丹尼尔（Daniel），多伦多大学（University of Toronto）的数学教授，同时也是一位执业数学家，分享了他对人工智能（AI）对数学影响的不断演变的看法。他强调，数学的目标“不是为了产出数学论文”，而是“为了产生某种理解”。尽管他承认人工智能的快速进步，但他强调了人工智能与人类数学推理之间的关键区别。 丹尼尔指出，人工智能驱动的爱尔兰单位距离问题（Irish unit distance problem）的解决方案（于五月中旬宣布）是他最喜欢的“完全自主的结果”。他认为它令人印象深刻，因为它具有创造性，引入了来自不相关领域的技术，并随后证明对其他开放问题也富有成效。他指出，许多人工智能生成的证明，尽管常被描述为“非人类的”，但实际上看起来与人类推理相似，这表明人工智能的成功可能源于对自然语言中“非正式推理”的规模化扩展，而不仅仅是像Lean这样的形式验证系统。 比较了Anthropic的Claude模型和OpenAI的模型，丹尼尔发现它们的能力“非常相似”，并指出它们只解决了“人类数学家所做工作的相对一小部分”。人工智能模型擅长应用已知技术、进行大量计算，或从海量来源中整合思想。然而，它们在“直觉”或“宏观视角”方面遇到了困难——那些“模糊的东西”，比如发展一种非严谨的哲学，或从零开始构建新理论，而这些正是丹尼尔自身在代数几何领域工作的核心。他将自己的数学活动归类为解决问题（其中问题衡量的是“对某事理解的不足”）和构建理论，这些活动受类比和识别基本概念驱动，而非审美美感。 对于他长期进行的项目，丹尼尔发现人工智能对“深层智力工作”的用处较小，主要充当“谷歌（Google）的替代品”，或辅助编程任务，在这些任务中它可以“并行处理一千个示例”。他观察到，人工智能目前更擅长提供反例的构造，而不是发展新理论或技术来证明复杂的猜想，后者通常需要“非常严肃的新思想”。 丹尼尔对数学界应如何适应表示担忧。他担心，当前专注于论文产出的学术激励机制，可能会导致一种“老虎机”式的方法，即人工智能生成大量可能质量低下的旧猜想证明，从而贬低真正的人类智力投入。他提到看到多个相同的人工智能生成的同一定理的证明同时出现，这表明在某些推理路径上存在“模式坍塌”（mode collapse）。他警告说，如果数学探索“从属于模型想要追求的东西”，这可能会扼杀历史上曾“让千百种花朵绽放”、从而拓展知识边界的多样化方法。 他进一步指出人工智能在检查非常长的证明方面的局限性。尽管人工智能可以生成更短、“巧妙的”证明，但丹尼尔怀疑这是因为更长、更枯燥的证明对于人类*和*人工智能都更难以可靠地验证。他举例提到了一个800页长、由人工智能生成的正特征下奇点消除（resolution of singularities in positive characteristic）的证明，他认为这个证明很可能不正确，并指出人工智能目前缺乏进行人类专家所使用的那种“模糊的”宏观检查的能力。 最终，丹 {"Daniel"}: 丹尼尔 {"University of Toronto"}: 多伦多大学 {"Anthropic"}: Anthropic {"Claude"}: Claude {"OpenAI"}: OpenAI {"Lean"}: Lean 尼尔希望人工智能能为复杂问题提供答案，让人类能够“学习这些答案”并加深他们的理解。他强调，即使在人工智能高度发达的未来，数学教育的核心价值依然是教导人们“清晰思考并更好地理解世界”。他认为，一个培养“更优秀思想家”的世界是最佳的，人类必须有意识地“利用”人工智能来“提升而非放弃”这些能力，从而避免陷入成为“更糟糕思想家”的陷阱。作为一名三岁孩子的父亲，他培养孩子对数学和清晰思考的热爱，认为这是一种适用于一个被人工智能深刻改变的世界的强大技能。

Daniel, a Professor of Mathematics at the University of Toronto and a practicing mathematician, shares his evolving perspectives on AI's impact on mathematics. He emphasizes that the goal of mathematics is "not to produce mathematics papers" but "to produce some kind of understanding." While acknowledging AI's rapid advancements, he highlights crucial differences between AI and human mathematical reasoning. Daniel points to the AI-driven solution to the Irish unit distance problem (announced mid-May) as his favorite "fully autonomous result." He finds it impressive due to its creativity, introducing techniques from unrelated areas and subsequently proving fruitful for other open problems. He notes that many AI-generated proofs, though often described as "inhuman," actually appear similar to human reasoning, suggesting AI's success may stem from scaling "informal reasoning" in natural language rather than just formal verification systems like Lean. Comparing Anthropic's Claude and OpenAI's models, Daniel finds their capabilities "pretty similar," noting they address a "relatively small portion of what human mathematicians do." AI models excel at applying known techniques, grinding out computations, or pulling together ideas from vast sources. However, they struggle with "intuition" or "big picture point of view" – the "fuzzy things" like developing a non-rigorous philosophy or building new theories from scratch, which are central to Daniel's own work in algebraic geometry. He categorizes his mathematical activity as problem-solving (where problems measure "failure to understand something") and theory-building, driven by analogies and identifying fundamental concepts rather than aesthetic beauty. For his long-standing projects, Daniel finds AI less useful for "deep intellectual work," primarily serving as a "substitute for Google" or aiding with coding tasks where it can "work through a thousand examples in parallel." He observes that AI is currently stronger at providing constructions for counterexamples than developing new theories or techniques to prove complex conjectures, which often require "very serious new ideas." Daniel expresses concerns about how the mathematics community should adapt. He fears that current academic incentives, focused on paper production, could lead to a "slot machine" approach where AI generates numerous, potentially low-quality, proofs of old conjectures, devaluing genuine human intellectual engagement. He mentions seeing multiple identical AI-generated proofs of the same theorem appearing simultaneously, suggesting a "mode collapse" on certain reasoning paths. He warns that if mathematical exploration is "subordinate to what the model want to pursue," it might stifle the diverse approaches that historically "let a thousand different flowers bloom," expanding the boundaries of knowledge. He further notes AI's limitation in checking very long proofs. While AI can produce shorter, "clever" proofs, Daniel suspects this is because longer, grindy proofs are harder for both humans *and* AI to reliably verify. He cites an 800-page AI-generated proof of resolution of singularities in positive characteristic as an example unlikely to be correct, arguing that AI currently lacks the ability to perform the "fuzzy" big-picture checking that human experts use. Ultimately, Daniel hopes AI will provide answers to complex questions, allowing humans to "learn the answers" and deepen their understanding. He emphasizes that the core value of mathematics education remains teaching "to think clearly and like better understand the world," even in an AI-advanced future. He believes that a world producing "better thinkers" is optimal, and humans must intentionally "leverage" AI to "improve those facilities rather than relinquish" them, avoiding the trap of becoming "worse thinkers." As a parent of a three-year-old, he instills a love for math and clear thinking, viewing it as a robust skill for a world profoundly changed by AI.