The Grothendieck Mystery

May 9, 2022 5:49 AM

While living in an internment camp in Vichy France, Alexander Grothendieck was tutored in mathematics by another prisoner, a girl named Maria. Maria taught Grothendieck, who was twelve, the definition of a circle: all the points that are equidistant from a given point. The definition impressed him with “its simplicity and clarity,” he wrote years later. The property of perfect rotundity had until then appeared to him to be “mysterious beyond words.” Rivka Galchen, in the New Yorker, on the life, work, thought, and spirit of Alexander Grothendieck, the great intellectual revolutionary barely known outside mathematics.
posted by escabeche (21 comments total)

I am quoted in this article but hopefully I'm right that it's not by virtue of that a self-link. I posted previously about Grothendieck when he died in 2014.
posted by escabeche at 5:51 AM on May 9, 2022 [5 favorites]

I love the idea of the Grothendieck Prime, "57"
posted by chavenet at 6:13 AM on May 9, 2022

Calculus, the most advanced math most people ever see, (and that subset far too small a minority) came a hundred years before the American Revolution. Now take all the math ideas from the next hundred years and abstract them into a single variable, no that makes no sense, not abstract enough. Maybe if one could just draw an arrow from one thing to another...
posted by sammyo at 6:16 AM on May 9, 2022 [2 favorites]

Grothendieck is featured, with a mix of fact and fiction, in the excellent book When We Cease To Understand The World.
posted by gwint at 7:45 AM on May 9, 2022 [3 favorites]

I am finding TFA rather tedious. The gosh-wow descriptions like "Imagine if math could be translated into poetry, and somehow it made sense to take the square root of a stanza" are worse than useless.

I realize that it's an item for, not just non-mathematicians, but mathematically illiterate people. But does that have any value at all?

It's like an essay about Picasso's paintings, for people who are blind.

Also, I am pretty sure that the assertion "[Grothendieck] rewrote definitions, even of things as basic as a point" is pure bullshit. Mathematics is based in part on a small number of "primitive" concepts, which is to say things that are undefined. The notion of "point" is probably the second-oldest of these, after the notion of "number."
posted by Aardvark Cheeselog at 7:53 AM on May 9, 2022 [1 favorite]

I enjoyed reading this; thanks for posting it.
posted by LobsterMitten at 8:34 AM on May 9, 2022

Also, I am pretty sure that the assertion "[Grothendieck] rewrote definitions, even of things as basic as a point" is pure bullshit. Mathematics is based in part on a small number of "primitive" concepts, which is to say things that are undefined. The notion of "point" is probably the second-oldest of these, after the notion of "number."

And yet, it's true.
posted by gleuschk at 9:44 AM on May 9, 2022 [16 favorites]

Mazur recalled that Grothendieck had met a family at the local train station with nowhere to stay, and he invited them to live in the basement apartment of his home. He had a machine installed that helped make taramosalata—a fish-roe spread—so that they could sell prepared food at the market.

This sounds so far-fetched that it curves back on itself, through "sure, why not?" and then approaches, as an asymptote, "definitely true."
posted by Caxton1476 at 10:09 AM on May 9, 2022 [2 favorites]

I think there was a long math journal version of this biography a few years ago that this article is heavily based on without crediting because I'm struck finding all the anecdotes very familiar.

But it was still a good read for a nonmathematician. The metaphorical bit about water levels reminds me of Terence Tao writing about how to solve math problems by pulling a string but it's under water (I forget the specifics but it's probably Googleable).

The Yoneda lemma metaphor is also neat because in sociology there's an adage (by Weber or Adorno, I forget): it's a mistake to analyze society in terms of individuals but in terms of society itself, or in Marxist terms, social relations. And furthermore, the whole eschewing of mathematical examples (and stretched to his own criticism of lived experience as a human philosophy/theory) seems to be explained by Grothendieck's particular philosophical commitment to category theoretic ideas.
posted by polymodus at 1:12 PM on May 9, 2022 [1 favorite]

I really wish I could understand the/ some math from this and Esc’s other neat post. For anyone feeling that way, I chanced upon this draft by Mumford:

https://www.dam.brown.edu/people/mumford/blog/2014/Grothendieck.html

It quickly got too difficult for me too fast, but at the bottom there’s a comment explaining one of the big ideas by Eric Zaslow. Also very tough for me, but with great effort it felt like I could just glimpse a tiny bit of the stakes in some part of Grotendieck. YMMV.
posted by rudster at 1:24 PM on May 9, 2022 [1 favorite]

One of my friends from college went on to be professional mathematician in differential geometry (specifically, generalized complex geometry) but when we were in undergrad, he had taken a course in (I think) algebraic geometry where he first encountered Grothendieck. I remember very clearly how my friend described Grothendieck's work as building up an enormous edifice of "structure" that eventually took on the form of a giant mech suit, all while miming being a giant robot clomping around and smashing things. To this day, I can't hear of Grothendieck without thinking of a guy in a giant robot suit made of mathematical structure going around smashing other math.

Meanwhile, from the article: "For a while, he still did occasional private mathematical work". I'm really, really curious what this "private mathematical work" entailed. Was it like being a private detective but instead of solving cases, he's proving theorems? Does a mysterious woman appear in his office one day, asking him to find the missing group from her homological chain complex, only to be subsequently pulled into a web of deceit, the seedy underbelly of the city, and/or a deadly game of cat and mouse?
posted by mhum at 1:34 PM on May 9, 2022 [7 favorites]

I liked this piece a lot. Grothendieck was a titan of mathematics, and I think this piece does him justice in that regard (if anything, I suspect it might understate how big of a deal his work was, at least according to my algebraic geometer friends). His eccentrism and abandonment of his family strike me as tragic. I did some snooping around, and was saddened to see that his son John, with whom he basically had zero meaningful contact, went on to do research for Raytheon. I imagine Grothendieck with his staunch anti-war views would've disapproved if he had cared about keeping a relationship with his family.

I actually appreciated the math metaphors a lot. I worked on mathematical statistics a little bit during my PhD, but have trended more and more towards applied work as time goes on. I felt the metaphors really captured the weight of his discoveries without getting too technical -- I'm ok at math compared to the average person, and certainly would not have grasped his work otherwise.

Also, I am pretty sure that the assertion "[Grothendieck] rewrote definitions, even of things as basic as a point" is pure bullshit. Mathematics is based in part on a small number of "primitive" concepts, which is to say things that are undefined. The notion of "point" is probably the second-oldest of these, after the notion of "number."

I went down the rabbit hole and found this link that gives some technical context as to how Grothendieck redefined space. Redefining commonly understood facts to have stronger, more generalizable/abstractable definitions is actually a holy grail within mathematics. Consider how students learn about numbers: first it's whole numbers, then maybe fractions or decimals. Then eventually negative numbers, then complex numbers, then imaginary numbers, etc. These developments were not always part of humanity's understanding of numbers, and finding out ways to strengthen/generalize definitions to be more broadly inclusive of edge cases is part of the magic and beauty of math.

It's like an essay about Picasso's paintings, for people who are blind.

I agree with this, although perhaps not in the way you intended. Why not describe Picasso's work for the visually impaired? His artistic impact was far more than just visual, and I didn't really appreciate how meaningful his work was until I read, yes, essays about it.
posted by bongerino at 4:38 PM on May 9, 2022 [6 favorites]

He had a machine installed that helped make taramosalata...

I could go for some of that right now if it was within reach...
posted by ovvl at 5:12 PM on May 9, 2022

This piece Can one explain schemes to hipsters? plays on the title of David Mumford’s piece Can one explain schemes to biologists which rudster mentioned above, and touches on the issues of communicating about mathematics to an audience which has little familiarity with the underlying concepts.

By the way, David Mumford was my father’s algebraic geometry Ph.D. thesis advisor.
posted by larrybob at 5:46 PM on May 9, 2022 [2 favorites]

archive link, fwiw...

speaking of categories[1,2] and language (and touching on the yoneda lemma!) tai-danae bradley (@math3ma) gives a nice introduction to her phd thesis here: At the Interface of Algebra and Statistics

Meanwhile, from the article: "For a while, he still did occasional private mathematical work". I'm really, really curious what this "private mathematical work" entailed.

bradley, for one, as a 'private mathematician' (for google's recently spun out sandbox) is helping explicate large language ML models[3]

Was it like being a private detective but instead of solving cases, he's proving theorems? Does a mysterious woman appear in his office one day, asking him to find the missing group from her homological chain complex, only to be subsequently pulled into a web of deceit, the seedy underbelly of the city, and/or a deadly game of cat and mouse?

or maybe just dropping papers (involving derivations of the operad of topological simplices ;) on A New Perspective of Entropy[4]

also btw, re:

Mazur suggests that it’s possible to glimpse the essence of Grothendieck’s approach to mathematics by looking at two concepts—categories and functors. A category can be thought of almost as a grammar: take triangles, perhaps, and understand them in terms of their relationship to all other triangles. The category consists of objects, and relationships between objects. The objects are nouns and the relationships are verbs, and the category is all the ways in which they can interact. Grothendieck’s discoveries opened up mathematics in a way that was analogous to how Wittgenstein (and Saussure) changed our views of language.

A functor is a kind of translation machine that lets you go from one category to another, while bringing along all the relevant tools...

@math3ma: "Functors in category theory and their names..." :P
posted by kliuless at 12:12 AM on May 10, 2022 [1 favorite]

No post about Grothendieck would be complete without a link to the Grothendieck Googling Twitter account.
posted by Johnny Assay at 4:59 AM on May 10, 2022 [1 favorite]

Calculus might be the most advanced math people ever see, but people use and rely on advanced, modern math all day every day.
posted by mhoye at 5:23 AM on May 10, 2022

"Using complex coordinates (z,w), a plane has four real dimensions and taking out a point, what's left is topologically a three dimensional sphere."

Huh. Why is that?
posted by bfields at 6:28 AM on May 10, 2022

"Using complex coordinates (z,w), a plane has four real dimensions and taking out a point, what's left is topologically a three dimensional sphere."

Huh. Why is that?

They're not homeomorphic, but they are homotopy equivalent.
posted by gleuschk at 9:16 AM on May 10, 2022 [1 favorite]

Very strange to read this article as these names and this terminology has always been a part of my life.

My dad is a retired algebraic geometry person. I spent time at IHES as a kid in the mid1970s and so many of the people mentioned in the article were around then. And those names were part of my dad’s everyday conversation for his entire career: Barry Mazur, David Mumford, Deligne, Weil, and so on.
posted by sciencegeek at 3:48 PM on May 10, 2022 [1 favorite]

Two Titans https://www.math.columbia.edu/~woit/wordpress/

I occasionally check Peter Woit's theoretical physics blog out of curiousity/procrastination, and today he also mentioned OP's New Yorker article as a segway into discussing Edward Witten's reflections on why he got involved in string theory.
posted by polymodus at 4:18 PM on May 12, 2022