人工智能时代的数学研究生教育：培养原创、独立且负责任的数学研究者
Graduate Mathematics in the Age of AI: Forming Mathematicians for Original, Independent, and Responsible Inquiry
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中文总结 AI 辅助
针对AI生成数学材料削弱学位证据的问题，提出以能力、判断、独立、责任四能力为核心的博士培养模型，通过贡献声明、指导支持与分阶段试点，实现独立工作与AI辅助研究的互补。
中文摘要 AI 辅助
人工智能日益能够以比研究生成长中理解和验证的速度更快的速度，生成看似合理且复杂的数学材料。因此，一个复杂的结果或论文草稿，作为学生自身数学发展的证据，其说服力变弱了。这造成了输出与个人能力之间的培养差距，以及有说服力的论证与有根据的接受之间的信任差距。即使学生理解了输出内容，培养差距也可能持续存在：理解一个提供的论证本身并不能确立发起和指导探究的能力。这些差距并非全部。人工智能还能帮助学生探索示例、比较方法、进入陌生领域并开展雄心勃勃的研究。任务是设计一种学徒制，在实现这些可能性的同时，发展实质性的数学掌控能力。数学博士学位的核心目的是培养能够进行原创、独立且负责任探究（包括使用人工智能进行的探究）的数学研究者。本文通过四个相互关联的能力来阐述这一目标：能力、判断力、独立性和责任感。它区分了一项工作对数学的贡献与它为学生培养所提供的证据；解释了已知答案如何能够开启而非终结创造性探究；并提出了在学习活动、评估、博士原创性、指导以及机构支持方面的变革建议。有目的的独立工作和雄心勃勃的人工智能辅助研究是该模型的互补部分。其建议包括相称的贡献声明、承认指导成本，以及分阶段试点，以评估数学能力和有效的人机协作。其目的不是保留既定的培训序列，而是在数学实践变化时改进数学培养。
英文摘要
Artificial intelligence can increasingly produce plausible, sophisticated mathematical material faster than a developing graduate student can understand or verify it. A sophisticated result or paper draft therefore becomes weaker evidence of the student's own mathematical development. This creates a formation gap between output and personal capacity, and a trust gap between a convincing argument and warranted acceptance. The formation gap can persist even when the student understands the output: understanding a supplied argument does not by itself establish the capacity to initiate and direct inquiry. These gaps are not the whole story. AI can also help students explore examples, compare approaches, enter unfamiliar areas, and undertake ambitious research. The task is to design an apprenticeship that realizes these possibilities while developing substantive mathematical command. The central purpose of a mathematics PhD is to form mathematicians capable of original, independent, and responsible inquiry, including inquiry conducted with AI. This document develops that objective through four connected capacities: competence, judgment, independence, and responsibility. It distinguishes a work's contribution to mathematics from the evidence it provides of a student's formation; explains how a known answer can initiate rather than end creative inquiry; and proposes changes in learning activities, assessment, doctoral originality, advising, and institutional support. Purposeful independent work and ambitious AI-assisted research are complementary parts of the model. Its recommendations include proportionate contribution statements, recognition of advising costs, and staged pilots evaluating both mathematical ability and effective human--AI collaboration. The aim is not to preserve an inherited sequence of training, but to improve mathematical formation as mathematical practice changes.
