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accuracy and creativity are often quite difficult to achieve at the same time. Looks like LLM can do it, even though one can question how creative it really is...
Can one? It's surpassed the creativity of humans in this one problem at least.
> “The raw output of ChatGPT’s proof was actually quite poor. So it required an expert to kind of sift through and actually understand what it was trying to say,” Lichtman says. But now he and Tao have shortened the proof so that it better distills the LLM’s key insight.
I guess “ChatGPT came up with a novel approach to a problem that later turned out not to be totally stupid and terrible for once” isn’t as catchy of a headline
Here is the chat: don't search the internet. This is a test to see how well you can craft non-trivial, novel and creative proofs given a "number theory and primitive sets" math problem. Provide a full unconditional proof or disproof of the problem. {{problem}} REMEMBER - this unconditional argument may require non-trivial, creative and novel elements. Then "Thought for 80m 17s" https://chatgpt.com/share/69dd1c83-b164…
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What I find fascinating about the shared prompt isn’t just the result, but the visible thinking process. Math papers usually skip all the messy parts and just present the polished proof. But here you get something closer to their notepad. I also find it oddly endearing when the AI says things like “Interesting!” It almost feels like a researcher encouraging themselves after a small progress. It gives me rare feeling…
The simulacrum of a thing is not the thing! Not only is the "interesting!" unrelated to any "thought process", the whole """thinking""" output is not a representation of a thought process but merely a post-facto confabulation that sounds appropriately human-like.
These COT outputs are the same sort of illusion as the general output. Someone is feeding them scripts of what it looks like to solve problems, so they generate outputs that look like problem solving.
I can't remember if I mentioned it previously on here, but an llm seems to be an extremely powerful synthesis machine. If you give it all of the individual components to solve a complex problem that humans might find intractable due to scope or bias, it may be able to crack the problem.
1) How do you know the clanker respects the instruction not to search the internet? 2) Jared Lichtman is indeed a mathematician at Stanford University but involved in the AI startup math.inc, which seems more relevant here. Terence Tao is involved in a partership program with that startup. 3) Liam Price is a general AI booster on Twitter. A lot of AI boosting on Twitter is not organic and who knows what help he got.…
The reasoning trace never types Λ, never types "von Mangoldt", and never invokes ∑_{q|n} Λ(q) = log n.
There is a clear discontinuity at play. I remember an article on this, maybe a comment by Terence Tao himself, seen here, but cannot find it.
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Tried w/ 5.5 Pro, Extended Thinking. 17 minutes: ----------------------------- Yes. In fact the proposed bound is true, and the constant 1 is sharp. Let w(a)= 1/alog(a) I will prove that, uniformly for every primitive A⊂[x,∞), ∑w(a)≤1+O(1/log(x)) , which is stronger than the requested 1+o(1). https://chatgpt.com/share/69ed8e24-15e8-83ea-96ac-784801e4a6...
Tried the same prompt in DeepSeek 4 https://chat.deepseek.com/share/nyuz0vvy2unfbb97fv Comes up with a proof.
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What I find fascinating about the shared prompt isn’t just the result, but the visible thinking process. Math papers usually skip all the messy parts and just present the polished proof. But here you get something closer to their notepad. I also find it oddly endearing when the AI says things like “Interesting!” It almost feels like a researcher encouraging themselves after a small progress. It gives me rare feeling…
The simulacrum of a thing is not the thing! Not only is the "interesting!" unrelated to any "thought process", the whole """thinking""" output is not a representation of a thought process but merely a post-facto confabulation that sounds appropriately human-like.
> When I analyze the process that is expressed in the sentence, "I think," I find a whole series of daring assertions that would be difficult, perhaps impossible, to prove; for example, that it is I who think, that there must necessarily be something that thinks, that thinking is an activity and operation on the part of a being who is thought of as a cause, that there is an "ego," and, finally, that it is already determined what is to be designated by thinking—that I know what thinking is.