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「Book version of the blog」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

It’s now been almost a year since I moved my “What’s new?” page from my home page to this blog. Since then, I’ve been quite happy with the directions this blog has been headed in (most of which I had not anticipated when I started), and also with the level of feedback, some of which has been extremely informative to me (and, I hope, to other readers as well).

Anyway, after discussing things with some of my friends and colleagues, I have decided to convert some of the posts here from 2007 into a book format, in order to place some of the mathematical content here in a more formal and traditional context (with accurate citations and references, etc.). After some thought, I decided not to transcribe all of my posts from last year (there are 93 of them!), but instead to restrict attention to those articles which (a) have significant m

已知结果和反例

I’ve converted a couple articles into a book format , and also created a table of contents, to see what it would look like. (The format comes from the American Mathematical Society, with whom I am planning to publish the book.) It looks like it will be relatively straightforward to convert the rest (at least compared to writing books from scratch, which I know from experience to be quite time-consuming!). I’ll of course post updates here on this blog when the book is closer t

At present, the structure and content of the book is still rather flexible; like all things related to this blog, it is an experiment. As such, I am open to suggestions on these matters. (For instance, I do not have any particularly imaginative title for the book, other than “What’s new – 2007”. )

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。

阅读和落地时建议先做的 5 件事

1. 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。
2. 找一个最小反例或边界情形，确认假设少一条会怎样。
3. 把证明拆成可独立检验的引理，每步只保留一个新想法。
4. 若涉及计算或形式化，先写可复现的小例子，再谈一般情形。
5. 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