I think the OpenAI model that resolved the Unit Distance Problem would be capable of solving a significant proportion of mathematics PhD thesis problems.
> Now if I know anything about math for the sake of math, and academics, these are the same people that lament the idea of intelligent people going to the finance sector or any other trade they just happen not to respect as much
IME a vastly more common sentiment among mathematicians regarding mathematical talent leaving the nest to apply their skills in other fields is that those other fields are lucky to get them!
To me, the most interesting feature of the OpenAI solution of the Unit Distance (Erdös) Problem is that the solution - using deep algebraic number theory as a source of extremal combinatorial/geometric constructions - is much more interesting than the problem’s elementary statement might lead one to expect.
Writing off Erdös’s problems as random, useless, or meaningless dismisses his mathematical intuition, second-to-none, and strikes me as somewhat uncharitable.
Finally, I agree that AI threatens mathematical training by rendering an entire class of acolyte-level research problems solvable by prompt. But the Unit Distance Problem is not of this class.
> much more interesting than the problem’s elementary statement might lead one to expect
This is reinforced by the immediate (human) use of the idea to resolve in the negative another significant problem, the sum-product conjecture on reals.
I don't think Erdos problems are useless myself, I put "useless" in quotes to emphasize that they are the sort of research that doesn't have an immediate application, and so their automated resolution should be weighed against the sociological cost.
I am not a mathematician and did not read the unit distance solution too carefully, but my impression was that it used a variation of a known technique to solve the problem. And that makes perfect sense to me, there are a lot of techniques and lot of less relevant problems, I am not surprised that one can solve some of them with known techniques that just nobody has tried [hard enough] before. I am much more sceptical when it come to the important unsolved problems where every known technique has probably been tried several times over. In those instances it will probably take a true leap in understanding to solve them and I am sceptical that large language models are well suited for that because of the way they work.
We're very fortunate to have had some very eminent mathematicians backfill the OpenAI proof with history, context, and a literature review [1]. Ideas behind the proof seem to have been "in the air". Indeed, looked at certain point of view, the OpenAI construction can be viewed as a high-dimensional generalization of a known low-dimensional one. In this vein see the remarks of Gowers, Sawin and Tsimerman in [1]. Are LLMs capable of "true leap[s] in understanding"? I have absolutely no idea. But LLMs keep surprising me.
Not sure about these books as a self-study curriculum — their unifying theme seems to be that they require a reasonable level of mathematical maturity going in. But, they absolutely comprise an excellent “greatest hits” list of math books in the most influential subdisciplines. You’re guaranteed to learn a tonne if you study any one of these books.
I don't buy the narrative that the article is promoting.
I think the machine learning community was largely over overfitophobia by 2019 and people were routinely using overparametrized models capable of interpolating their training data while still generalizing well.
The Belkin et al. paper wasn't heresy. The authors were making a technical point - that certain theories of generalization are incompatible with this interpolation phenomenon.
The lottery ticket hypothesis paper's demonstration of the ubiquity of "winning tickets" - sparse parameter configurations that generalize - is striking, but these "winning tickets" aren't the solutions found by stochastic gradient descent (SGD) algorithms in practice. In the interpolating regime, the minima found by SGD are simple in a different sense perhaps more closely related to generalization. In the case of logistic regression, they are maximum margin classifiers; see https://arxiv.org/pdf/1710.10345.
The article points out some cool papers, but the narrative of plucky researchers bucking orthodoxy in 2019 doesn't track for me.
Yeah this article gets a whole bunch of history wrong.
Back in 2000s, the reason why nobody was pursuing neural nets was simply due to compute power, and the fact that you couldn't iterate fast enough to make smaller neural networks work.
People were doing genetic algorithms and PSO for quite some time. Everyone knew that multi dimentionality was the solution to overfitting - the more directions you can use to climb out of valleys the better the system performed.
To the left of the "detailed spaceship" I think I see a distortion pattern reminiscent of a cloaked Klingon bird of prey moving to the right. Or I'm just hallucinating patterns in nebular noise.
Two schools of thought here. One posits that models need to have a strict "symbolic" representation of the world explicitly built in by their designers before they will be able to approach human levels of ability, adaptability and reliability. The other thinks that models approaching human levels of ability, adaptability, and reliability will constitute evidence for the emergence of strict "symbolic" representations.
TLDR: Browser vendors made Shadow DOM for themselves.
Browser implementors use Shadow DOM extensively under the hood for built-in HTML elements with internal structure like range inputs, audio and video controls, etc. These elements absolutely need to work everywhere and be consistent, so extreme encapsulation and fixed api for styling them is an absolute must.
The Shadow DOM API is the browsers exposing, to developers, a foundational piece of functionality.
If you’re thinking about whether Shadow DOM is appropriate for your use case, consider how/why the vendors use it —- when an element’s API needs to be totally locked down to guarantee it works in contexts they have no control over. Conversely, if your potential use case is scoped to a single project, the encapsulation imposed (necessarily!) by Shadow DOM is probably overkill.
Web components are a decent way to make reusable UI, but if they don’t have strong encapsulation needs, you might avoid Shadow DOM.
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