros. Perhaps he had a romantic memory of the Pyrenees, seeing them from the infested, hot, flat plain of the wretched camp; perhaps he had fantasized about escaping to these mountains.
Grothendieck spent time on and off with his group in their new compound in the Pyrenees, but he also maintained a house in the town of Villecun, where he lived from 1973 to 1979. In 1977, because he participated in an incident involving people the French government called “hippies,” he was accused and put on trial based on an outdated wartime French law against “consorting with foreigners.” He was given a suspended sentence of six months’ imprisonment and forced to pay a steep fine. This incident infuriated him even further.
Grothendieck then took up an academic position at his alma mater, the University of Montpellier—not too far from the Pyrenees—but he didn’t lecture much. More and more often, he retreated alone to the mountains, and, over months of absences, his mail would pile up at the mathematics department of the university.
By the mid-1980s very few people even saw Grothendieck, as he would spend much of his time hiding in the Pyrenees. Then, in August 1991, he suddenly burned twenty-five thousand pages of his mathematical manuscripts at the University of Montpellier and abruptly left again for the mountains. Eventually he gave up his position at the university and set up a post-office box. After 1993 even the post-office box was discontinued, and Grothendieck no longer has a listed address. He seems to have disappeared without a trace.
Around the turn of the twenty-first century, two youngish researchers from a mathematics institute in Paris decided to search for the reclusive mathematician. They used clever tricks to locate a small town in the Pyrenees that they suspected might be close to Grothendieck’s hideout. Then they laid a trap for the aging mathematician on a day they thought he might have to visit a location in the town in order to buy food. They succeeded, and when they encountered him, he agreed to talk and didn’t run away. They made pleasant conversation for a while and even informed Grothendieck that a conjecture he had once made had just been proved by another mathematician. Very few people, if any, have seen him since. According to Cartier:
If I can believe his most recent visitors, he is obsessed with the Devil, whom he sees at work everywhere in the world, destroying the divine harmony and replacing 300,000 km/sec by 299,887 km/sec as the speed of light!14
Grothendieck’s thoughts and writings have become increasingly bizarre. In a meditation called “The Key to Dreams,” the reclusive mathematician talks about people he calls the mutants. These “mutants,” whom Grothendieck describes over hundreds of pages, comprise eighteen individuals—some well known, others not—including Charles Darwin, Sigmund Freud, Walt Whitman, Mahatma Gandhi, Pierre Teilhard de Chardin, and the mathematician Bernhard Riemann. Gr
othendieck believes that the mutants are exceptional people who represent humanity’s potential rise above the general moral decline he sees in society. He views himself as their disciple.
I spent part of the summer of 2005 trying to find Grothendieck, using clues given by the pair of French mathematicians who had found him—but to no avail. Grothendieck must be hiding very well. When I contacted Grothendieck’s relatives, I was told, “[Grothendieck] is probably dead.”15 But at this writing, the great mathematician is alive.
NOTES
CHAPTER 1
1. Heath, A History of Greek Mathematics, 128; Plutarch, Lives of the Noble Grecians and Romans, chap. 3 (“Solon”).
2. Ibid.
3. Heath, 129.
4. Ibid.
5. Heath, 137.
6. Ibid., 137–38.
7. Heath, 4.
8. Ibid.
9. Ibid., 5.
10. Boyer, A History of Mathematics, 48.
11. Ibid.
12. Ibid.
13. Heath, 77.
14. Heath, 75.
CHAPTER 2
1. Boyer, A History of Mathematics, 72.
2. This is proved by contradiction. Assume that the square root of 2 is rational and equal to a/b, where a and b are integers and the fraction is in lowest terms (it can’t be simplified). Then 2 = a2/b2 and therefore a2 = 2b2. This means that a2 is an even number, and hence a must also be even (when you square an odd number you get an odd number). If a is even, then you can write it as a = 2k, where k is some integer, and we get 2 = a2/b2 = (2k)2/b2 = 4k2/b2 and thus b2 = 2k2, which means that b2 is even and hence b is even. But if both a and b are even, then the fraction a/b could not have been in lowest terms and this is a contradiction, establishing the claim that the square root of 2 is irrational.
3. Boyer, A History of Mathematics, 71.
4. Bell, Men of Mathematics, 29.
5. Ibid.
CHAPTER 3
1. Boyer, A History of Mathematics, 178.
CHAPTER 4
1. Boyer, A History of Mathematics, 210.
2. Plofker, Mathematics in India, 319.
3. Ibid., 73.
4. Boyer, 238.
5. Ibid., 242.
6. Escofier, Galois Theory, 14.
CHAPTER 5
1. Boyer, A History of Mathematics, 199.
CHAPTER 6
1. Boyer, A History of Mathematics, 256.
2. In our modern notation, for the cubic equation x3 + px + q = 0, a solution is:
3. Ibid., 283.
4. Escofier, Galois Theory, 15.
5. Ibid.
CHAPTER 7
1. Boyer, A History of Mathematics, 303.
2. Ibid., 312.
3. Ibid., 329.
CHAPTER 8
1. Baillet, Vie de Monsieur Descartes, 69.
2. Bell, Men of Mathematics, 38.
3. Mahoney, The Mathematical Career of Pierre de Fermat, 26.
CHAPTER 9
1. Robert, Leibniz, vie et oeuvre, 23.
2. Ibid., 11.
3. Ibid., 14.
4. Belaval, Leibniz critique de Descartes, 67.
5. Robert, 29.
6. Bell, Men of Mathematics, 120.
7. Robert, 71.
8. Ibid., 31.
9. Ibid.
10. Belaval, 81.
11. Bell, 90.
12. Ibid., 97.
13. Ibid., 105.
14. Ibid., 108.
15. Ibid.
CHAPTER 10
1. Boyer, A History of Mathematics, 415–416.
2. Ibid.
3. Bell, Men of Mathematics, 139.
4. Ibid., 143.
5. Ibid., 145–146.
6. Ibid.
7. Ibid., 149.
8. Ibid.
9. Ibid., 147.
10. Ibid., 152.
11. Ibid., 221.
12. Ibid., 222.
13. Ibid., 223.
14. Ibid., 228–229.
15. Ibid., 229–230.
16. Ibid., 231.
17. In the early part of the twentieth century, Albert Einstein found non-Euclidean geometries crucial in developing his general theory of relativity.
18. Bell, 232.
19. Ibid., 234.
20. Ibid., 243.
CHAPTER 11
1. Boyer, A History of Mathematics, 471.
2. Bell, Men of Mathematics, 153.
3. Ibid.
4. Ibid., 154.
5. Ibid.
6. Ibid., 155.
7. Boyer, 470.
8. Bell, 166.
9. Ibid.
10. Ibid., 172.
11. Ibid., 173.
12. Ibid., 174.
13. Ibid., 182.
14. Boyer, 495.
CHAPTER 12
1. Escofier, Galois Theory, 220.
2. Ibid., 221.
3. Ibid., 222.
4. Ibid.
5. Boyer, A History of Mathematics, 581.
6. Ibid., 592.
7. Reprinted in Baez, “The Octonions,” 145.
8. Ibid.
CHAPTER 13
1. Dauben, Georg Cantor, 275–276.
CHAPTER 14
1. Kanigel, The Man Who Knew Infinity, 168.
2. Neville, “Srinivasa Ramanujan,” 292–295.
3. Bell, Men of Mathematics, 526.
CHAPTER 15
1. Weil, The Apprenticeship of a Mathematician, 97.
2. Weil’s work forms the mathematical appendix to Claude Lévi-Strauss’s book The Elementary Structures of Kinship (Boston: Beacon Press, 1971), 221–233.
3. Boas, “Bourbaki and Me.”
4. Personal communication from Sylvie Weil, André Weil’s daughter, 2010.
5. Personal communication from Pierre Cartier, 2006.
6. Cartier, “A Mad Day’s Work,” 389–408.
7. Scharlau, “Who Is Alexander Grothendieck?” 93–94.
8. Jackson, “Comme Appelé du Néant—As If Summoned from the Void,” 1038–1056; 1196–1212.
9. Ibid.,1039.
10. Cartier, 391.
11. Ibid., 393.
12. Ibid.
13. Ibid., 392.
14. Ibid., 393.
15. Interviews with Grothendieck’s relatives, 2005.
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