The current state-of-the-art in artificial intelligence is impressive, especially in terms of mastery of language, but not so much in terms of mathematical reasoning. What could be missing? Can we learn something useful about that gap from how the brains of mathematicians go about their craft? This essay builds on the idea that current deep learning mostly succeeds at system 1 abilities -- which correspond to our intuition and habitual behaviors -- but still lacks something important regarding system 2 abilities -- which include reasoning and robust uncertainty estimation. It takes an information-theoretical posture to ask questions about what constitutes an interesting mathematical statement, which could guide future work in crafting an AI mathematician. The focus is not on proving a given theorem but on discovering new and interesting conjectures. The central hypothesis is that a desirable body of theorems better summarizes the set of all provable statements, for example by having a small description length while at the same time being close (in terms of number of derivation steps) to many provable statements.

翻译：当前人工智能的最新技术成果令人印象深刻，尤其在语言掌握能力方面表现突出，但在数学推理领域仍显不足。这种落差因何产生？我们能从数学家的大脑运作方式中汲取哪些有益启示？本论文立足于以下观点：当前深度学习主要擅长系统1能力——对应直觉和习惯性行为——但在系统2能力（包括推理和稳健不确定性估计）方面仍存在显著缺失。本文以信息论视角探讨何为有意义的数学命题，这或将为未来构建AI数学家指明方向。研究重点并非证明特定定理，而是发现新颖有趣的猜想。核心假设在于：理想的理论体系应能更有效地概括所有可证明命题，例如通过较小的描述长度，同时与众多可证明命题在推导步数保持相近距离。