遇到了一个有趣的问题，正好落在 AI 模型的能力边界处：试证明不可能把平面分成无穷个圆的无交并。
在我尝试的所有模型里，只有 GPT 5 thinking model 成功做了出来（虽然花了点时间）。
有趣的不是这个结论，而是观察它们的思路。所有失败的模型都有个共同点：它们的思考基本上是从文字到文字的。它们会调用自己脑海中各种已有的定理和知识，然后漫无目的地试图拼凑出一个证明，但所有这些定理，不管是拓扑的还是几何的还是测度的，对它们来说都是纯粹字面意义上的陈述。Qwen 的思考过程最典型：它滔滔不绝想了很久，但很显然从头到尾它都并不真的理解它在说什么。圆也罢，开集闭集也罢，Baire 纲定理也罢，对它来说都是纯粹的概念，给人的感觉是它甚至并不真的知道「圆是圆的」。
微妙之处在于，这种「没有几何直觉的几何思考」在某些时候其实未必是一种劣势。现代数学早已挣脱了对三维现实想象的依赖，大部份数学思考本来也确实是在纯粹的概念思辨空间中进行（特别是当问题进入代数乃至范畴论的领域的时候，这时从概念到概念的思考就变成了一种必然）。有的时候，几何直觉甚至反而会成为一种束缚，特别是当思考高维空间的时候，基于低维现实的直观常常是有误导性的。在这些问题上，AI 的「盲目」反而带来了自由，使得它不必受困于视觉直觉。——当然，人类的视觉直觉可能会渗透进人类的文本语料里，在某种程度上「污染」AI，但这是另一个问题。
然而对原问题来说，因为这是一个低维问题，几何直觉在这里不但有用，而且能大大缩短思考搜索的难度。在这一点上，一个把圆只作为抽象概念来理解的 AI 就会有巨大的劣势，因为它无法享受到几何直觉带来的跳步。这种直觉使得人可以一眼「看出」关键的构造，而这种构造在文本层面被搜索出来是困难的。
考虑到 AI 的应用毕竟大多数情况下还是为了解决世界现实问题而不是思考高维几何，有几何直觉的 AI 会在大多数问题上显得聪明得多。于是一个现实问题是，这种直觉是只有依赖多模态的训练才能获取，还是可以通过精巧的文本训练就能实现？这有点像是 AI 领域的玛丽房间问题。这是一个经典的知识论思想实验：一个从出生就生活在黑白房间里、精通颜色物理与神经机制的科学家玛丽，当她第一次走出房间看到红色时，她是否获得了新的知识？
今天大多数 AI 领域的困难都可以归结于此。人类是自己感官的奴隶，我们听到、看到、闻到，我们体会身体激素的涨落，我们想象、困惑、愤怒，然后试图把这一切投射在文字空间里。AI 则正好相反，它们在文字里理解这一切，但最终需要努力地——有时候是徒劳地——明白，一个圆在什么意义上是圆的。
Circle
I came across an interesting problem that happens to sit right at the boundary of an AI model’s capabilities:
Prove that it is impossible to partition the plane into an infinite (uncountable) disjoint union of circles.
Among all the models I tried, only the GPT-5 thinking model managed to solve it (though it took some time).
What’s interesting is not the conclusion itself, but watching the thought process. All the unsuccessful models shared a common feature: their “thinking” was essentially text-to-text. They would call up all sorts of theorems and knowledge they had stored, then aimlessly try to patch together a proof. But all those theorems — whether from topology, geometry, or measure theory — were, for them, nothing more than purely verbal statements. Qwen’s process was the most typical: it rambled on for quite a while, but it was obvious that it didn’t truly understand what it was saying. Whether it was circles, open or closed sets, or the Baire category theorem, for it these were all purely abstract notions — one got the sense it didn’t actually “know” in what sense “a circle is a circle.”
The subtlety here is that this lack of “geometric intuition” isn’t necessarily a weakness in all situations. Modern mathematics has long since freed itself from reliance on imagining three-dimensional reality; most mathematical reasoning is indeed carried out in purely conceptual space (especially when problems move into the realms of algebra or category theory, where concept-to-concept thinking becomes inevitable). In some cases, geometric intuition can even be a hindrance — particularly when thinking about high-dimensional spaces, where low-dimensional, real-world intuition is often misleading. For these problems, an AI’s “blindness” can, in theory, actually grant it freedom — it need not be trapped by visual intuition. Of course, human visual intuition can seep into human-produced text corpora, “contaminating” the AI to some degree, but that’s another issue.
However, for the original problem which is low-dimensional, geometric intuition is not only useful but can greatly reduce the difficulty of the search. In such cases, an AI that understands a circle only as an abstract concept is at a major disadvantage, because it cannot enjoy the shortcut that geometric intuition provides. Such intuition lets a human “see” the key construction at a glance — something that is much harder to search for in purely textual space.
Given that AI is, in most real-world applications, meant to solve practical problems rather than think about high-dimensional geometry, an AI with geometric intuition would appear far more intelligent in most scenarios. This raises a real question: can such intuition only be acquired through multimodal training, or could it be achieved through cleverly designed text-only training? It’s somewhat like the “Mary’s Room” problem in AI. This is the classic thought experiment in epistemology: a scientist named Mary, who has lived her whole life in a black-and-white room and knows everything about the physics and neuroscience of color, steps outside and sees red for the first time — does she gain new knowledge?
Most of today’s hardest challenges in AI can be traced back to this. Humans are slaves to their senses: we hear, see, smell, feel the ebb and flow of our hormones, imagine, get confused, feel anger — and then we try to project all of this into the space of language. AI is the exact opposite: it understands all these things within the space of language, but must then work — sometimes in vain — to truly grasp in what sense a circle is a circle.