陶哲轩博客写数学问题时，通常先把对象定义清楚，再给直觉、反例和证明轮廓。把「Article in the New York Times, and maths education」改写成可阅读的中文笔记，重点是：问题在问什么、已知到哪一步、下一步最容易走偏在哪。原站广告、分享条和导航已去掉。
问题在问什么
I’ve received quite a lot of inquiries regarding a recent article in the New York Times , so I am borrowing some space on this blog to respond to some of the more common of these, and also to initiate a discussion on maths education, which was briefly touched upon in the article.
Most of the feedback I received, though, concerned the issue of maths education. I mentioned in the article that I feel that the skill of thinking in a mathematical and rigorous way is one which can be taught to virtually anyone, and I would in the future hope to be involved in some project aimed towards this goal. I received a surprising number of inquiries on this, particularly from parents of school-age children. Unfortunately, my maths teaching experience is almost comple
已知结果和反例
( Update , March 13: A bad link has been repaired. Also, I can’t resist a somewhat political plug: for Californian readers, there is an open letter in support of California’s K-12 education standards , together with some background information .)
证明或构造的主线
先写出对象、假设和失败的例子，再进入证明。没有反例的直觉，很容易把局部技巧当成一般定理。
阅读时建议盯住的点
先写出对象、假设和失败的例子，再进入证明。没有反例的直觉，很容易把局部技巧当成一般定理。
值得单独记下的条目
- The video for the talk “ Structure and randomness in the prime numbers ” mentioned in the article can be found here (requires RealPlayer ). The slides can be found here . My other expository and research material on number theory can be fou
- I don’t have any specific advice regarding gifted education, though some articles on my own experiences can be found here . I do however have some thoughts on career advice at the undergraduate level and beyond.
- I have some responses to several other common queries (e.g. regarding books, interviews, invitations, etc.) at my contact information page .
阅读和落地时建议先做的 5 件事
- 用自己的语言重写定义和结论，不看原文能不能说清对象是什么。
- 找一个最小反例或边界情形，确认假设少一条会怎样。
- 把证明拆成可独立检验的引理，每步只保留一个新想法。
- 若涉及计算或形式化，先写可复现的小例子，再谈一般情形。
- 记下尚未解决的缺口：缺估计、缺构造，还是缺正确的范畴。
和智能体、形式化工具怎么接
龙虾PRO做 OpenClaw 落地时，数学笔记最有用的部分往往是「可检验的步骤」：定义、反例、引理边界。智能体适合帮忙展开计算和检索，不适合代替你决定哪条假设能扔。
本文侧重全链路风控方法论。落地时请用自身业务单据做回放验证，不要把示例阈值直接当生产策略。 相关：风控体检 · 方案资源
常见问题 FAQ
什么是AI智能系统？
「AI智能系统」可概括为：I’ve received quite a lot of inquiries regarding a recent article in the New York Times, so I am borrowing some space on this blog to respond to some of the more common of these, a 本文从定义、方法与实践要点展开说明。
为什么要关注AI智能系统？
关注AI智能系统，是因为它直接影响效率、风险与可复制性。文中指出：I’ve received quite a lot of inquiries regarding a recent article in the New York Times , so I am borrowing some space on this blog to respond to some of the more common of these, and also to initiate a discussion on maths education, which was br…
如何落地AI智能系统？有哪些关键步骤？
建议按以下路径推进AI智能系统：1) I have some responses to several other common queries (e.g. regarding books, in…；2) 用自己的语言重写定义和结论，不看原文能不能说清对象是什么。；3) 找一个最小反例或边界情形，确认假设少一条会怎样。；4) 把证明拆成可独立检验的引理，每步只保留一个新想法。；5) 若涉及计算或形式化，先写可复现的小例子，再谈一般情形。。细节见正文对应章节。
AI智能系统适合哪些人或团队？
AI智能系统更适合：产品/技术负责人、运营与增长团队、需要落地智能体或自动化的中小团队、关注「AI智能系统」方向的读者。若你只需要单次聊天式问答，可先读概念；若要上生产，请重点看步骤、权限与风控相关段落。
关于「问题在问什么」，本文给出了什么结论？
在「问题在问什么」部分，要点是：the New York Times , so I am borrowing some space on this blog to respond to some of the more common of these, and also to initiate a discussion on maths education, which was briefly touched upon in the article. Most of
关于「已知结果和反例」，本文给出了什么结论？
在「已知结果和反例」部分，要点是：ogether with some background information .) 证明或构造的主线 先写出对象、假设和失败的例子，再进入证明。没有反例的直觉，很容易把局部技巧当成一般定理。 阅读时建议盯住的点 先写出对象、假设和失败的例子，再进入证明。没有反例的直觉，很容易把局部技巧当成一般定理。 值得单独记下的条目 The video for the talk “ Structure and randomness in the