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 speech for the American Academy of Arts and Sciences」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

Next month, I am scheduled to give a short speech (three to five minutes in length) at the annual induction ceremony of the American Academy of Arts and Sciences in Boston. This is a bit different from the usual scientific talks that I am used to giving; there are no projectors, blackboards, or other visual aids available, and the audience of Academy members is split evenly between the humanities and the sciences (as well as people in industry and politics), so this will be a

My chosen topic is on the future impact of internet-based technologies on academia (somewhat similar in theme to my recent talk on this topic ). I have a draft text below the fold, though it is currently too long and my actual speech is likely to be a significantly abridged version of the one below [ Update , Oct 12: The abridged speech is now at the bottom of the post.] In the spirit of the theme of the talk, I would of course welcome any comments and suggestions.

已知结果和反例

For comparison, the talks from last year’s ceremony, by Jim Simons , Peter Kim , Susan Athey , Earl Lewis , and Indra Nooyi , can be found here . Jim’s chosen topic, incidentally, was what mathematics is, and why mathematicians do it.

[Update, Nov 3: Video of the various talks by myself and the other speakers (Emmylou Harris, James Earl Jones, Elizabeth Nabel, Ronald Marc George, and Edward Villela) is now available on the Academy web site here .]

证明或构造的主线

If I had to name the most significant technological development in recent decades, I would have to say it would be the internet. By this, I mean not just the physical architecture of the internet per se , which was already available to academics and government agencies since the 1960s, but also all the innovative technologies that flourished once the internet matured, from tools as humble as the email mailing list to such unreasonably effective services as modern search engin

As the internet has become more integrated into the mainstream of modern life, it has disrupted and revolutionised one sphere of human activity after another. We read in the news about how online media is thriving as “old” media stumbles; how online medical information is transforming patient-doctor relationships; how blogs, tweets, and online videos are tipping the balance in closely fought elections; and so forth.

阅读时建议盯住的点

But to most of us in academia, there is a temptation to view these changes with a certain detachment: sure, established for-profit companies may well face competition (as they ought to) from lower-cost internet-based rivals, and it is only reasonable in a democracy that politics should be influenced by popular debate, both offline and online, but we, by contrast, should be secure in our ivory towers from any internet revolution, with our tenure, our unique expertise, and our

Even when new technologies do hit close to home – by threatening the profit model of the academic journal system, say, or by greatly facilitating the ability for students to cheat on their homework (and also for professors to detect such cheating!) – we can still rationalise away these developments as requiring only superficial changes to adapt to – switching from physical journals to online journals, perhaps, or placing more safeguards on our homework formats. We still perfo

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