I would say this is why formal proofs (and things like the Lean 4 libs) are so important, so that you can deconstruct the tower provably back into pieces you can understand. It shouldn't be possible to construct a formal proof you cannot destructure like this.
As a (crude) analogy, it's a bit like how you can prove the healthiness of a git tree because it's a graph of content hashes and the tree graph pointers are part of the hash. Imagine this but with a tree of knowledge.
> Because the core of the issue is that it may well not have solved it, but instead plagiarised the significant step of the result from other researchers
It's also true however that I haven't seen a single write up trying to discern what did more of the work in those AI chats - the prompts or the responses - bubble to the surface, also since we don't have access to them.
For example, if I prompt Codex with "Make me a website about strawberry cake" and nothing else, and OpenAI announces they have the best strawberry cake minutes before I launch, I'm not sure they plagiarized anything.
We just don't know if this is quibbling over "who prompted first" or if the researchers came up with anything strikingly original by themselves.
The researchers apparently spend a year or so working on this, and it builds off significant previous work, so it seems like it was a pretty significant amount of work that OpenAI may have trained on
I'd love to see an in depth analysis of how much OpenAI actually did, but I suspect we'll never see that because it would indicate at least some plagiarism which undermines a lot of what OpenAI is putting out in public
The American Mathematical Society credits the Spanish researchers Diego Córdoba and Luis Martínez‑Zoroa with the breakthroughs that eventually led to this solution, and which were published from ~2023 onwards.
This is a good summary:
> In broad outline, the pair’s technique relies on creating an infinite sequence of “layers,” each of which is a non-singular solution to the equation they are studying. (They’ve applied similar techniques to both the Euler and Navier-Stokes equations, as well as to other related systems.) They then combine those solutions in what Martínez-Zoroa calls an “infinite cascade” to produce a new solution.
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> That new solution, they showed, contains the desired singularity. However, even though each individual layer relies on a smooth forcing function, combining them together can cause the forcing function to have undesirable mathematical properties. That’s why their solution fell short of satisfying the Millennium Prize criteria. The remaining hurdle was to figure out how to create a similar infinite cascade that resulted not only in a singularity, but also in a smooth forcing function.
>
> That’s the step that both competing AI groups appear to have had success with.
The question is whether OpenAI started out from that published and well known research exclusively, or they also had some insight into the ongoing work of Tristan Buckmaster and Levent Alpöge.
On the one hand, OpenAI have already admitted that they only launched their massive effort after hearing rumours that this particular problem had been solved.
On the other, progress in mathematics research has accelerated significantly over the past months thanks to the availability of newer and more capable AI models. Alpöge himself presented a counterexample to the Jacobian conjecture on July, found with Claude Fable. So if model capability was a bottleneck, that gives credibility to the idea that an even more powerful unreleased model with massive compute would be able to make even faster progress.
It's worth noting that the case is that your input is being used to train their AI, and that's more important than whether it materially contributed, it cannot be denied or attributed accurately, it cannot be said with certainty which way it happened, and that's what's important.
> People seem to be talking about anything except the actual results with this particular announcement.
To be fair, most people have a fairly good handle on "Does opting out my prompts from training runs actually work?", but not on Navier-Stokes. They discuss what more immediately affects them.
Additionally, I'm no physicist but I suspect the possibility of singularities in NS equations is probably one of those 'true but not meaningful' facts. If it took our brightest minds 175 years to craft such a scenario, how relevant can it be in practice? Especially when turbulence exists. Maybe I'm wrong or it has some consequences for pure math though.
Yeah and the production numbers from the start seemed to indicate that Apple knew it wouldn't be extremely popular.
FWIW, I bought one knowing this, and have zero regrets. Amazing hardware, it's the best personal display on the market. I use it for the majority of my movie watching.
>>The Mac Mini is an entry level product I see quite lot of people pick up. How is it niche?
It's a stationary computer that requires a standalone monitor, keyboard and mouse/touchpad. People just buy a laptop instead, a dedicated "computer" space at home is increasingly a thing of the past. And especially now if you can get the Neo for very similar money I have no idea why you'd buy the mini if your entire computer use is Facebook and maybe some word editing from time to time.
Interesting how the first beat of the curiously-low-bitrate, kinda-ugly video presentation was admitting that the agent's highest priority is to get me to buy more stuff through its store front.
It's certainly honest.
Is this marketing to end-users or to stakeholders?
As a (crude) analogy, it's a bit like how you can prove the healthiness of a git tree because it's a graph of content hashes and the tree graph pointers are part of the hash. Imagine this but with a tree of knowledge.
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