不要害怕学习领域之外的知识
不要害怕学习领域之外的知识
试着对一切事物都了解一点，对某一事物了解一切。 — 托马斯·赫胥黎
数学恐惧症在更广泛的社群中是一个普遍存在的问题。不幸的是，它有时也存在于专业数学家之中（连同它的远亲——数学势利）。
如果事实证明，为了在你的问题上取得进展，你必须学习一些外部的数学知识，这是一件好事——你自身的数学范围将会扩大，你将获得一些新工具，你的工作将变得更加有趣，无论是对你所在领域的人还是对外部领域的人都是如此。
如果一个数学领域有很多活动，通常值得去了解它为什么如此有趣，人们在那里试图解决什么样的问题，以及该领域产生了哪些”酷”或令人惊讶的见解、现象和结果。（另见我关于什么是好的数学的讨论。）这样，如果你在自己的工作中遇到类似的问题、障碍或现象，你就知道该去哪里寻找解决方案。
学习领域之外知识的一个好方法是参加领域之外的讲座和会议。另一个方法是进行智能的文献搜索，以找到某个学科中的关键研究论文、综述或书籍。在这方面，我发现一个特别有用的技巧是：从一篇你知道相关的论文开始，要么通过查看论文的引言，在参考文献中找到该学科中一篇重要的早期论文，要么进行引用搜索（例如通过Mathscinet）找到引用你已经知道的那篇论文的该学科中一篇重要的未来论文。（如果有很多这样的论文，我经常发现按它们自身的引用次数对引用论文进行排序很有启发性，因为这往往会在排序列表的顶部突出特别关键的论文。）将这些过程迭代几次后，通常最终会得到一个很好的关键论文列表，然后你可以仔细阅读这些论文来了解该学科。
另见”学习和重新学习你的领域”。
Don’t be afraid to learn things outside your field
Try to learn something about everything and everything about something. — Thomas Huxley
Maths phobia is a pervasive problem in the wider community. Unfortunately, it sometimes also exists among professional mathematicians (together with its distant cousin, maths snobbery).
If it turns out that in order to make progress on your problem, you have to learn some external piece of mathematics, this is a good thing – your own mathematical range will increase, you will have acquired some new tools, and your work will become more interesting, both to people in your field and also to people in the external field.
If an area of mathematics has a lot of activity in it, it is usually worth learning why it is so interesting, what kind of problems people try to work on there, and what are the “cool” or surprising insights, phenomena, results that that field has generated. (See also my discussion on what good mathematics is.) That way if you encounter a similar problem, obstruction, or phenomenon in your own work, you know where to turn for the resolution.
One good way to learn things outside your field is by attending talks and conferences outside your field. Another is to make an intelligent literature search to locate key research papers, surveys, or books in a subject. One specific trick I have found very useful in this regard is to start with a paper that you know to be relevant, and either look through the introduction of the paper to find an important earlier paper in that subject in the references, or to do a citation search (for instance via Mathscinet) to find an important future paper in the subject that cites the paper you already know about. (If there are many such papers, I often find it illuminating to sort the citing papers by the number of citations that they themselves have, as this tends to highlight particularly pivotal papers at the top of this sorted list.) After iterating these procedures a couple times one usually ends up with a good list of key papers that one can then read carefully to get a feel for the subject.
See also “Learn and relearn your field”。