AI 中文总结
本文提出“重新连线论题”，认为概念革命是数学问题间转移关系的大规模重组，并以振动弦历史和压力测试论证，进而提出人工数学创造力的更强标准：发明“下一个傅里叶变换”。
AI 中文摘要
什么将一个困难数学问题的解决与概念革命区分开来？一些数学创新不仅仅是确立新结果：它们重组了问题之间的关系。我们提出通过数学的转移结构的变化来描述此类事件，即决定哪些问题能够自然地相互启发、哪些方法能够在它们之间移动、以及哪些新问题变得可及的模式。十八世纪关于振动弦的争论提供了我们的核心历史案例。达朗贝尔发现了著名的波传播模型，伯努利提出了一种模态解释，欧拉拓宽了可容许函数的类别，拉格朗日构建了基于模态分解的有限维策略。后来，傅里叶分析给出了三角语言，使得所有先前的描述能够相互作用。这激发了我们的“重新连线论题”：概念革命是数学问题之间转移关系的持久、大规模重组。生产性表示是能够产生这种重新连线的一种机制，但一些压力测试（涉及伽罗瓦、黎曼、勒贝格-施瓦茨和怀尔斯）表明它并非唯一机制。音乐记谱法提供了一个独立案例，其中表示在操作上变得具有生成性，尽管在信息上并不完备。由此产生的框架为人工数学创造力提出了一个比单纯解决问题更强的标准。相关问题不仅仅是“人工智能能否解决一个著名的猜想”，而是“它能否认识到继承的概念组织本身就是一个障碍，并产生一个新的组织，同时重组多个问题”。在这个意义上，决定性的成就可能是发明“下一个傅里叶变换”。
英文摘要
What distinguishes the solution of a difficult mathematical problem from a conceptual revolution? Some mathematical innovations do more than establish new results: they reorganize the relations among problems. We propose to describe such events through changes in the transfer structure of mathematics, namely the pattern determining which problems can naturally inform one another, which methods can move between them, and which new questions become accessible. The eighteenth-century controversy over the vibrating string provides our central historical case. D'Alembert found a famous model for wave propagation, Bernoulli proposed a modal interpretation, Euler broadened the class of admissible profiles, and Lagrange constructed a finite-dimensional strategy based on modal decomposition. Later, Fourier analysis gave the trigonometric language in which all the previous descriptions can interact. This motivates our Rewiring Thesis: a conceptual revolution is a persistent, large-scale reorganization of transfer relations among mathematical problems. The productive representation is one mechanism capable of producing such rewiring, but some stress tests (involving Galois, Riemann, Lebesgue-Schwartz and Wiles) show that it is not the only one. The musical notation provides an independent case in which the representation becomes operationally generative without being informationally complete. The resulting framework suggests a stronger criterion for artificial mathematical creativity than problem solving alone. The relevant question is not merely "whether AI can solve a famous conjecture", but "whether it can recognize that an inherited conceptual organization is itself an obstruction and produce a new one that reorganizes several problems at once". In this sense, the decisive achievement may be the invention of "the next Fourier transform".
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