Alexander Grothendieck
theguardian.com
theguardian.com
The story of Grothendieck is a tragedy about a generational genius, not unlike Godel’s story. It’s deep and far reaching enough to stand on its own without AI hype making it appear more relevant.
So it seems valuable to me in that respect, especially if they can achieve what they want (logical inference-rule-invariant NN blocks, PL semantics invariant stuff, etc).
Anyway, I'm interested in your perspective/objections! (if it's technical that's fine too, I have a lot of maths background)
This idea has been well studied in mathematical physics, going back to Poincaré, where you work with Lie groups and Lie group actions on your action space. The reason this works, however, is you get a Lie algebra/Lie algebra action that more-or-less behaves like the tangent bundle so the same basic theory around optimization works.
The main problem is they’re generalizing in the wrong direction. Everything still works when you move to Lie groupoids/Lie algebroids. You still get something like a tangent bundle, so ideas like gradient descent or Euler-Lagrange equations still make sense. Thats not the case with a generic monad - in fact the authors don’t seem to acknowledge the fact that there is some work to do regarding compatibility between the monad and derivative to ensure that gradient-based optimization will still make sense.
So, basically, anyone who is familiar with the basics of optimization on manifolds or Lie groups will immediately recognize this approach as hopelessly naive. All they’ve _really_ managed to do is draw some diagrams and say “wouldn’t it be cool if these things were preserved by gradient descent.”
Thanks for taking the time to reply. Coincidentally, I'm doing a project with Arnold's book on CM at the moment, so that all makes perfect sense to me.
Edit: I need to think more before I comment, “A new kind of data science” was right there.
The original topos theory developed by Grothendieck and his collaborators in the 60s is quite pragmatic and served to define cohomology theories for varieties over finite fields. Later other people, coming from mathematical logic, distilled "elementary" axioms from there and developed another kind of topos theory that is pretty much divorced from algebraic geometry.
It’s a bit too long for the HN title submission but the actual article title in the Guardian is
“ ‘He was in mystic delirium’: was this hermit mathematician a forgotten genius whose ideas could transform AI – or a lonely madman?”
[1] https://en.wikipedia.org/wiki/Isaac_Newton%27s_occult_studie...
https://en.wikipedia.org/wiki/Alexander_Grothendieck#Manuscr...
Never heard of him before, RIP but this reads like the beginning of neal stephenson novel… interesting
Thought it was important to state this, as Grothendieck also wrote rather esoteric texts, Récoltes et Semailles as well as La Clé des Songes, which haven't been translated or even published.
Source: https://www.jp-petit.org/Nouvelles/Grothendieck.htm
- I know the story. He was a man viscerally opposed to militarism. He once said "he'd rather be shot than wear a uniform." One day a letter arrived at IHES in Bures, where the army's scientific services, called DRET at the time (Direction of Research and Technical Studies), now DGA (Delegation for Armament Applications), offered a grant of four thousand francs (650 euros). When he came across this paper, he flushed red, saying "No way we're accepting a penny from these people!" His colleagues tried to change his mind: "Listen, Alexandre, don't be so rigid. It'll pay for photocopies..."
- And then?
- He said, "It's not difficult, we'll put it to a vote. The IHES scientific council will decide whether or not to accept money from the soldiery. But if you accept this grant, I solemnly warn you: you'll have my resignation in your hands within the minute that follows."
- And what happened?
- They didn't take his threat seriously. The vote took place and the four thousand francs were accepted by a majority of one vote. His face then turned grey, hard as marble. He took a letterhead paper and simply wrote: "I have the honor to tender my resignation" then handed it to the council members and turned on his heel. The next day he didn't show up at his office, nor the day after. Paperwork began to pile up. There were letters from all over the world.
- He was a Fields Medal winner.
- His reputation was such that he attracted the greatest mathematicians on the planet to the Institute. For everyone, he was the beacon of Algebraic Geometry, illuminating the entire planet with all its light. At first, people thought it was depression or a disappearance. At IHES, he occupied an official apartment. After a week, they ended up calling a locksmith to open the door. The apartment was empty. Masses of his papers were found in a trash can. He had thrown everything away, his notes, his books, his reports, his correspondence.
- Incredible! ...
- Wait, weeks and months passed without anyone knowing where he had gone. You can't imagine the panic at the Institute. Scientists started calling from all corners of the world. They had to answer and admit that he had resigned. People wanted to know why he had acted this way, under what circumstances this had happened, where he had gone, what he was doing now. The most unbelievable rumors were circulating. At one point, they thought he had committed suicide, but as some people had met him, they had to face the facts: he was apparently still alive. We have a letter from him dated two years after his resignation from IHES addressed to a company providing organic fertilizers, where he complains that these do not meet the specified standards. It was indeed his signature and, it must be said, his style.
- And since then?
- Since then, nothing. The world's greatest mathematician simply vanished one fine day. He simply let the scientific community know that he wanted nothing more to do with this environment. He announced, through a letter he addressed to one of his former students, his decision to withdraw completely. As people had finally located him in this small village near Carpentras where he had rented a small farm, they hoped to flush him out by offering him a new prize, the Crafoord Foundation prize. This must have been in the early eighties. The amount was around forty million francs.
Alexander Grothendieck was indeed awarded the Crafoord Prize, which he rejected. (It was never worth 40 mil francs as the translation above claims. The original French put it at "40 briques" = 400,000, currency not specified, which is much closer to the more accurate 800K SEK ~ 800K FFr that he would have received. The fact that the full amount, 1,6M SEK, would be split between him and Pierre Deligne, whom Grothendieck had denounced, might have contributed to his decision.) Grothendieck's rejection letter was remarkably lucid and articulate: https://www.fermentmagazine.org/quest88.
"Although fifty-seven is not prime, it is jokingly known as the Grothendieck prime after a legend according to which the mathematician Alexander Grothendieck supposedly gave it as an example of a particular prime number." [1]
[1] https://en.wikipedia.org/wiki/57_(number)#In_mathematics
but the fun part is how the same happens to these four numbers -30
so 23, 29 twin with 31 then 37
my problem is that I don't know why this would be important
Something that caught my attention recalling Arthur Schopenhauer’s philosophy on the need for conceptual or apriori knowledge based proofs than empirical or derived in math.
[1] french, https://www.youtube.com/watch?v=V8BbFTEyvIw