이미지: The Decoder
Summary
- Fields Medalist Timothy Gowers and mathematician Peter Sarnak have pointed to the limits of LLMs' mathematical creativity
- Gowers diagnosed that LLMs lack the intuition to choose productive paths within a vast search space
- Google DeepMind researcher Tom Zahavy reached the same conclusion in his paper "LLMs Can't Jump"
- 발언자
- 티모시 가워스(필즈상 수상자), 피터 사르나크
- 가워스 진단
- 기존 방법 조합·다중 경로 탐색엔 능하나 생산적 경로 선택할 직관 부족
- 사르나크 진단
- 기존 이론에서 결과 도출은 가능, 새 추상화 개발은 불가
- 관련 연구자
- 구글 딥마인드 톰 자하비
- 논문 제목
- LLMs Can't Jump
- 병목 개념
- manipulative abduction(언어적 선례 없는 새 기초 가정을 발명하는 능력)
- 제안된 대안
- 세계모델(world model)
- 출처
- 미국수학회(AMS) 인터뷰
They can compute, but they can't come up with ideas
Proving a new theorem isn't about calculation — it's closer to imagining "a first step no one has ever tried." That's the diagnosis Fields Medalist Timothy Gowers and mathematician Peter Sarnak offered in a recent interview with the American Mathematical Society (AMS). While acknowledging that large language models (LLMs) show considerable ability in mathematics, both pointed to a fundamental limitation in generating genuinely new ideas.
Gowers assessed that current models are strong at combining already-known methods and trying multiple search paths simultaneously. The problem lies in what comes next. He noted that models lack the intuition to pick out which path within a vast search space is actually productive. In other words, they lack the sense of "this direction is right" that human mathematicians develop only after decades of training.
Sarnak explained this with a more concrete example. He said AI can derive results from already-established theory, but starting from a very basic question, it cannot on its own create the new abstractions that underpin major proofs. The weight of this statement becomes clearer when you think back to when the great theorems of number theory first appeared. Those theorems weren't the result of skillfully applying existing tools — they were the result of creating concepts that didn't exist before.
A DeepMind researcher reaches the same conclusion
A similar diagnosis has come from within the industry itself. Google DeepMind researcher Tom Zahavy gave this bottleneck a name in his paper "LLMs Can't Jump." He explains the core of the problem through the concept of "manipulative abduction" — the ability to invent new foundational assumptions on one's own, with no linguistic precedent whatsoever. This is different from rearranging existing sentences or extending patterns; it is, quite literally, creating something that did not previously exist.
Zahavy proposed world models — AI models that learn the structure of the actual environment, such as visual and physical information, rather than language — as a candidate for overcoming this limitation. The logic is that if learning confined to language is the problem, models that directly learn structure outside of language could be an alternative. However, whether this proposal will actually solve the problem of mathematical creativity remains unverified.
Benchmarks keep rising — so why doesn't creativity?
This diagnosis connects to a question that has been recurring across the industry lately: LLMs keep setting new records on various test scores, but the ability actually felt in practical work and research settings hasn't grown to match. In the PerceptionBench results Moonshot AI released on the 15th, not a single one of 16 frontier models exceeded 60% accuracy on pure visual perception, with the top model reaching only 59.7%. This suggested that models celebrated for their reasoning ability stumble even at the basic stage of accurately reading images.
Gowers and Sarnak's point runs in the same direction. There's a gap between benchmark scores and actual creative ability, and that gap doesn't close on its own just because models get bigger. That said, both mathematicians acknowledge LLMs' computational and combinatorial abilities, so it would be an overstatement to read these remarks as claiming "AI is useless." What the two pointed to is a limitation in a specific domain — the creation of fundamentally new mathematical concepts.
Editor's view
What makes these remarks interesting is that neither of the two is an AI skeptic. Gowers has long held a position closer to believing AI can assist mathematical research, and Sarnak, too, doesn't deny AI's usefulness as a computational tool. The fact that both of them separately pointed to a "wall of creativity" is itself a signal that this wall is starting to become visible even to optimists.
Anyone who has actually worked with LLMs in practice will find this familiar. Models are remarkably fast and accurate at applying known theorems or filling in specific steps of a proof. But reframing a problem — the kind of thinking that says "we should look at this problem through a completely different lens from the start" — rarely emerges. Something similar shows up in code review or drafting reports: models quickly combine existing patterns to produce plausible-looking output, but proposals that overturn the problem definition itself are rare. This is likely not because models are lazy, but because data for learning that kind of reframing is simply scarce to begin with.
The lesson for domestic research and education settings is clear. Using LLMs as "an assistant that fills in specific steps of a proof" remains valid and efficient today. But expecting them to "suggest new research directions" is still asking too much. Rather than adopting AI brainstorming outright in graduate seminars or R&D planning stages, limiting its use to quickly combining and verifying known methods reduces the chance of failure.
In the coming months, this debate is likely to grow alongside research announcements in the world-model space. If models that learn structure outside of language, as Zahavy proposed, actually demonstrate something close to "manipulative abduction," the diagnosis offered by these two mathematicians could turn out to be a short-lived provisional conclusion. Conversely, if such attempts remain stalled, the frame of "LLMs are calculators" is likely to harden into the industry's default assumption for the time being.
