Alexander Grothendieck Revolutionized 20th-Century Mathematics | Hacker News Reader

Alexander Grothendieck Revolutionized 20th-Century Mathematics

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152 pointsanujbans32 comments

One of my favourite Grothendieck stories from <https://www.ams.org/notices/200410/fea-grothendieck-part2.pd...>:

> 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."

I had to follow your link to get it: I hadn't realized that 57 is not prime. At least I'm in good company.

Take 111 as an example.

It looks like a prime, but can be caught with the second-simplest test: sum of the digits is 12, which is divisible by 3. Hence it's divisible by 3.

(The simplest test being of course if the number is even and bigger than 2)

Edit: now that I think about it, probably should not have tried to impose ordering to the simplicity of tests. There's of course the divisibility by 5 test, which is even simpler.

In fact, most 2 digit numbers not divisible by 2, 3, or 5 are prime. [1] The only one that's likely to ruin your day is 7 * 13 == 91, but that's self-defeating because after you think about it long enough 91 falls victim to [2].

[1] https://til.andrew-quinn.me/posts/most-2-digit-numbers-not-d...

[2]: https://en.wikipedia.org/wiki/Interesting_number_paradox

John H Conway proved that the smallest number which looks prime, but isn’t is 91. https://youtu.be/S75VTAGKQpk?si=fCGilXECmCOy7T7R

“This is an important theorem, and a result I’m very proud of.”

I just noticed that it's 60-3 without any divisibility tests.

Tao's 27 prime was much more embarassing but understandable as he's no a calculator.

Savants are for things like remembering the first million primes. Someone like Tao or Grothendieck can't remeber them beyond 20, but it doesn't mean they can't actuly reason about them.

What's Tao's 27 prime again?

Was mentioned in a twin thread:

"27 is a Tao prime. Terence Tao suggested 27 was a prime number on The Colbert Report in 2014. He was likely very nervous."

there was some interview where he illustrated the idea of a twin prime pair using 27 and 29

It's referred to as the Grothendieck Prime for this reason.

27 is a Tao prime. Terence Tao suggested 27 was a prime number on The Colbert Report in 2014. He was likely very nervous.

As a curiosity, the contrast between Grothendieck and Ramanujan is very striking. One famous story about Ramanujan from Wikipedia (https://en.wikipedia.org/wiki/1729_(number)):

"Hardy stated that the number 1729 from a taxicab he rode was a "dull" number and "hopefully it is not unfavourable omen", but Ramanujan remarked that "it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways"."

They, of course, were very different personalities, doing very different mathematics with very different impacts on the field. I always found it interesting that Ramanujan seemed to be very comfortable with numbers, their properties, patterns (continued fractions) and Grothendieck was very comfortable with structures and their rhythms without paying attention to concrete examples.

For anyone interested in Grothendieck's opinions on kimchi …

https://mikepierce.github.io/grothendieck-kimchi/translation...

all mention of aekjeot (fish sauce) and saeujeot (tiny shrimp) seems to be elided here. kimchi in coastal regions (and most commercialized korean kimchi) has a strong tendency to have that in, so if you take grothendieck's recipe as is, you won't have the exported korean taste. northern kimchis have diverged materially in addition, due to north korea being fucked up. aekjeot you can just add but the procedure to add saeujeot traditionally is pretty fiddly

he mentions using whatever herbs he had in a european setting like juniper and rosemary but the canonical herb to add is korean chives and dropwort. never seen juniper or rosemary, frankly.

he does mention that the pepper is a specific variety in korea without exception. this is the korean chili, sun-dried and flaked. it's a very distinctive varietal, the taste will be very different without it

Likely because Grothendieck was a vegetarian, at least later in his life.

If anyone's interested in Grothendieck's writing, which is primarily in French, I threw his "Séminaires de Géométrie Algébrique" (SGA, Algebraic Geometry Seminars) and "Éléments de Géométrie Algébrique" (EGA, Elements of Algebraic Geometry) into an LLM to translate it to English. It's spotty in some sections, so I intend to do another pass, but it's better than my remedial French.

EGA: https://github.com/jcreinhold/ega (https://jcreinhold.github.io/ega/)

SGA: https://github.com/jcreinhold/sga (https://jcreinhold.github.io/sga/)

I maintain a website of scans of things Grothendieck wrote, pictures, etc. https://wstein.org/sga/

Interestingly, the way in which Grothendieck conceived of equality is nowadays being questioned, especially due to the rise of formalized mathematics and Lean. More concretely, there is this fun paper by Kevin Buzzard which deconstructs it: https://arxiv.org/abs/2405.10387

Money quote:

> In this paper I argue that the first assertion above is false, the second is dan- gerous, and the third is meaningless.

Happy to see that it's got the obligatory monk/wizard photo.

For more life and times stuff I also suggest Labatut's Cease to Understand the World book and https://theanarchistlibrary.org/library/konstantinos-foutzop...

That book is fiction with a factual veneer. I liked it a lot until I started realizing that many of the details were made up. Then I couldn't read any more. It was like when TwoSetViolin described what it was like for them to watch movies with musician characters played, unrealistically, by non-musician actors. You'd be watching the perfectly fine movie until you noticed that the bananas were blue instead of yellow, with nobody mentioning it. After that the movie made no sense any more.

I hated the movie Oppenheimer for the same reason.

> That book is fiction with a factual veneer.

Definitely, but do check the link.. I dug it up originally by trying to track down detail about the nonfiction background that the book is pulling from. Seems like the best short source, but I'd love to hear recs for a good biography. The autobiography that Groth is careful to say is not an autobiography is on my shelf and also in pdf form. Haven't read it yet, but I'm not sure it's the type of thing that's going to cover the descent into madness properly.

https://web.ma.utexas.edu/users/slaoui/notes/recoltes_et_sem...

There is an incredible (alas unfinished) multipart Grothendieck biography by the German mathematician Winfried Scharlau: Wer ist Alexander Grothendieck? Anarchie, Mathematik, Spiritualität, Einsamkeit. I think an English translation exists, at least for the first and in my opinion most interesting volume: Anarchy. It mostly deals with Grothendieck’s childhood and his parents, who lived unbelievably fascinating lives as anarchists in pre-war Berlin.

The other article looks interesting, though it too has "blue banana" errors. Laurent Schwartz's autobiography "A Mathematician Grapples With His Century" has a more believable account of Schwartz and Dieudonné's work with Grothendieck. The Grothendieck Circle website (https://webusers.imj-prg.fr/~leila.schneps/mitanni/grothendi... since they haven't done a good job renewing domains) has a lot of Grothendieck's own writings (mostly in French) and Wilfried Scharlau's biography (in German). I tried to read those some years back but my language skills weren't up to it. Machine translation is much better now than it was then, so I might try the lazy approach.

Interestingly, von Neumann's daughter was kind of shocked by the research the author did for the book The MANIAC; as a kid she carried graph paper in her pocket and Labatut had somehow found this out in his research and put it into the book, really blew her away I guess.

I read and enjoyed that book out of a general interest in the history of ideas, but admit I am not able to judge the underlying mathematics. Is the "fiction" part only related to descriptions of his mathematical contributions, or are there problems with the biographical information as well?

Personally, just from the implausible level of first-person detail throughout I thought it's immediately clear the book is exceptionally well-researched but still technically qualifies as historical-fiction. We don't necessarily know what these people thought/ate/did minute by minute on the important days at that level.

But I think the biggest "sin" in terms of mixing fact/fiction was mostly implied and not actually stated. What's implied is that Groth saw inside mathematics some kind of terrible truth that motivated him to stop working and withdraw from the world. I don't think it's stated explicitly, but due to proximity with other topics in the book, reader is invited to conclude that there was a discovery of some kind inevitable doom, possibly a super weapon, etc.

We don't know that, but in a lot of ways it might be more surprising if he never thought along those lines. My understanding is that the other limited sources really do say he was talking to God in dreams, preoccupied with apocalyptic visions, became more interested in physics, politics, religion, the problem of evil, hostile entities ambiguously demonic, etc

Very interested in reading your list of blue bananas in Oppenheimer.

Peter Woit (https://www.math.columbia.edu/~woit/wordpress/) has occasionally posted about Grothendieck's life and work. E.g.

Articles on his life: https://www.math.columbia.edu/~woit/wordpress/?p=7335

Two Titans (Grothendieck and Witten) - https://www.math.columbia.edu/~woit/wordpress/?p=12868

AMS Math articles on Grothendieck - https://www.math.columbia.edu/~woit/wordpress/?p=78

Super. I always wanted to learn about sheaves and schemes and the like, and this gives a simple introduction that really motivates digging deeper into the details.

It is also immediately clear why this plays a role in semantics for logics: although a ring is not that important in logic (I would think), the idea to study a theory through its syntactical consequences turned into semantics is very natural, and exactly what I do for abstraction logic as well, in particular via "valuation spaces". And it has the same property, once you set up everything the right way, things like completeness just automatically flow out of it.

Not one mention of why he chose to leave those top universities.... It was because of their connection to the military industrial complex.

Field medals are still handed out to people who deign to look upon his prophetic ramblings. I'm half-convinced the religious stuff probably unveils the geometric structure of the universe.

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