> The author of the post is Terence Tao that is the best live mathematician. If he says it's a "breakthrough", it's a breakthrough.
I think it's pretty silly to say that Tao is the best living mathematician, but even if he were I don't think that this would be a useful way to think about things.
> I think it's pretty silly to say that Tao is the best living mathematician
I agree. I was going to write "one of the best" or "probably the best" or something like that. It's like discussing if Messi or Maradona were the best football players. [1]. Anyway, all three of them are quite good.
> but even if he were I don't think that this would be a useful way to think about things.
I also agree. It's just and argument by authority. For a decition that would change the lives of millons of persons and has a lot of subtle tradeoff and unknown unknowns, I'd ask for a better justification and evidence. But deciding if this is a breakthrough or not, may only change the lives of a few graduate students and a few grown up mathematicians, so I'm happy to take the word of Tao.
No matter what you think, best living mathematician is what other mathematicians say about him.
But I'll humor you. How would you prefer we say things?
- Highest IQ on record.
- One of only 3 people to score over 700 on the Math SAT before he was 8. He had the highest score of the three with 760.
- At ages 10, 11, and 12 he set the record for winning Bronze, Silver, and Gold respectively in the International Math Olympiad. After that he lost interest.
- PHD from Princeton at 21.
- Full professor at UCLA at 24. This is a record.
- He is a respected leader in at least a half-dozen areas of mathematics. He regularly publishes in many more. It is unusual for a mathematician to have significant publications in 2 areas.
- Wikipedia lists 28 major prizes for him. Excepting Australian of the Year in 2022, all are major mathematics prizes. No other mathematician comes close.
- Once you exclude junk journals, Tao publishes papers faster than any other mathematician. And his are all good.
- Tao's papers show up in the popular press more often than the next 3 mathematicians combined.
And so on.
At what point is there a better way to think about this than, "best live mathematician"?
(And yeah, until I began noticing Tao, I would have also thought that a silly way to think...)
The idea that Tao has accomplished more than, say, Serre because the latter, who won the Fields medal at 27, only received his PhD at 25 and his bachelor's at 22 while the former received his PhD at 21 and his bachelor's at 16 is so absurd that it can be refuted merely by alluding to it.
Serre is indeed a top mathematician. (I'm actually surprised to find out that he's still alive!)
At this point Tao only has 3/4 his number of publications, similar numbers of textbooks, a similar number of awards (using https://mathshistory.st-andrews.ac.uk/Biographies/Serre/ to count awards), and so on. I'd count Tao as having more of what I see as major breakthroughs, but that is subjective. But then again, Tao is half of Serre's age.
Yeah. I still think it is fair to put Tao in the same tier as Euler, Gauss and Hilbert.
Sorry, I think this style of hagiography is completely goofy, there's nothing else to say about it. And I'm sure it's not even true that he has the "highest IQ on record."
To make a claim like "greatest living mathematician" it would be more appropriate to talk about his actual research accomplishments and how they compare to canonical figures like Gromov, Milnor, Serre, Smale, Yau, among many younger counterparts.
But, personally, I think defending any such claim for any particular person is kind of a juvenile exercise. I am a mathematician and among the slight minority of mathematicians I know who would take the question seriously, a minority of them would select Tao. Which of course isn't to say that he isn't universally considered a top mathematician.
> And over and over I come away with Hinton explaining mathematically and conceptually whats going on in LLMs
Every time I see Hinton talking about LLMs he's just anthropomorphizing whatever 'mathematics' is going on there. He's a great researcher but tbh I think he's a really silly guy
Sometimes people act like guys like Bill Gates or Elon Musk are coming from deep personal scientific knowledge and accomplishment, but they're absolutely nothing compared to Simons. His contributions to geometry in the 60s and 70s, from minimal surfaces to Berger's classification of special holonomy to Chern-Simons theory, were fundamental and are still well-remembered. His name would be known even if he'd never gone into finance or philanthropy.
Bill Gates did graduate study in math and computer science as a first-year undergraduate at Harvard College, and published the fastest pancake-sort algorithm (held the record for 30 years), before dropping not to start Microsoft. And of course he invented Microsoft's early technology, which advanced the state of the art. He was highly likely to be a great computer scientist if he chose to stay in school.
I can read both the Gates-Papadimitriou paper and Simons' work, and it doesn't compare. Maybe Gates could have been a great scientist, who knows, but no matter how many advanced classes he took, he never was. It doesn't even matter if he got good grades in them.
I was classmates with several people who did well in Math 55, and knew some people who were teaching fellows for it. Very smart folks but they themselves would not have compared themselves to Simons as mathematicians, esp. at age 18.
Friend, there's a world of difference between being a smart guy who completed a hard math class and being one of the world's best researchers in differential geometry.
> Bill Gates had the aptitude to be a leading researcher at probably anything if he set out to do it. He was already doing coding whilst working on math courses, so his attention was divided. Simons focused 100% on math until doing trading.
Ok, I guess you're right. A smart guy who completes a hard math class and is even also doing coding can probably do anything he wants.
Bill Gates had the aptitude to be a leading researcher at probably anything if he set out to do it. He was already doing coding whilst working on math courses, so his attention was divided. Simons focused 100% on math until doing trading.
"In many cases, AlphaFold predictions matched experimental maps remarkably closely. In other cases, even very high-confidence predictions differed from experimental maps on a global scale through distortion and domain orientation, and on a local scale in backbone and side-chain conformation. We suggest considering AlphaFold predictions as exceptionally useful hypotheses."
Yep, I know Paul Adams (used to work with him at Berkeley Lab) and that's exactly the paper he'd publish. If you read that paper carefully (as we all have, since it's the strongest we've seen from the crystallography community so far) they're basically saying the results from AF are absolutely excellent, and fit for purpose.
(put another way: if Paul publishes a paper saying your structure predictions have issues, and mostly finds tiny local issues and some distortion and domain orientation,r ather than absolutely incorrect fold prediction, it means your technique works really well, and people are just quibbling about details.)
I also worked with the same people (and share most of the same biases) and that paper is about as close to a ringing endorsement of AlphaFold as you'll get.
I don't know Paul Adams, so it's hard for me to know how to interpret your post. Is there anything else I can read that discusses the accuracy of AlphaFold?
I can't find the link at the moment but from the perspective of the CASP leaders, AF2 was accurate enough that it's hard to even compare to the best structures determined experimentally, due to noise in the data/inadequacy of the metric.
A number of crystallographers have also reported that the predictions helped them find errors in their own crystal-determined structures.
If you're not really familiar enough with the field to understand the papers above, I recommend spending more time learning about the protein structure prediction problem, and how it relates to the epxerimental determination of structure using crystallography.
Thanks, those look helpful. Whenever I meet someone with relevant PhDs I ask their thoughts on AlphaFold, and I've gotten a wide variety of responses, from responses like yours to people who acknowledge its usefulness but are rather dismissive about its ultimate contribution.
The people who are most likely to deprecate AlphaFold are the ones whose job viability is directly affected by its existence.
Let me be clear: DM only "solved" (and really didn't "solve") a subset of a much larger problem: creating a highly accurate model of the process by which real proteins adopt their folded conformations, or how some proteins don't adopt folded conformations without assistance, or how some proteins don't adopt a fully rigid conformation, or how some proteins can adopt different shapes in different conditions, or how enzymes achieve their catalyst abilities, or how structural proteins produce such rigid structures, or how to predict whether a specific drug is going to get FDA approval and then make billions of dollars.
In a sense we got really lucky because CASP has been running so long and with some many contributors that it became recognized that winning at CASP meant "solving protein structure prediction to the limits of our ability to evaluate predictions", and that Demis and his associates had such a huge drive to win competitions that they invested tremendous resources and state of the art technology, while sharing enough information that the community could reproduce the results in their own hands. Any problem we want solved, we should gamify, so that DeepMind is motivated to win the game.
this is very astute, not only about deepmind but about science and humanity overall.
what CASP did was narrowly scope a hard problem, provided clear rules and metrics for evaluating participants, and offered a regular forum in which candidates can showcase skills -- they created a "game" or competition.
in doing so, they advanced the state of knowledge regarding protein structure.
how can we apply this to cancer and deepen our understanding?
specifically, what parts of cancer can we narrowly scope that are still broadly applicable to a complex heterogenous disease and evaluate with objective metrics?
[edited to stress the goal of advancing cancer knowledge, not to "gamify" cancer science but to create structures that inivte more ways to increase our understanding of cancer.]
Important caveat: it's only about 70% accurate. Why doesn't the press release say this explicitly? It seems intentionally misleading to only report accuracy relative to existing methods, which apparently are just not so good (30%, 50% in various settings).
https://www.fastcompany.com/91120456/deepmind-alphafold-3-dn...
They also had a headline for Alphazero that convinced everyone that they crushed Stockfish and that classical chess engines were stuff of the past, when in fact it was about 50 elo better than the Stockfish version they were testing against, or roughly the same as how much Stockfish improves each year.
I think Alphazero is a lot more interesting than Stockfish though. Most notably it lead me to reevaluate positional play. Iirc A0 at around 2-3 ply is still above SuperGM Level which is pretty mind-blowing. Based on this I have increased my strategy to tactics ratio quite a bit. FWIW Stockfish is always evolving and adapting and has incorporated ideas from A0.
Hmm: NNUE was introduced in 2018, the AlphaZero preprint 2017, AlphaGo 2015-2016. I checked this because my memory claimed that it was AlphaGo's success that sparked the new level of interest in NN evaluation.
Wouldn't surprise me if AlphaZero's improvements had no influence in that timeline, but for AlphaGo it would.
The original NNUE paper cites AlphaZero[0]. The architectures are different because NNUE is optimized for CPUs and uses integer quantization and a much smaller network. I don't think one could credibly claim that it would have come about if not for Google making so much noise about their neural network efforts in Go, Chess and Shogi.
For whatever it's worth, the NNUE training dataset contains positions from Leela games and several generations of self-play. Stockfish wouldn't be where it is if not for Google's impact. AlphaFold will likely have a similar impact on our understanding of protein structure. I don't know why everyone is so offended by them puffing their chests out a little bit here, the paper's linked in the article.
The first thing I'd recommend is constantly evaluating positions from a strategic POV ("Evaluate like a GM" is a good book, alternatively look at a lot of positions and evaluate like you were an engine and then engine check).
Second (or first if you lack even the basics to do said evaluation) is understand strategic concepts. A good starting point would be "Simple Chess" the next step would be pawn structures ("Power of Pawns" -> "Chess Structures" would be my recommendations, the latter is probably the greatest chess book in recent times imo). There's also many Chessable courses, I'm quite fond of "Developing Chess Intuition" by GM Raven Sturt and the "Art of..." series by CM Can Kabadayi for lower rated players. The sky is the limit, there's good books all the way up, for example "Mastering Chess Strategy" usually recommended for 2000+ ELO
Third study great positional players like Carlsen, Karpov, Petrosian etc.
I'd say the most important thing to realize is that just like tactics puzzles, there's strategic puzzles but they are not as obvious.
It definitely deserved a lot of praise, but the testing situation wasn't really against a fully fledged stockfish running on similar hardware, but one that, among other things, had no opening library.
The issue is not whether alphazero was impressive, but that we should be careful about the specific claims of the press releases, as they are known to oversell. The whole thing would have been impressive enough if the games had been against the last release of stockfish with good hardware, just for the way it played.
And then what happened is AlphaZero changed the professional game in various interesting ways, and all its ideas were absorbed into Stockfish. A little bombast is forgivable for technology that goes on to have a big impact, and I don’t doubt it’s the same story here.
That's not true at all, Stockfish still uses only human heuristics for search and NNUE for eval, a completely different architecture than alphazero and derived from the Yu Nasu Shogi engine.
It's a neural network trained on self-play games (many of them lifted from Leela Zero). I get that it's a different shape of network, but people really seem touchy about crediting Google with the kick up the bum that led us here. AlphaZero had a massive effect on chess globally, whatever people think about its press releases. My main point is that people should update the heuristic that wastes energy arguing about bold claims when clearly something amazing has happened that everyone in the industry will react to and learn from.
I don't have any particular thoughts about DeepMind's board game algorithms or how they were advertised, but even if I happened to think it was the most innovative and influential research in years, I'd still ask for honest communication about the work. It's part of being a healthy research community - although clearly the AI community falls well short on this, and nobody could say it's only DeepMind's fault.
Where do the evaluations come from? The idea that Stockfish isn't benefiting hugely from Google having created and advertised AlphaZero is preposterous, can we please just stop?
Okay, well, no sale I guess. Stockfish's training dataset is mostly self-play games from an engine directly inspired by AlphaZero. It moved to neural network evaluation after a fork based on a paper that cites AlphaZero. It plays chess more like AlphaZero than Stockfish 11. Yes, it's extremely interesting that it continues to edge out Leela with a fast, rough approximation of the latter's evaluation but much faster search. But it (and human chess) wouldn't be where it is today without AlphaZero, and I was originally responding to someone dismissing it based on the perceived over-zealousness of its marketing, as people seem to want to do with TFA. I merely submit that both of these Google innovations are exciting and impactful, and we should forgive their presentation, which nevertheless has been kind enough to link to the original papers which have all the information we need to help change the world.
Computational models have been shown to improve with computing power though, right?
It's a tongue in cheek comment about how fast models have been improving over the last few years, but I forgot HN scrutinizes every comment like it's a scientific paper.
The ROI exponentially decreases with language models. At this point, each percentage point of accuracy costs you tens of billions, and the projections show a solid wall approaching with the current implementation tricks.
It's an off the cuff comment, at this point though, HN apparently needs to bully everyone who refuses to go along with the zeitgeist, as if being negative near a thing would destroy it.
Just "Alpha-- models" in general?? That's not a remotely reasonable way to reason about it. Even if it were, why should it stop DeepMind from clearly communicating accuracy?
The way I think about this (specifically, deepmind not publishing their code or sharing their exact experimental results): advanced science is a game played by the most sophisticated actors in the world. Demis is one of those actors, and he plays the games those actors play better than anybody else I've ever seen. Those actors don't care much about the details of any specific system's accuracy: they care to know that it's possible to do this, and some general numbers about how well it works, and some hints what approaches they should take. And Nature, like other top journals, is more than willing to publish articles like this because they know it stimulates the most competitive players to bring their best games.
(I'm not defending this approach, just making an observation)
I think it's important to qualify that the relevant "game" is not advanced science per se; the game is business whose product is science. The aim isn't to do novel science; it's to do something which can be advertised as novel science. That isn't to cast aspersions on the personal motivations of Hassabis or any other individual researcher working there (which itself isn't to remove their responsibilities to public understanding); it's to cast aspersions on the structure that they're part of. And it's not to say that they can't produce novel or important science as part of their work there. And it's also not to say that the same tension isn't often present in the science world - but I think it's present to an extreme degree at DeepMind.
(Sometimes the distinction between novel science and advertisably novel science is very important, as seems to be the case in the "new materials" research dopylitty linked to in these comments: here https://www.404media.co/google-says-it-discovered-millions-o...)
Initial previews of Republic in 2000 focused upon the purported level of detail behind the game's engine, the "Totality Engine". Described as "the most advanced graphics engine ever seen, (with) no upper bound of on the number of polygons and objects", it was claimed the game could "render scenes with an unlimited number of polygons in real time".[14] Tech demonstrations of Republic at this time showcased a high polygonal level of detail,[21] with the claim that players would be able to zoom smoothly from the buildings in Novistrana to assets such as flowers upon the balconies of buildings with no loss of detail.[22] The game was further purported to have artificial intelligence that would simulate "approximately one million individual citizens" at a high level of detail,[23][19] each with "their own unique and specific AI" comprising "their own daily routine, emotions, beliefs and loyalties"
I feel like it's always worth bearing in mind when he talks about upcoming capability.
Thanks, I'd never heard about this before. I definitely think it helps in understanding his commentary. It's really a shame that DeepMind picked up his communication style.
Which specific AlphaX model evolved like that? Most of the ones that were in the press had essentially a single showing, typically very good, but didn't really improve after that.
That reflects kind of badly on NeurIPS if true. I first became aware of him over 10-15 years ago when he was pushing his "mathematical universe" idea, which is quite simply the most vapid and contentless idea I've ever seen in physics. (I don't think it's at all surprising that someone like him would be drawn to the AI community, or vice versa!)
He has some PhD students doing what seems to me to be reasonable work that gets published at NeurIPS. Somewhat speculative, of the flavor you’d expect from theoretical physicists doing AI work, but it’s at least speculation about concrete technologies that exist, not pure metaphysics. For example, there’s a recent paper proposing and validating in small models a possible mechanism to explain some power laws seen in NN scaling curves [1]. May turn out to be wrong, but doesn’t seem nutty to me. On the other hand, I’d guess these papers specifically are probably not what got him famous in AI circles. For that his general self-created role as AI futurist is probably more responsible [2]. I tend to avoid that kind of stuff, but staking out a debate position on questions like “how smart could AI get? Will it kill us all?” is the kind of stuff the general public likes to hear.
I think that paper is pretty weak, but at least in terms of content it's still night and day compared with the nature of Tegmark's clout-chasing in his physics days. But it's pretty grim if that's the kind of paper that made him popular at NeurIPS.
Brute searching all possible mathematical constructs, theorems, etc. to see which one fits the problem would probably take you practiacally an infinite amount of time.
This works tbh, how I see it, very closely to how a human does - via "instinct" it gathers relevant knowledge based on the problem and then "brute searches" some combinations to see which one holds. But this "intuition" is the crucial part where brute search completely fails and you need very aggressive compression of the knowledge space.
> Brute searching all possible mathematical constructs, theorems, etc. to see which one fits the problem would probably take you practiacally an infinite amount of time.
I think it's pretty silly to say that Tao is the best living mathematician, but even if he were I don't think that this would be a useful way to think about things.