Notably, the NYU professors have made a statement. It has a few notable facts which do not reflect well on OpenAI They were not willing to give a direct answer as to whether or not their user...
Notably, the NYU professors have made a statement. It has a few notable facts which do not reflect well on OpenAI
They were not willing to give a direct answer as to whether or not their user prompts were used for training in the model.
They wanted Levent removed from the NYU paper, seemingly for the sole reason that Levent works at Anthropic.
They threatened Tristan with “ruining his career” if he didn’t give in to one of 2 options they proposed for him.
I think it’s fully possible that their user prompts had little realistic influence in the model, however the threats and random personal attack on one of the coauthors seemingly for just being associated with Anthropic look pretty childish regardless.
We (the researchers and the agents) did not see any of their work through any means until they released it publicly — in particular, no specific user data was accessed in order to solve this problem. While unlikely, we cannot rule out that de-identified data derived from their usage of our products helped improve our models. However, our proofs differ significantly and even the precise results proved are different in the Euler case (forced vs unforced).
Our system produced an analytical proof and a Lean formalization that an initially smooth fluid at rest can develop a singularity in a finite time. The fluid has a smooth force applied to it, and its energy remains finite through the entire dynamics, from rest to the formation of the singularity. This resolves the Navier–Stokes Millennium Prize problem by establishing statement “C” (and also “D”) in the official Millennium Prize formulation.
The solution is a vortex, a spinning swirl of fluid, that spirals inward and gets increasingly elongated, like spaghetti. This central region shrinks while it speeds up in such a way that its energy still stays finite, as required by the laws of physics. The technical challenge is for the equations to develop the breakdown through the motion of the fluid itself, rather than, for example, us putting in an infinite force by hand. More mathematically, the terms in the Navier–Stokes equations that describe the motion—acceleration, pressure gradients, momentum transfer, viscosity—must both become big yet cancel in a precise way. This detailed balance leaves a smooth external force even as the velocity of the fluid grows without bound.
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Since August 28 we have been training a new internal model that has exhibited unprecedented performance in our benchmarks, including mathematics. This model’s training is ongoing and its performance continues to improve.
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On Tuesday, September 1, we heard rumors that two Millennium Prize problems had been resolved. Inspired by these rumors and by the step change in performance of our internal model, we launched an effort to evaluate it on all open Millennium Prize problems and a few other high-impact problems.
We used a system of coordinating agents powered by our internal model. The agents had access to tools such as the ability to read from a cached version of the internet and the ability to run code. Agents were subdivided into groups with the ability to communicate within the group. The groups varied in size, and the group that produced the Navier–Stokes resolution involved on the order of 10,000 concurrent agents. At all times we maintained the same strict safeguards that we apply to all our frontier model evaluations, including monitoring and isolation.
For each problem, we prompted different groups of agents with different variants of the problem statement, covering all variants of the problem. For the Navier–Stokes problem, we suggested versions “A” and “B” (particular forms of the Navier–Stokes problem which would result in a proof) and versions “C” and “D” (which would result in a disproof) to separate groups of agents.
[...]
The agents arrived at their resolution on Saturday, September 5, about 88 hours after the first agents were launched. Lean formalization and verification took an additional 17 hours via GPT‑6 Astra.
Across all attempted problems, the agents sent 4.9 million messages and used about 300 billion output tokens. In the process of resolving the Navier–Stokes problem, the agents sent 2.7 million messages and used approximately 130 billion output tokens.
[...]
Our effort began on September 1st after hearing a rumor which we later realized was related to Levent Alpöge, an Anthropic employee, and Tristan Buckmaster, a math professor at NYU. After the completion of our full project and Lean verification (on September 6th), believing from the rumor they also had a solution of Navier–Stokes, we reached out to them to offer a concurrent release of our result and to recognize their priority in a joint announcement. At that point we found out that they had a resolution of the forced Euler problem. In these discussions we offered them visibility into all of the prompts we used and later to see the proof. We recognize the priority of their work on forced Euler and congratulate them on their remarkable mathematical achievement.
[...]
Our goal in releasing this result is to report on the substantial progress of our AI models. We do not intend to claim the Millennium Prize for this result.
This seems promising! It's always been unclear which problems in the world are "difficult" and which are "difficult for humans", so it's good to see empirical evidence of that here. It's...
This seems promising! It's always been unclear which problems in the world are "difficult" and which are "difficult for humans", so it's good to see empirical evidence of that here. It's unfortunate that many of our present day problems are caused by poor governance and resource allocation, rather than mathematics, so this doesn't seem like it necessarily helps a tonne. Still neat though!
So now that a counterexample has been found, I guess the follow-up questions would be: why did the solution work, and what's the mathematical/physical implications? We've always known the...
So now that a counterexample has been found, I guess the follow-up questions would be: why did the solution work, and what's the mathematical/physical implications?
We've always known the Navier-Stoke equation is not an accurate description of real fluids, and so is bound to deviate at microscopic scale. Personally, I think of this problem as an instance of a more general question: whether an approximation of physical systems based on continuous assumptions (which is a common approach for mafhematically modeling reality) can be guaranteed to be well-bahaved and not produce infinite values, or if any continuous models of reality would always be in some mathematical sense "unphysical".
The details are beyond my understanding so I look forward to the various blogs and videos that will help explain this results in layman terms at some point.
The human drama side of this story kinds of reminds me of the AI vs. artist debate since the beginning. I still don't think AI will replace human but the lines between human and machine contributions are increasingly blurred. Likely hyperbolic but this feels like the introduction of steam engine, except for intellectual labors.
The very loose, extremely outdated, ELI5 explanation I'd received about this years ago is that: we expected that, in order to arrive at this result, human mathematicians would need to develop...
[...] and what's the mathematical/physical implications?
The very loose, extremely outdated, ELI5 explanation I'd received about this years ago is that: we expected that, in order to arrive at this result, human mathematicians would need to develop understanding/tools which would allow them to build the frontiers of our understanding further. Now that an LLM has (seemingly, we still need to wait for peer review) produced a result, it's unclear whether the overall endeavour will be fruitful in the expected way.
More directly, here's an excerpt from a random redditor talking about the problems.
[Q:] A common trope in mathematics is about single proofs or theorems that can change the world forever. [...] What about the others problems, what would proving/refuting them mean?
[R:] Any proof that Navier-Stokes is regular, or blows up, would probably generalize to make progress on a lot of other PDEs. But I am also skeptical that people doing water simulations would learn very much from it.
... and in reading that, I found an old blog post by Terence Tao about the importance behind that Millennium Prize problem.
To say that it goes over my head would be an understatement. The interested reader should set out on a clear autumn night, travelling at least an hour past the furthest reaches urban light, aiming for any mountain clearing of at least 1200m in elevation. Then -- with the assistance of a dedicated amateur astronomer, armed with a high power optical telescope mounted on a gyroscopically assisted tripod, observing towards the southern sky -- they would stand a decent chance of catching a glimpse of my understanding of Tao's blog post in the rough orbital vicinity of Neptune.
But my broad takeaway is that it backs the notion that solving these problems was supposed to result in more than just kudos and a novelty sized cheque, and instead for it to signify that the scaffolding had now been set for further expansion into novel mathematical problems. Not -- to directly quote world renound mathematician Terence Tao, since I don't want to come off as an incompetent fool, spouting off condescension towards Sam Altman's precious child -- "a mere viral social media post advertising some benchmark progress".
Notably, the NYU professors have made a statement. It has a few notable facts which do not reflect well on OpenAI
They were not willing to give a direct answer as to whether or not their user prompts were used for training in the model.
They wanted Levent removed from the NYU paper, seemingly for the sole reason that Levent works at Anthropic.
They threatened Tristan with “ruining his career” if he didn’t give in to one of 2 options they proposed for him.
I think it’s fully possible that their user prompts had little realistic influence in the model, however the threats and random personal attack on one of the coauthors seemingly for just being associated with Anthropic look pretty childish regardless.
Yea, it’s a mess…
Here is an update from the OpenAI researcher.
They did write this in the blog post:
From the article:
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[...]
[...]
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This seems promising! It's always been unclear which problems in the world are "difficult" and which are "difficult for humans", so it's good to see empirical evidence of that here. It's unfortunate that many of our present day problems are caused by poor governance and resource allocation, rather than mathematics, so this doesn't seem like it necessarily helps a tonne. Still neat though!
So now that a counterexample has been found, I guess the follow-up questions would be: why did the solution work, and what's the mathematical/physical implications?
We've always known the Navier-Stoke equation is not an accurate description of real fluids, and so is bound to deviate at microscopic scale. Personally, I think of this problem as an instance of a more general question: whether an approximation of physical systems based on continuous assumptions (which is a common approach for mafhematically modeling reality) can be guaranteed to be well-bahaved and not produce infinite values, or if any continuous models of reality would always be in some mathematical sense "unphysical".
The details are beyond my understanding so I look forward to the various blogs and videos that will help explain this results in layman terms at some point.
The human drama side of this story kinds of reminds me of the AI vs. artist debate since the beginning. I still don't think AI will replace human but the lines between human and machine contributions are increasingly blurred. Likely hyperbolic but this feels like the introduction of steam engine, except for intellectual labors.
The very loose, extremely outdated, ELI5 explanation I'd received about this years ago is that: we expected that, in order to arrive at this result, human mathematicians would need to develop understanding/tools which would allow them to build the frontiers of our understanding further. Now that an LLM has (seemingly, we still need to wait for peer review) produced a result, it's unclear whether the overall endeavour will be fruitful in the expected way.
More directly, here's an excerpt from a random redditor talking about the problems.
... and in reading that, I found an old blog post by Terence Tao about the importance behind that Millennium Prize problem.
To say that it goes over my head would be an understatement. The interested reader should set out on a clear autumn night, travelling at least an hour past the furthest reaches urban light, aiming for any mountain clearing of at least 1200m in elevation. Then -- with the assistance of a dedicated amateur astronomer, armed with a high power optical telescope mounted on a gyroscopically assisted tripod, observing towards the southern sky -- they would stand a decent chance of catching a glimpse of my understanding of Tao's blog post in the rough orbital vicinity of Neptune.
But my broad takeaway is that it backs the notion that solving these problems was supposed to result in more than just kudos and a novelty sized cheque, and instead for it to signify that the scaffolding had now been set for further expansion into novel mathematical problems. Not -- to directly quote world renound mathematician Terence Tao, since I don't want to come off as an incompetent fool, spouting off condescension towards Sam Altman's precious child -- "a mere viral social media post advertising some benchmark progress".