问HN：数学中的阈限概念
在数学教育中，我们可以提到*阈限概念*。这些是数学概念，具有以下特点：
1) 直观上、概念上和计算上都非常难以理解；
2) 对于深化一个人的数学理解至关重要；
3) 最终变得相当简单（“我之前怎么会不理解呢？”），因为我们使用它们的频率很高，任何数学家经过一段时间后都会内化这些概念。
在这三个标准的要求下，我想到四个阈限概念：
- 基础代数（众所周知，许多孩子在从算术过渡到代数时，在中学数学上会遇到很多困难）；
- 微分和积分（据我所知，对于大多数学生来说，微分似乎比这两者中的积分更难，因为它要求他们以新颖的方式思考图形）；
- 实分析中的德尔塔-埃普西龙论证（99%的数学本科生都能确认这一点）；
- 高级集合论中的强制性（我自己对此一无所知，但我在多个地方读到它可能会让人感到非常棘手；但我不确定它是否满足第（3）条）。
你知道数学中还有其他这样的阈限概念吗？
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In mathematics education we can find mention of *threshold concepts*. These are mathematical concepts that:<p>1) are notoriously difficult to grasp, both intuitively, conceptually and computationally;
2) are central and critical for furthering one's mathematical understanding;
3) eventually become quite easy ("How is it I could not understand it before?"), because we employ them so much that any mathematician will internalize them after a while.<p>Setting bar for these three criteria high, four threshold concepts come to my mind:<p>- basic algebra (it is well-known that many children struggle a lot with middle school maths when transitioning for arithmetics to algebra);
- differentiation and integration (AFAIK, differentiation seems more difficult of these two for most students, because it makes them think about graphs in a novel way);
- delta-epsilon arguments in real analysis (as 99% undergraduate students in maths can confirm :));
- forcing in advanced set theory (I know nothing about it myself, but I have read several places that it can be a backbreaker; but I am not sure whether it satisfies (3)).<p>Do you know other examples of such threshold concepts in mathematics?