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「A draft version of the blog book」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

A few months ago, I announced that I was going to convert a significant fraction of my 2007 blog posts into a book format. For various reasons, this conversion took a little longer than I had anticipated, but I have finally completed a draft copy of this book, which I have uploaded here ; note that this is a moderately large file ( 1.5MB 1.3MB 1.1MB), as the book is 374 pages 287 pages 270 pages long. There are still several formatting issues to resolve, but the content has a

It may be a while before I hear back from the editors at the American Mathematical Society as to the status of the book project, but in the meantime any comments on the book, ranging from typos to suggestions as to the format, are of course welcome.

已知结果和反例

[ Update , April 21: New version uploaded , incorporating contributed corrections. The formatting has been changed for the internet version to significantly reduce the number of pages. As a consequence, note that the page numbering for the internet version of the book will differ substantially from that in the print version.]

[ Update , April 21: As some readers may have noticed, I have placed paraphrased versions of some of the blog comments in the book, using the handles given in the blog comments to identify the authors. If any such commenters wish to change one’s handle (e.g. to one’s full name) or to otherwise modify or remove any comments I have placed in the book, you are welcome to contact me by email to do so.]

证明或构造的主线

[ Update , April 23: Another new version uploaded , incorporating contributed corrections and shrinking the page size a little further.]

[ Update , May 8: A few additional corrections to the book.]

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