Valerio Capraro
I think this is the most important concept to understand right now: LLMs can’t jump. OpenAI says an internal version of Astra, its next major model, has solved ten major open problems in mathematics and theoretical computer science. This is extraordinary. And I don’t use this word lightly. During my PhD in mathematics, I worked on a problem closely related to Gromov’s conjecture. For decades, constructing a non-sofic group—an infinite group whose finite pieces cannot be approximated by finite groups—was a dream shared by many of us. Now Astra appears to have done it. This is not a toy result. This is serious mathematics. But it is not “the most significant day in the history of mathematics”, as some have claimed. This claim confuses scale with kind. Astra has solved difficult problems inside existing conceptual worlds. Calculus, topology, and scheme theory created new conceptual worlds. They did not merely answer questions. They changed which questions mathematics could ask. Induction finds patterns. LLMs are extraordinary at it. Deduction follows implications. Using symbolic AI tools, LLMs are becoming formidable at traversing chains of logic that humans missed, abandoned, or could never afford to search. But abduction is different. Abduction invents the right concept, the right representation, the right question. That is the jump. Current LLMs can fill the gaps of knowledge left by humans. But they can’t jump beyond the external boundary of existing knowledge. * Full paper in the first reply
