为什么突然感觉数学学习依靠积累而非智商?

While Working on my Notes

作者：小平邦彦（Kunihiko Kodaira）

时间：1980年8月

对我来说，没有什么比数学书（和论文）更难读的了。读一本从头到尾有几百页的数学书是一项艰巨的任务。当你打开这样一本书时，首先遇到的是一些定义和公理，接着是定理和证明。由于一旦有了洞察力，数学就变得平淡而容易，你努力通过阅读定理来领会其中的洞察力，并尝试自己证明定理。很可能，你的思维并不能让你走得很远，所以你别无选择，只能读书中的证明，但通过一两次的阅读却无法理解它。这就是为什么你把证明抄到你的笔记本上。但是这一次，你会突出那些你不喜欢的证明的部分。你会问自己是否可能有另一种证明方法，最理想的情况是立即找到一个，但如果找不到，就需要花费很长时间才能放手。经过一个月才艰难地完成一章的学习后，你已经忘记了开头，现在你必须进行复习。这一次，章节中材料的安排开始困扰你，并使你想到像“在证明定理 3 之前先证明定理 7，会不会更好”这样的想法，然后你重新写了整个章节。现在你觉得自己终于理解了第一章，但与此同时也感到困扰，因为这一切花费了你太多的时间。单单时间就几乎不可能让你从一本书的数百页读到最后一章。如果有人知道更快速的阅读数学文本的方法，我希望他们能教教我。

有人可能会说，也许直接读到最后而不考虑任何非必要的东西可能更好。这当然是正确的，我可以更快、更顺利地阅读与我自己领域无关的数学书籍（尽管我很少阅读与我领域无关的数学文本）。我可能已经读过它们，但是否有任何洞察力是非常可疑的。一个人在什么阶段可以说自己已经阅读并理解了一篇数学著作呢？逐步验证了证明并同意没有错误是否足够了？如果我读了一本来自陌生领域的数学书，我注意到那些我不理解的定理即使在验证了它们的证明之后也会困扰我。证明可能是正确的，但整体画面却是模糊的、朦胧的。另一方面，当我理解来自我自己领域的定理时，即使忘记了证明，我也能完全理解它，就像理解 2 + 2 = 4 的事实一样：对 2 + 2 = 4 背后数学真理的本能把握产生了理解，而不是来自证明。在类似的思路中，理解任何定理似乎意味着对其提供的基本数学真理有一种感觉。我认为，逐步跟踪证明更像是吸收定理直觉的一种好方法，而不是验证证明中的论证是否正确。（一个著名定理的证明是正确的这一事实应该是如此清晰，以至于不需要每个人都去检查它。）这就是为什么为了深入理解一个定理，仅仅读一遍证明是不够的，更有益的是一遍又一遍地阅读它，将其抄写到你的笔记本上，并尝试将其应用于各种问题。写出证明并不是为了记住它，而是为了花时间详细观察定理背后的数学思想是如何建立起来的。通过这种方式完全理解了定理后，完全没有反对忘记整个证明（除非你还没有从大学毕业，在这种情况下，最好为了考试把它记住）。如果以后需要忘记的证明，并对其进行复习，那么它甚至可能会看起来像是对定理的一个不自然的标签，而定理本身就像 2 + 2 = 4 一样清晰明了。

数学是一个高度技术性的学科。获得人们称之为技术性的任何东西都需要大量反复练习。例如，任何想成为钢琴家的人都别无选择，只能从小就每天练习数小时。数学也有类似的一面，我认为你需要每天花费许多小时做重复的练习才能掌握它。这就是你对数学真理的直观把握如何发展的方式。阅读与你自己领域无关的数学书籍，即使在验证了定理的证明之后仍然不理解定理，这表明你对该领域的直觉尚未成熟。

原文：

Nothing is more unreadable to me than books (and papers) on mathematics. Reading a book on mathematics with hundreds of pages cover to cover is an arduous task. When you open such a book, you rst encounter some denitions and axioms, followed by theorems and proofs. Since mathematics becomes plain and easy once the insight is there, you make an eort to get to it just by reading the theorems - and try to produce the proofs on your own. Most likely, your thinking doesn't get you very far, so you have no choice but to read the proof in your book, but you can't make sense of it by looking at it once or twice. That is why you copy the proof to your notebook. But this time, the parts of the proof you dislike leap out. You ask yourself if there could be another proof, and ideally nd one immediately, but if you don't, it takes quite some time until you let go. After you have spent an entire month getting nally through one chapter this way, you have forgotten the beginning, which you now have to revise. This time, the arrangement of the material in the chapter starts to bother you, and makes you think something like Wouldn't it be better to prove theorem 7 before theorem 3, and you rewrite the entire chapter. Now you feel condent that you have nally understood chapter 1, but at the same time troubled, since this all cost you so much time. Time alone can make it next to impossible to get through hundreds of pages in a book to the last chapter. If anybody knows a faster method of reading mathematical texts, I would like them to teach me.

Somebody might say that perhaps it is better to read straight to the end without thinking about anything non-essential. This is certainly true and I can read books from elds of mathematics other than my own faster and more smoothly (although I rarely read mathematical texts unrelated to my eld). I may have read them, but it is acutely dubious whether there is any insight. At what stage can one say that one has read and understood a written piece of mathematics? Is having veried the proof step by step and having agreed that there is no error enough? If I read some book on mathematics from an unfamiliar eld, I notice that the theorems that I don't understand puzzle me even after having veried their proofs. The proof may be correct, but the overall picture is fuzzy and foggy. When I understand a theorem from my own eld on the other hand, I comprehend it perfectly even after forgetting the proof, as perfectly as the fact that 2 + 2 = 4: The understanding behind 2 + 2 = 4 comes from an instinctive grasp of the mathematical truth behind 2 + 2 = 4, and not from a proof. In a similar vein, understanding any theorem seems to mean having a sense of the underlying mathematical truth that it provides. I think that following a proof step by step serves more as one good method for absorbing a theorem intuitively than for verifying that the arguments in the proof are correct. (The fact that the proof of a famous theorem is correct should be so clear that it doesn't have to be checked by everybody.) This is why in order to understand a theorem well, reading the proof just once won't suce, but it is more benecial to read it again and again, copy it to your notebook, and try to apply it to various problems. Writing the proof out is not something you do for memorizing it, but for taking your time to look in detail at what builds up to the mathematical idea behind the theorem. Once you have gained a complete understanding of the theorem in this way, there is no objection at all to forgetting the entire proof (except if you haven't graduated from university, in which case you better have it memorized for the exam). If you happen to need the forgotten proof later and review it, it may then even seem like an unnatural tag on the theorem which itself is as clear as 2 + 2 = 4.

Mathematics is a highly technical subject. Acquiring anything that people call technical needs extensive and repetitious practice. For example, anybody who wants to become a pianist has no choice but to practice every day for hours from childhood on. Mathematics also has a similar aspect to this and I think you need to spend many hours every day doing repetitious exercises in order to master it. This is how your intuitive grasp for mathematical truth develops. Reading books on mathematics from a eld unrelated to your own and not understanding theorems even after having veried the proofs is a sign that your intuition for that area isn't yet mature.

Suugaku Seminaa, August 1980. Translation: Florian Sprung.