Comme Appelé duNéant - As If Summoned from the Void: The Life of Alexandre Grothendieck - Part 2

Aug 26, 2026

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Second part of a two-part article on Alexandre Grothendieck’s life, following an earlier installment published in the October 2004 issue of the Notices. The text highlights Grothendieck’s distinctive mathematical mindset: intense attention, precise calibration of generality, and a preference for thinking that strips away enough structure without becoming empty. It contrasts his low dependence on examples with other styles, discusses caution in assuming exceptional beauty, and explains his view that chosen problems should feel almost obvious before one is ready to work. It also emphasizes the role of evocative naming in discovery, including concepts like étale and topos.

LanguageEnglish

CategoryLiterature

FormatPDF

Pages17

Reading Time26 min

AccessFree to download

A Different Way of Thinking

Attention, abstraction, and generality

Thinking with few examples

“Almost obvious” problems and natural preparation

Evocative naming: étale and topos

Comme Appelé duNéant—As If Summoned from the Void: The Life of Alexandre Grothendieck Allyn Jackson This is the second part of a two-part article about the life of Alexandre Grothendieck. The first part of the article appeared in the October 2004 issue of the Notices. A Different Way of Thinking Dans le travail de découverte, cette attention intense, cette sollicitude ardente sont une force essentielle, tout comme la chaleur du soleil pour l’obscure gestation des semences enfouies dans laterre nourricière, et pour leur humble et miraculeuse éclosion à la lumière du jour. In the work of discovery, this intense attention, this ardent solicitude, are an essential force, just like the warmth of the sun for the obscure gestation of seeds covered in nourishing soil, and for their humble and miraculous blossoming in the light of day. —Récoltes et Semailles, page P49 Grothendieck had a mathematical style all his own. As Michael Artin of the Massachusetts Institute of Technology commented, in the late 1950sand 1960s “the world needed to get used to him, to his power of abstraction.” Nowadays Grothendieck’s point of view has been so thoroughly absorbed into algebraic geometry that it is standard fare for graduate students starting in the field, many of whom do not realize that things were once quite different. Nicholas Katz of Princeton University said that when as a young mathematician Allyn Jackson is senior writer and deputy editor of the Notices. Her email address [is](is axj@ams.org)[ axj@ams.org](is axj@ams.org). he first encountered Grothendieck’s way of thinking, it seemed completely different and new. But it is hard to articulate what the difference was. As Katz put it, the change in point of view was so fundamental and profound and, once adopted, so completely natural “that it’s sort of hard to imagine the time before you thought that way.” Although Grothendieck approached problems from a very general point of view, he did so not for generality’s sake but because he was able to use generality in a very fruitful way.“It’s a kind of approach that in less gifted hands just leads to what most people would say are sterile generalities,” Katz commented.“He somehow knew what general things to think about.” Grothendieck always sought the precise level of generality that would provide precisely the right leverage to gain insight into a problem.“He seemed to have the knack, time after time, of stripping away just enough so that it wasn’ta special case, but it wasn’t a vacuum either,” commented John Tate of the University of Texas at Austin. “It’s streamlined; there is no baggage. It’s just right.” One striking characteristic of Grothendieck’s mode of thinking is that it seemed to rely so little on examples. This can be seen in the legend of the so-called “Grothendieck prime”. In a mathematical conversation, someone suggested to Grothendieck that they should consider a particular prime number.“You mean an actual number?” Grothendieck asked. The other person replied, yes, an actual prime number. Grothendieck suggested,“All right, take 57.” 1196 NOTICES OF THE AMS VOLUME 51, NUMBER 10 But Grothendieck must have known that 57 isnot prime, right? Absolutely not, said David Mumford of Brown University.“He doesn’t think concretely.”Consider by contrast the Indian mathematician Ramanujan, who was intimately familiar with properties of many numbers, some of them huge. That way of thinking represents a world antipodal to that of Grothendieck.“He really never worked on examples,”Mumford observed.“I only understand things through examples and then gradually make them more abstract. I don’t think it helped Grothendieckin the least to look at an example. He really got control of the situation by thinking of it in absolutely the most abstract possible way. It’s just very strange. That’sthe way his mind worked.” Norbert A’Campo of the University of Basel once asked Grothendieck about something related to the Platonic solids.

FAQs

What makes Grothendieck’s approach to discovery distinctive?

He combines intense attention with ardent solicitude and seeks the precise level of generality that gives the right leverage to gain insight into a problem.

How does Grothendieck’s style differ from mathematicians who rely on examples?

The text describes a contrast: while others understand through examples and then abstract, Grothendieck is portrayed as not benefiting from working through examples in the same way, instead controlling the situation by thinking in a highly abstract manner.

What guidance does the article attribute to Grothendieck about choosing what to prove or work on?

He is said never to try to prove anything that is not almost obvious; if one cannot see the “almost obvious” nature of what is being worked on, one is not ready yet. His approach is also described as preparing so that the result eventually seems completely straightforward.

Aug 26, 2026

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