Even this result was very much a human-AI collaboration. While the AI system found the proof on its own, human mathematicians verified the result. Other humans came up with better-written proofs that extended the AI's initial ideas.
大多数人可能认为AI能够独立解决人类无法解决的数学问题，表明人类数学家角色将被削弱，但作者强调这仍然是人机协作的结果。因为作者指出，人类数学家不仅验证了结果，还改进和扩展了AI的初步想法，表明在可预见的未来，人类在数学研究中仍将发挥关键作用。
The AI constructed a grid in a high-dimensional space and then projected this more complex structure into two dimensions. And instead of using a whole-number grid with points like (1,3) or (-3,6), the AI construction used something called algebraic integers to build this more complicated grid.
大多数人认为AI在数学领域的突破需要全新的思维方式和人类尚未掌握的技术，但作者认为AI的解决方案实际上是通过巧妙组合现有数学概念实现的。这挑战了人们对AI创新能力的认知，表明AI的优势在于跨领域知识整合而非创造全新理论。
An internal OpenAI model has disproved this longstanding conjecture, providing an infinite family of examples that yield a polynomial improvement.
大多数人认为解决数学难题需要人类数学家的直觉和创造力，但作者认为AI模型能够独立解决长期存在的数学猜想，并取得多项式改进。这挑战了数学研究必须由人类主导的传统观念，展示了AI在纯数学领域的突破性能力。
Tools such as AlphaEvolve are giving mathematicians very useful new capabilities. For optimization problems in particular, we can now quickly test potential inequalities for counterexamples, or to confirm our beliefs in what the extremizers are, which greatly improves our intuition about these problems and allows us to find rigorous proofs more readily.
大多数人认为数学证明需要人类直觉和创造力，但作者认为AI工具可以显著加速数学发现过程，甚至帮助人类找到更严谨的证明。这挑战了数学研究作为纯粹人类智力活动的传统观念，暗示AI可能成为数学家的真正合作伙伴而非简单工具。
Liam Price just cracked a 60-year-old problem that world-class mathematicians have tried and failed to solve. He's 23 years old and has no advanced mathematics training.
大多数人认为解决复杂的数学问题需要深厚的专业训练和多年经验，但作者认为一个没有高级数学训练的23岁年轻人仅凭AI工具就能解决困扰顶级数学家60年的问题，这挑战了数学领域的专业壁垒认知。
Liam Price just cracked a 60-year-old problem that world-class mathematicians have tried and failed to solve. He's 23 years old and has no advanced mathematics training.
大多数人认为解决复杂的数学难题需要深厚的专业训练和多年经验，但这个案例表明，一个没有高级数学训练的23岁年轻人仅通过AI工具就解决了困扰顶尖数学家60年的问题，挑战了专业知识在数学突破中的必要性。
GPT‑5.5 found a proof of a longstanding asymptotic fact about off-diagonal Ramsey numbers, later verified in Lean. The result is a concrete example of GPT‑5.5 contributing not just code or explanation, but a surprising and useful mathematical argument in a core research area.
大多数人认为AI在数学研究领域仅能辅助计算或提供解释，无法独立进行创造性数学推理。但作者展示GPT-5.5能够发现并证明数学定理，这一突破挑战了数学研究作为纯粹人类活动的传统观念，暗示AI可能成为真正的'研究伙伴'而非仅是工具。
Norbert Wiener was a mathematician with extraordinarily broad interests. The son of a Harvard professor of Slavic languages, Wiener was reading Dante and Darwin at seven, graduated from Tufts at fourteen, and received a PhD from Harvard at eighteen. He joined MIT's Department of Mathematics in 1919, where he remained until his death in 1964 at sixty-nine. In Ex-Prodigy, Wiener offers an emotionally raw account of being raised as a child prodigy by an overbearing father. In I Am a Mathematician, Wiener describes his research at MIT and how he established the foundations for the multidisciplinary field of cybernetics and the theory of feedback systems. This volume makes available the essence of Wiener's life and thought to a new generation of readers.