The mathematician Timothy Gowers doesn’t mince words about what large language models can and cannot do. They’re strong calculators, he says, capable of combining known methods and grinding through thousands of search paths. What they lack is the intuition to spot the handful of routes actually worth pursuing. That’s the bottleneck—not raw computation, but the judgment that tells a human mathematician which of a million possible directions might lead somewhere genuinely new.
Peter Sarnak, another heavyweight in the field, sees the same ceiling from a different angle. LLMs can derive results from existing theory. Ask them to build the abstractions that underpin a major proof—starting from an elementary question with no obvious scaffolding—and they stall. DeepMind researcher Tom Zahavy reached a similar conclusion in his paper “LLMs Can’t Jump,” pinning the failure on what he calls “manipulative abduction”: inventing foundational assumptions that have no linguistic precedent. An LLM can only remix what language already encodes. A truly novel assumption, by definition, isn’t in the training data.
This isn’t just an academic quibble. It cuts to the heart of whether LLMs are becoming more versatile or simply better at benchmarks and familiar problem spaces. A model that aces a math Olympiad test has mastered pattern-matching within a known distribution. The question is whether it can ever step outside that distribution. Gowers and Sarnak are skeptical, and their skepticism carries weight—these are Fields Medal and Abel Prize territory, not AI hype merchants.
Zahavy floats world models as a possible way forward. Instead of processing language alone, an AI that builds internal representations of how systems actually behave might develop the kind of grounded intuition that pure text prediction cannot produce. It’s a compelling idea, but it remains exactly that—an idea. For now, the mathematicians’ verdict is blunt: LLMs are excellent students of existing mathematics and poor inventors of new mathematics. The gap between those two skills may be the real measure of progress in AI.
