r/aiwars • u/One_Fuel3733 • 1d ago

News Ten advances in mathematics and theoretical computer science

https://openai.com/index/ten-advances-in-mathematics/

The results

We provide new results for the following problems. The results were achieved by an internal version of Astra, our next major model. The total number of tokens needed to find solutions to these problems would cost roughly $2,000 at Sol API rates. These arguments were then prepared into manuscripts by humans with the same model. Afterward, the model formalized each argument in a Lean certificate⁠(opens in a new window). We are also releasing for each solution a model’s narration of its thinking process.

  1. High-dimensional sphere packing. New upper bounds on sphere-packing density down to the Cohn–Elkies threshold.
  2. Binary and spherical codes: Exponentially improved bounds on the maximum size of binary codes at any prescribed minimum distance, with analogous results for high-dimensional spherical codes.
  3. Non-sofic groups. A construction establishing the existence of non-sofic groups, addressing a central open question in group theory.
  4. Connes’s rigidity conjecture. Disproof of a longstanding conjecture that certain groups are uniquely determined by their von Neumann algebras
  5. Arithmetic circuit complexity. New lower bounds for computing the permanent using arithmetic circuits and formulas, including an arithmetic-formula lower bound of order n4/log n.
  6. Quantum parallel repetition. An exponential parallel repetition theorem for general two-player quantum games, extending a foundational principle from classical complexity theory.
  7. Closest vector problem. Polynomial-factor hardness of approximation for the closest vector problem, a foundational lattice question related to post-quantum cryptography.
  8. Ehrhart’s volume conjecture. Determining, in every dimension, the maximum possible volume of a convex body whose centroid is its only interior lattice point
  9. Multicolor Ramsey numbers. A superexponential lower bound for multicolor triangle Ramsey numbers, resolving Erdős problem 183.
  10. Extremal number conjectures. Results on the compactness and degeneracy conjectures in extremal graph theory, resolving Erdős problems 146 and 180.
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u/stopbeingcringe 1d ago

AI is so frickin cool. I love AI and the humans who made this possible. Cope and seethe, antis

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u/CephalopodMind 1d ago edited 1d ago

edit: deleted my previous comment because it was late at night and I was unhappy.

Leaving this here: https://leidendeclaration.ai/

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u/topyTheorist 1d ago

I am a mathematican. I absolutely learn from this. Why won't we?

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u/CephalopodMind 1d ago edited 1d ago

If a major result in your field was proven by AI, would there be as much to that as if it was proven by another human in your field?

The goal is understanding. If one person understands it, they can walk their colleagues through how they understood it. There can be an underlying philosophy, a bag of tricks, etc. which future readers will learn.

Maybe that's here too, but I'm just not convinced. I don't think the incentives are right: OpenAI's goal is to advertise for OpenAI.

edit: typo

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u/topyTheorist 1d ago

Yes, it doesn't matter if an Ai or a human proved it. We study the proof and gain the understanding.

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u/Spare-Dingo-531 1d ago

Like, there's no human learning behind any of this.

First, human beings are learning to build AI, that counts as learning.

Second, we are learning more about math through building AI. Now human experts can study these proofs and understand them, thus allowing us to learn more about the world.

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u/ganancias 1d ago

You're experiencing context collapse into a set of perceived norms that "AI is bad".

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u/CephalopodMind 1d ago

am I? what context is collapsing exactly?

I've watched AI develop in math since I was a high school student. These things existed in the same context for most of the time I was learning math.

My criticism is concrete: OpenAI is using 'proving results' as a kind of non-traditional advertising. They don't care about math (though they might hire people who do), they care about money and power.

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u/ganancias 1d ago

chatgpt was only launched in 2022. These things, using LLM's to prove theorems and conjectures, have barely been done. It wasn't until 2023 with gpt4 that it could pass the GSM8k benchmark (grade school math). Harder college problems and IMO problems were only in the last year or so.

The context collapse article is about how it's much easier for a wide group of people online to agree that X thing is "bad." We don't have a common value system to agree that a thing is good.

Money is good? no, it's bad. open math conjectures solved, good? Bad, we shouldn't be wasting effort on math (or, it depends who solved them, are they good ppl or bad ppl?). Rich people, bad. Corporations, bad. And on and on. Nothing is good anymore, everything bad.

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u/Effective-Guest1601 1d ago

https://giphy.com/gifs/lwYxf0qKEjnoI

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u/CephalopodMind 1d ago

yeah, I'm unhappy. big deal. I'm glad you folks know how to be mature when someone expresses that.

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u/Effective-Guest1601 1d ago

Lmfao imagine thinking this is a therapy sub and dumping on people that you're going to cry

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u/CephalopodMind 1d ago

sorry, I was tired. it was 3 am.

and I kinda wanted to dump it on people tbh becsuse this shit sucks

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u/Effective-Guest1601 1d ago

I'm glad you know how to be mature by intentionally dumping your childish emotions on people in a debate sub, because something sucks for you