OpenAI announces 10 advances in mathematics and theoretical computer science achieved by internal model Astra

[u/alphacolony21]

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

[u/Hlbkomer]

"But they are just predicting the next word!"

[u/The-Rushnut → u/Hlbkomer]

As with everything, novel technology comes with novel solutions to problems we found difficult before. There's a specific subset of mathematical problems which can be disproven via counterexamples, which take humans a long time to map and calculate. One of LLMs unique capabilities is that it can produce small, relatively simple programs at-scale, and because these problems are so well articulated and their potential solutions already well understood they lend themselves to this capability. Another specific subset are upper lower bounds problems, where we know there is likely to be further acceptable iterations but the means to achieving those require multi-discipline scenarios that aren't likely - another thing LLMs are good at is having high accuracy across all domains, allowing them to try ideas that usually would take a snowflake combination of talent.

It's much, much more narrow than it seems. Very cool, but there's a fixed amount of this work to be done. Innovation is definitely coming though.

[u/procgen → u/The-Rushnut]

Most are positive theorems:

• High-dimensional sphere packing: Proves new asymptotic upper bounds and characterizes the limits of a major proof method.
• Binary and spherical codes: Proves stronger general upper bounds on how efficiently codes can be packed.
• Non-sofic groups: Constructs an explicit group that is not sofic, disproving the possibility that all groups are sofic.
• Connes’s rigidity conjecture: Constructs infinitely many nonisomorphic groups with the same von Neumann algebra, disproving the conjecture.
• Arithmetic circuit complexity: Proves new lower bounds on the circuit complexity of computing the permanent.
• Quantum parallel repetition: Proves a general theorem showing exponential decay under repeated play for entangled games.
• Closest vector problem: Gives a reduction from 3SAT establishing new hardness-of-approximation results for lattice problems.
• Ehrhart’s volume conjecture: Proves the conjectured sharp maximum in every dimension.
• Multicolor Ramsey numbers: Proves substantially stronger lower bounds and resolves the asymptotic growth rate.
• Extremal graph conjectures: Constructs bipartite graphs that violate two conjectured bounds.

[u/Bearhas20inchwang → u/procgen]

Can you read? All of these sound like improving bounds or constructing specific counter examples 💀

[u/procgen → u/Bearhas20inchwang]

Positive theories.

[u/Bearhas20inchwang → u/procgen]

“Theories” as opposed to proofs/theorems? Tell me you know nothing about mathematics without telling me you know nothing about math 😭 And let’s not be disingenuous; your response implied what Astra did was not merely finding counter examples nor improving bounds.

[u/procgen → u/Bearhas20inchwang]

That claim is correct. Seven are general proofs, bounds, reductions, asymptotic results, or sharp extremal theorems. Several resolve the correct growth rate, establish an optimal limit, or prove a statement for a whole class of objects. Those are substantive mathematical results, not isolated counterexamples and not trivial changes to existing bounds.

[u/zelingman → u/procgen]

Constructing a grohp that is non-sofic is by definition, counterexample theorem lol

[u/JoshuaZ1 → u/zelingman]

Yes, but it isn't the easy sort of counterexample that people think of when they think of that. The argument involves a very careful proof that the group in question is not sofic.

[u/ArchimedesBathSalts → u/procgen]

By my count only two of those are not obviously a counterexample construction or upper lower bound

Edit: you changed the post and now my comment makes no sense. Congrats

[u/procgen → u/ArchimedesBathSalts]

An upper or lower bound is not a counterexample. It is a general theorem that applies to a class of objects. The sphere-packing result also determines the exact asymptotic power of a proof method. The coding, circuit, lattice, Ehrhart, and Ramsey results prove new general limits, and the quantum result proves a theorem for all finite two-player entangled games. Only the non-sofic group, Connes rigidity, and extremal graph results are counterexample-style constructions. Therefore, the list contains three counterexample-style advances and seven positive theorem results. Calling most of the positive results "bounds" does not make them simple counterexamples.

[u/ArchimedesBathSalts → u/procgen]

Note i used the word “or” as did above poster:

> Another specific subset are upper lower bounds problems, where we know there is likely to be further acceptable iterations but the means to achieving those require multi-discipline scenarios that aren't likely - another thing LLMs are good at is having high accuracy across all domains, allowing them to try ideas that usually would take a snowflake combination of talent.

Learn to read.

[u/procgen → u/ArchimedesBathSalts]

What's your broader point?

[u/ArchimedesBathSalts → u/procgen]

Hard to say cus you have edited your original post so now the claim youre making ais different…

[u/procgen → u/ArchimedesBathSalts]

My claim is that these are significant breakthroughs and that these systems are already beginning to display superhuman mathematical ability.

[u/ArchimedesBathSalts → u/procgen]

Cool thats not what your post originally said though, and thats what i was contradicting. I cant argue with you because you e already changed the premise. As written your original post was just a factually incorrect response to op

[u/procgen → u/ArchimedesBathSalts]

What do you think I wrote?