Editor’s note

编辑说明

This piece uses an interview structure to translate a mathematically rigorous discussion into a public explanatory framework: how “chaos” is understood in dynamical systems, how unpredictability becomes a provable result, and why mathematics does not seek a single, final definition in this field. This mathematical perspective underpins how modern technological systems confront risk, uncertainty, and the limits of control.

这篇文章以采访为结构工具，将一个高度数学化的讨论转译为可公共理解的解释框架： 动力系统意义上的“混沌”如何被理解、不可预测性如何成为可证明的结论， 以及数学为何并不执着于给出一个终极定义。 这种数学视角，构成了当代技术系统面对风险、不确定性与控制边界时的理论底座。

Key questions

核心问题框架

• What does “chaos” mean in mathematics, and why is a universal definition not the main issue?
• 在数学语境里，“混沌”意味着什么？为何“统一定义”并不是最关键的问题？
• How did “Period Three Implies Chaos” introduce chaos as a mathematical term—and what exactly did it prove?
• 《周期三意味着混沌》如何把“混沌”变成一个严格的数学术语？它究竟证明了什么？
• How did Lorenz’s “butterfly effect” get translated into a rigorous theorem (and why did the proof almost get lost)?
• 洛伦兹的“蝴蝶效应”如何被数学化为可证明的定理？这段证明为何几乎被埋没？
• What is “weak chaos,” and why does it remain an open mathematical challenge?
• 什么是“弱混沌”？它为何至今仍是数学家面前的难题？

Selected excerpts

文章节选

Excerpt 1 · Definition without one definition: how mathematics frames “chaos”.

节选 1 · 不是一句话的定义：数学如何理解“混沌”。

混沌的情形也类似。它的数学概念首次由李天岩和约克在《周期三意味着混沌》（Period Three Implies Chaos)中进行了定义。实际上，那篇仅8页的短文并没有按教科书的形式给出混沌的正式定义。它仅仅对具有“周期三点”的那类连续函数，证明出了一个出乎意料的结论：比全体自然数还要多的“初始点”，从它们各自的位置出发进行迭代，所产生的无穷点列，其最终走向是无法预测、乱七八糟的状态。于是他们提出了混沌的概念，第一次给出了“混沌”这个数学术语。

The mathematical idea of chaos was first introduced by Tian-Yuan Li and James Yorke in “Period Three Implies Chaos.” That eight-page note did not offer a textbook-style formal definition. Instead, it proved an unexpected result for continuous functions with a “period-three point”: there exist more initial points than all natural numbers, and for those starting points, iterating the function generates infinite sequences whose eventual behavior is unpredictable—seemingly a mess. From this, they proposed “chaos” as a mathematical term for the first time.

Excerpt 2 · How a proof travels: rejection, revision, and the theorem’s second life.

节选 2 · 证明如何被传播：退稿、修改，以及定理的“第二次生命”。

文章出来以后他们马上投稿给《美国数学月刊》，这是在全世界读者人数最多的数学期刊。但文章很快被退了回来，编辑说：“你们这篇文章写得太深了，我们这个杂志的主要读者是大学生，”但又补充道，“如果你们还是想投稿我们杂志的话，请改到大学生能看懂的地步。”当时他们还没意识到这篇文章真正的潜在意义，加上又很忙，文章被搁置了一年都没处理。 1974年5月底，当时最有名的科学家之一罗伯特·梅被邀请来马里兰大学数学系做了一个礼拜的报告。……报告结束后，约克送他去机场，路上把《周期三意味着混沌》拿给他看。梅恍然大悟，并在随后的夏季赴欧访问时，到处宣传李-约克定理。……约克发现了这篇文章真正的价值，它可以解释种群动力学的迭代问题，并且对气象学也有帮助，解释了洛伦兹的发现，即长期天气预报是不可能的。所以两周后，文章改好，三个月后被《美国数学月刊》接受，并于第二年12月发表。

After finishing the paper, they submitted it to The American Mathematical Monthly, the world’s most widely read mathematics journal. It was quickly rejected. The editor said, “Your article is too deep—our main readers are undergraduates,” but added, “If you still want to publish with us, rewrite it so undergraduates can understand it.” At the time, they had not realized the paper’s potential significance; busy with other work, they left it untouched for a year. In late May 1974, Robert May—one of the best-known scientists of the time—visited the University of Maryland’s math department for a week of talks. After the final talk, Yorke drove him to the airport and showed him “Period Three Implies Chaos” on the way. May immediately grasped its importance and began promoting the Li–Yorke theorem widely during his summer trip to Europe. Yorke then recognized the paper’s real value: it could explain iterative problems in population dynamics, and it also helped clarify Lorenz’s finding that long-term weather prediction is impossible. Two weeks later they revised the manuscript; three months later it was accepted by The American Mathematical Monthly and published the following December.

Excerpt 3 · “Weak chaos”: why nature stays unstable but not catastrophic.

节选 3 · “弱混沌”：为何世界不稳定，却不至于崩塌。

混沌通常与“初始误差呈指数式增长”的行为相关。在现实中，指数式增长意味着极其快速的增长，在坏的情况下会导致可怕的结果，如疫情中感染人数的指数式增长。我想戴森所说的弱混沌，意思是它的不规则程度没那么强、那么大。 在力学里，一列兵士在经过一座桥时，如果步伐太一致可能会引起共振现象而让桥倒塌。但弱混沌确保了混沌运动保持有界而不会产生猛烈变化的不稳定性，这也是为什么天体虽然存在混沌现象，它的轨道经常有小的摆动，但没有大到致使星体碰撞、宇宙毁灭。蝴蝶效应也是如此，蝴蝶扑闪一下翅膀会造成两个礼拜后另一个地区的暴雪，好像是部灾难片。但最后也不会让广东出现-30°的严寒。弱混沌对于我们是好事，说明大自然还是仁慈的。

Chaos is often associated with “initial errors growing exponentially.” In real life, exponential growth means extremely rapid increase, and in the worst cases it can lead to frightening outcomes—like infection numbers rising exponentially during a pandemic. By “weak chaos,” Dyson likely meant that the irregularity is not so strong or so extreme. In mechanics, if a column of soldiers crosses a bridge in perfect step, resonance can collapse the bridge. Weak chaos, however, keeps chaotic motion bounded, preventing violent instability. That is why celestial bodies can exhibit chaotic behavior—small oscillations in their orbits—without escalating into collisions and the destruction of the universe. The butterfly effect is similar: a butterfly’s wingbeat may contribute to a snowstorm elsewhere two weeks later, like a disaster movie, but it will not drive Guangdong to −30°C. Weak chaos is good news for us: it suggests nature is, in a sense, merciful.