先讲 Gromov 手稿里的一个小场景。

一个四五岁的孩子，看见有人把一根棍子立在指尖上保持平衡，就会自己去试。没人教，没人表扬，他试了一次又一次，最后学会了。Gromov 接着问：到底是什么，在驱动孩子做这种事？

大人一般不会这样。大人做事先问“有什么用”。可孩子学会说母语、学会走路，靠的并不是“有什么用”这套逻辑。

今天训练一个 AI，几乎总要先写下一个目标：要么是一个要降低的误差，要么是一个要拿到的奖励。Gromov 的想法正好反过来：学习最根本的那部分，是没有目标的。

Gromov 是当代最重要的几何学家之一，2009 年获得阿贝尔奖，1982 年起在法国高等科学研究所（IHES）任教，现为那里的荣休教授。大约从 2010 年起，他花了大量时间想另一件事：人到底是怎么学会东西的。他为此写了几百页手稿，管这件事叫 ergo 计划。

我最早是在一期播客里听 Gromov 说起 life-long playing，一辈子都在玩，才去翻了他的手稿。这个词在他的书面文字里我没有找到，但手稿里通篇都是这个意思。

名字来自笛卡尔那句“我思故我在”，拉丁文是 cogito ergo sum。Gromov 盯住了中间那个不起眼的 ergo，意思是“所以”。

大脑每时每刻收到的，是一大团乱糟糟的电信号和化学信号。从这团乱麻“所以”出一个清清楚楚的世界，这个转换本身，在他看来才是最有结构、最值得研究的东西。他把做这件事的那部分心智，叫作 ergo。

另一部分叫 ego，也就是你平时感觉到的那个“我”。它务实，讲常识，要目标，替你和你的基因活下去。ergo 不管这些，它只被“有趣的结构”吸引。游戏、艺术、科学和数学，都从这里来。

在大人身上，ergo 藏在下面，自己感觉不到。在小孩身上，两者几乎还没分开，所以看小孩怎么玩，最容易看见 ergo。在 2013 年底的一次杂志访谈里，他说得更直白：孩子身上的那套程序，是冲着学习和理解去的，不是冲着活下去；活下去，是父母的事。

他还有一句话说得很传神：

这种认真，ego 看不懂。Gromov 举过一个例子：编一道毫无用处的国际象棋排局题，对 ergo 是享受；一丝不苟地填报税表，ego 觉得必要，ergo 只觉得无聊透顶。

他还有一个很妙的说法。想赢，是 ego 的事；在游戏的结构里，输赢其实很次要。一个纯粹的 ergo 不会拼命去赢，反而会把自己的水平降下来，去配合比它弱的对手，好让这盘棋尽可能有意思。陪孩子下过棋的大人，大概都干过这种事。

为什么说学习没有目标？Gromov 的论证很短。

人学会的东西千差万别：母语、走路、下棋、数学。如果学习背后是一套通用的机制，它就不能依赖某个具体的目标，因为目标和目标之间没有共同的结构，学说话的目标和学走路的目标毫无相似之处。所以，最根本的那部分学习机制，必须和目标无关，也不依赖外部的奖惩。他眼里最好的证据，是每个孩子都学会了母语，而没有谁按词给孩子发奖励。

他打了一个物理的比方。牛顿说，物体不受力，也会一直运动下去，这叫惯性。学习就像惯性：它自己会走。奖励和惩罚像外力，能改变它的轨迹，却不是它动起来的原因。他甚至说，奖惩与其说在引导学习，不如说在限制学习。

另一个比方是扔石头。

猎人只关心打中什么，瞄一块吃不了的东西，在他看来就是犯傻。物理学家看的是另一样东西：从出手到落地，中间那条由运动定律决定的抛物线。靶子五花八门，规律只有一条。Gromov 说，研究学习也该像物理学家那样，不去管学习者各自想达到什么，而是去找那条从“输入”通向“结果”的普遍规律。

要注意，他并不是说目标没用。他明确写过，目标、指令和外界刺激都重要，只是退到了次要的位置。他想找的，是不管目标是什么都在起作用的那一部分。

没有目标，那 ergo 靠什么决定往哪儿走？靠“有趣”。

Gromov 对“有趣”的刻画很朴素。太单调的东西无聊，因为没什么可预测、可学的。完全随机的东西也无聊，因为你对它毫无把握。有趣的东西夹在两者中间：既有新意，又让你觉得自己能抓住它。

第三串的信息量比第二串少得多，却最“有内容”：里面藏着词，等着被发现。可见“有趣”不等于信息量大。香农的信息论只数比特，不管结构；Gromov 对这一点很警惕，他认为拿概率和熵去描述语言，本身就要打个折扣。

Gromov 反复用到一个例子：一只在叶子上爬的虫子。

设想这只虫子只分得清两种颜色。在叶子中间爬，周围处处一样，它很快就无聊了。爬到边上，颜色突然变了，它惊奇起来，慢慢学会了沿着边走。沿着边走久了，又无聊了，直到碰上叶尖或者叶柄，它才又来了兴致。

有意思的是，虫子觉得有趣的地方，正好是这张图里信息所在的地方。叶子里面处处一样，几乎不用存；整片叶子的形状，全写在边上。Gromov 粗略估过：存整片叶子的代价跟面积走，存边界的代价只跟周长走。我们照这个思路算了一下：画到 1024 × 1024，逐格存要一百多万比特，只存边界约 5,400 比特，差了将近 200 倍。

在他看来，“理解”在相当程度上就是这样一种压缩：找出重复的、能预测的部分，把真正的结构留下来。他列过理解的几个特征：能压缩，能预测，能借它学到新东西，还有一条容易被忽略，能提出问题。他顺便说，最擅长提问的两种人，是科学家和小孩。

Gromov 最看重的例子是学母语。

婴儿听到的话，词和词之间其实没有停顿，一句话就是一长串连着的音节。他们是怎么知道一个词从哪里开始、到哪里结束的？

Gromov 在手稿里提过一个办法：词，就是那些出现得比“碰巧”频繁得多的音节串；到了词的边界，下一个音节会突然变得难以预测，就像虫子爬到了叶子边上。他自己说，这只是已有统计方法的一个“很粗的版本”。这个思路至少可以追到语言学家 Harris 1955 年的工作。

更妙的是，心理学家真在婴儿身上测过。1996 年，Saffran、Aslin 和 Newport 给 8 个月大的婴儿放了大约两分钟的录音：四个三音节的“假词”随机连读，没有停顿，也没有语调上的提示。一个词里的音节总是一起出现；一个词后面，接的是另外三个词之一。听完以后，婴儿对“词”和“跨着词边界拼出来的音节串”反应不一样，说明他们已经从统计里把词切了出来。没人奖励他们，他们只是在听。

回到开头那张图。Gromov 在《Memorandum Ergo》开篇，把心智画成一张有限的图：ergo 是核心，由一些圈组成；ego 是外围，是一棵棵树，每棵树只在一个点上接到核心。紧接着，他在脚注里补了一句：当然，所有有限图都能这样分解。

这是数学家的一句玩笑，也是一个真定理。

剩下的部分，图论里叫这张图的“2-核”。既然对任何图都成立，这个分解本身并不说明心智有什么特别。但它给了一个好记的画面，下面是我的读法，不是 Gromov 的原话。

在树上，不管往外怎么走，最后都会走到一片叶子，停下来。在核里，从任何一点出发，都能一直走下去，永远走不到头。目标像树：做 A 是为了 B，做 B 是为了 C，一路走到某个终点，就结束了。玩耍像圈：没有终点，转一圈回来，还想再转一圈。

Gromov 写这些手稿时（2011 到 2015 年），他认为最接近自己想法的，是“好奇心驱动”的学习机器。Schmidhuber 从 1991 年起就提出，让机器从“预测上的进步”里得到内在的奖励；Oudeyer、Kaplan 和 Hafner 在 2007 年做出了会自己去找“有学头”的事情来探索的机器人。Gromov 说，他和他们的区别，在于更看重“结构”，而不是“行为”。

这里有个容易说错的地方。Gromov 的“没有目标”，不等于“没有任何准绳”。他自己也写过，ergo 在某种意义上是在尽量提高自己的预测能力，只是他给“尽量提高”这几个字打上了引号。他反对的，是外部强加的具体目标和奖惩，不是一切内在的尺度。

今天的大语言模型，训练的第一步是读海量的文字，学着预测下一个词。没有人告诉它要完成什么任务，这一点很像 Gromov 设想的“读一座图书馆，再把它压缩成一本词典”。区别也很明显：“预测下一个词”本身是一个写死了的目标；模型读过的文字，也比一个孩子一辈子听到的多出好几个数量级。这一段是我的对比，不是 Gromov 的说法。2026 年春天，他在 IHES 开课，题目从细胞一路讲到大语言模型，但没有留下讲义。

下棋也一样。AlphaZero 只靠自己跟自己下，就超过了最强的国际象棋程序，可它一开始就被告知了规则，每盘棋结束还要按胜、平、负打分，靠这个分数一点点改进。Gromov 设想的“ergo 棋手”是另一种东西：只看人类的棋谱，自己猜出规则，觉得某些局面有意思，而且不急着赢。

读 Gromov 的这些手稿，要记住几件事。

第一，这是一个纲领，不是一个理论。他自己说，我们知道这样的学习程序存在，因为我们脑子里就有，但对它是什么毫无头绪。手稿里没有算法、没有定理、没有实验，有的章节只有标题，到处是问号。

第二，“结构”和“有趣”都没有严格的定义，他承认这里有循环。

第三，关于孩子和动物的不少例子是轶事，没有给出处。比如他说，玩耍没有公认的进化上的解释。这是他的立场；研究动物行为的人提出过好几种解释，比如玩耍是在为意外情况做练习。

第四，他给出的那几块数学，分词、聚类、熵、压缩，大多是已有的工具。他做的，是把它们摆到一起，指出一个方向。

那么，ergo 计划留给我们什么？

我觉得是一个看待学习的角度：学习首先不是为了什么，而是因为世界上有有趣的结构，而人天生会被它吸引。目标、考试和奖励可以给这个过程加上方向，可如果它们太多、太早、太硬，就会把它掐断。Gromov 在手稿里不止一次批评过教育：太爱指挥的老师，会挡住孩子自己的学习；理想的老师只需要把材料按难易排好，剩下的交给孩子。

他在手稿里把数学叫作 ergo 最晚出生的孩子，又说科学是一门“不懂”的艺术。总有地方不懂，人才会一直想玩下去。这大概就是 life-long playing 的意思：学习不是一段为了毕业而熬过去的时间，而是一种可以一直持续下去的玩法。

这样看，一个数学家去研究孩子怎么玩，一点也不奇怪。他是在研究自己每天都在做的事。

学习首先不是为了什么，
是因为世界很有趣。

Let's start with a small scene from Gromov's manuscripts.

A child of four or five sees someone balancing a stick upright on a fingertip, and goes off to try it. Nobody teaches the child and nobody gives any praise. The child tries again and again, and in the end gets it. Then Gromov asks: what is it, exactly, that drives a child to do this?

Adults usually don't do this. Before doing anything, an adult asks “what's it for?” Yet children do not learn their mother tongue, or learn to walk, by the logic of “what's it for.”

Training an AI today almost always starts with writing down a goal: an error to drive down, or a reward to collect. Gromov's idea runs the other way: the most basic part of learning is goal-free.

Gromov is one of the most important geometers of our time: winner of the Abel Prize in 2009, on the faculty of the Institut des Hautes Études Scientifiques (IHES) in France since 1982, and now a professor emeritus there. Since around 2010 a different question has taken up a great deal of the geometer's time: how do people actually learn things? It has filled hundreds of pages of manuscripts, and Gromov calls the whole effort the ergo program.

I first heard Gromov talk about life-long playing, playing all your life, on a podcast, and that sent me to the manuscripts. I could not find the phrase anywhere in them, but the idea runs all through them.

The name comes from Descartes' “I think, therefore I am,” in Latin cogito ergo sum. Gromov fixed on the unremarkable word in the middle, ergo, which means “therefore.”

What the brain takes in at every moment is a huge, messy tangle of electrical and chemical signals. To “therefore” a clear, orderly world out of that tangle: in Gromov's view, this step is itself the most structured thing there is, and the most worth studying. The part of the mind that does it gets the name ergo.

The other part is ego, the “I” you normally feel yourself to be. It is practical, sticks to common sense, wants goals, and keeps you and your genes alive. Ergo cares about none of this. It is drawn only to “interesting structure.” Games, art, science and mathematics all come from here.

In adults, ergo is buried deep, and we cannot feel it ourselves. In small children the two have barely separated, so watching children play is the easiest way to see ergo. In a magazine interview at the end of 2013 Gromov put it more bluntly: the program in a child is aimed at learning and understanding, not at survival; survival is the parents' job.

There is also a line that captures it perfectly:

Ego cannot make sense of this seriousness. Gromov gives an example: composing a perfectly useless chess problem is a pleasure for ergo, while filling in a tax form with scrupulous care is something ego finds necessary and ergo finds utterly boring.

The manuscripts have another lovely way of putting it. Wanting to win is ego's business; in the structure of a game, winning and losing are really secondary. A pure ergo would not fight hard to win. It would lower its own level to match a weaker opponent, so that the game is as interesting as possible. Most adults who have played chess with a child have done exactly this.

Why should learning be goal-free? Gromov's argument is short.

People learn wildly different things: their mother tongue, walking, chess, mathematics. If one general mechanism lies behind learning, it cannot depend on any particular goal, because goals share no common structure: the goal of learning to speak and the goal of learning to walk have nothing in common. So the most basic part of the learning mechanism must be independent of goals, and must not rely on rewards and punishments from outside. The best evidence, in Gromov's eyes, is that every child learns a mother tongue, and no one rewards a child word by word.

One analogy comes from physics. Newton said that a body with no force acting on it keeps moving; this is inertia. Learning is like inertia: it goes on by itself. Rewards and punishments are like outside forces. They can bend its path, but they are not why it moves. Gromov even says that rewards and punishments do less to guide learning than to restrict it.

Another analogy is throwing stones.

The hunter cares only about what the stone hits, and aiming at something you cannot eat seems plain foolish. The physicist looks at something else: the parabola from release to landing, fixed by the laws of motion. The targets vary endlessly, but there is only one law. Gromov says the study of learning should be like the physicist's: ignore what each learner is trying to achieve, and look for the universal law that leads from “input” to “outcome.”

Note that this is not a claim that goals are useless. The manuscripts say plainly that goals, instructions and outside stimuli all matter; they just take second place. What Gromov wants to find is the part that is at work whatever the goal may be.

With no goal, how does ergo decide where to go? It goes by what is “interesting.”

Gromov's account of “interesting” is simple. Something too monotonous is boring, because there is nothing in it to predict or learn. Something completely random is boring too, because you can get no grip on it. Interesting things sit between the two: they are new, yet you feel you can grasp them.

The third string carries far less information than the second, yet it has the most “content”: words are hidden in it, waiting to be found. So “interesting” is not the same as “lots of information.” Shannon's information theory counts bits and ignores structure. Gromov is wary of this, and thinks probability and entropy only go so far in describing language.

Gromov keeps coming back to one example: a bug crawling on a leaf.

Suppose the bug can tell only two colors apart. In the middle of the leaf everything around it looks the same, and it soon gets bored. At the edge the color suddenly changes; it is surprised, and slowly learns to walk along the edge. After following the edge for a while it gets bored again, until it reaches the tip of the leaf or the stalk, and its interest comes back.

Here is the nice part: the places the bug finds interesting are exactly where the information in this picture sits. The inside of the leaf is the same everywhere and needs almost no storage; the shape of the whole leaf is written on its edge. Gromov made a rough estimate: the cost of storing the whole leaf grows with its area, the cost of storing the edge only with its perimeter. We ran the numbers along those lines. At 1024 × 1024, storing every pixel takes over a million bits, and storing only the boundary about 5,400 bits, nearly 200 times less.

In Gromov's view, “understanding” is to a large extent this kind of compression: find the parts that repeat or can be predicted, and keep the real structure. The manuscripts list several marks of understanding: it compresses, it predicts, it lets you learn new things, and, one that is easy to overlook, it lets you ask questions. And a remark in passing: the two kinds of people best at asking questions are scientists and small children.

The example Gromov cares about most is learning a mother tongue.

In the speech babies hear, there are in fact no pauses between words; a sentence is one long, unbroken string of syllables. How do they know where a word begins and where it ends?

The manuscripts suggest one way: words are the strings of syllables that turn up far more often than chance would allow, and at a word boundary the next syllable suddenly becomes hard to predict, like the bug reaching the edge of the leaf. By Gromov's own account, this is only a very crude version of existing statistical methods. The idea goes back at least to work by the linguist Harris in 1955.

Better still, psychologists have actually tested this on babies. In 1996, Saffran, Aslin and Newport played 8-month-old infants about two minutes of recorded speech: four made-up three-syllable “words” strung together at random, with no pauses and no cues from intonation. The syllables of a word always came together; a word was followed by one of the other three. Afterwards the infants responded differently to “words” and to syllable strings spliced together across word boundaries, showing that they had already carved the words out of the statistics. Nobody rewarded them. They were only listening.

Back to the picture at the start. In the opening pages of Memorandum Ergo, Gromov draws the mind as a finite graph: ergo is the core, made of cycles; ego is the periphery, a collection of trees, each attached to the core at a single vertex. A footnote right after adds: of course, every finite graph decomposes this way.

It is a mathematician's joke, and also a true theorem.

In graph theory, what remains is called the “2-core” of the graph. Since this holds for every graph, the decomposition by itself says nothing special about the mind. But it gives a picture that is easy to remember. What follows is my reading, not Gromov's words.

On a tree, however you walk outward, you end up at a leaf and stop. In the core, you can start anywhere and keep walking forever, never reaching an end. Goals are like trees: you do A for the sake of B, and B for the sake of C, until you reach some end point, and then it is over. Play is like a cycle: there is no end point; you go around once, come back, and want to go around again.

Writing these manuscripts between 2011 and 2015, Gromov saw “curiosity-driven” learning machines as the closest thing to the ideas in them. From 1991 on, Schmidhuber proposed letting a machine earn an intrinsic reward from its own “progress in prediction”; in 2007, Oudeyer, Kaplan and Hafner built robots that pick out, on their own, what is “worth learning” and explore it. Gromov says the difference lies in focus: the ergo program cares more about “structure” than about “behavior.”

This is an easy place to go wrong. Gromov's “no goal” does not mean “no yardstick at all.” The manuscripts themselves say that ergo in some sense tries to maximize its predictive power, though with “maximize” in quotation marks. What Gromov objects to is specific goals, rewards and punishments imposed from outside, not every inner measure.

Today's large language models begin their training by reading vast amounts of text and learning to predict the next word. Nobody tells them what task to accomplish, which is a lot like what Gromov imagined: reading a whole library, then compressing it into a dictionary. The differences are just as plain. “Predict the next word” is itself a hard-coded goal, and a model reads several orders of magnitude more text than a child hears in a lifetime. This comparison is mine, not Gromov's. In the spring of 2026 Gromov taught a course at IHES whose topics ran all the way from cells to large language models, but left no lecture notes.

Chess is the same story. AlphaZero, playing only against itself, overtook the strongest chess programs. But it was told the rules from the start, and every game ended with a score for a win, a draw or a loss, which it used to improve bit by bit. Gromov's imagined “ergo chess player” is something else: it only watches human games, guesses the rules for itself, finds some positions interesting, and is in no hurry to win.

A few things are worth keeping in mind when reading these manuscripts.

First, this is a research program, not a theory. As Gromov says, we know such a learning procedure exists, because we have one in our heads, but we have no idea what it is. The manuscripts contain no algorithms, no theorems and no experiments; some chapters are only titles, and question marks are everywhere.

Second, neither “structure” nor “interesting” has a rigorous definition, and the geometer admits there is some circularity here.

Third, many of the examples about children and animals are anecdotes, given without sources. For instance, play is said to have no accepted evolutionary explanation. That is Gromov's position; people who study animal behavior have proposed several explanations, such as play being practice for the unexpected.

Fourth, the pieces of mathematics on offer (word segmentation, clustering, entropy, compression) are mostly existing tools. What Gromov does is put them side by side and point in a direction.

So what does the ergo program leave us?

I think it leaves a way of looking at learning: learning is not, first of all, for anything. It happens because the world has interesting structure, and people are born drawn to it. Goals, exams and rewards can give the process a direction, but if there are too many of them, too early, too rigid, they cut it off. More than once in the manuscripts Gromov criticizes education: a teacher who directs too much gets in the way of a child's own learning, and the ideal teacher only needs to arrange the material from easy to hard and leave the rest to the child.

The manuscripts call mathematics ergo's youngest child, and science the art of “not understanding.” There is always something we do not understand, and that is why we keep wanting to play. This is probably what life-long playing means: learning is not a stretch of time to get through until graduation, but a way of playing that can go on and on.

Seen this way, it is not strange at all for a mathematician to study how children play. It means studying the very thing the mathematician does every day.

Learning isn't, first of all, for anything.
It happens because the world is interesting.