数学2020people.math.harvard.edu
数学中的一些基本定理（2019）
Some Fundamental Theorems in Mathematics (2019)
进阶 · 约 180 分钟读完前置知识：微积分与线性代数、数学分析基础、抽象代数与拓扑基础
导读
哈佛数学家 Oliver Knill 撰写的一份长篇讲义，系统梳理了数学各分支（代数、分析、拓扑、概率等）中的一百多个基本定理，以清晰的陈述形式呈现。它源于作者在哈佛扩展学院讲授的一门面向多学科的课程，目标是让读者（包括作者自己）快速纵观数学全貌、达到接近研究生一年级的水平。
HN 关注点：HN 读者喜欢它作为一份浓缩的数学地图：非数学专业的人不用读几十本书就能看到各领域最重要的定理长什么样，讨论也围绕其准确性和学习价值展开。
HN 评论精选
这几条评论里：支持 2 质疑 1 补充 1
支持这份文档让非数学专业的人无需系统学习就能纵观数学重要定理，性价比极高。
我非常喜欢这份文档。作为一个热爱数学但本科没读数学专业的人，我错过了太多东西，而且已经没有时间从本科教材开始补三年基础了……这份文档里清晰的定理陈述至少让我看到有哪些东西值得学。它确实很长，但比起要读几十本书才能遇到这些定理，它其实非常短！文档末尾有一些有用的背景说明：这个项目的动机来自作者在哈佛扩展学院教的一门多学科数学课程，试图勾勒数学的主要版图；另一个目标是让自己恢复到研究生一年级的水平（很多年过去会忘掉很多东西），也许还能碰运气通过资格考试。
ivan_ah查看英文原文
Wow I really like this document. As someone who really like mathematics but didn't get to do this as my major in undergrad, I'm missing out on so much, but also there is not enough time to start with undergrad books and do three years of basics... This document with all the clear statements at least allows me to see what is there to learn. Yes it's long, but compare that to dozens of the other books I would have to read to encounter these theorems, it's actually very short! There is some useful "about this document" info near the end: > The motivation to try such a project came through teaching a course called Math E 320 at the Harvard extension school. This math-multi-disciplinary course is part of the “math for teaching program”, and tries to map out the major parts of mathematics and visit some selected placed on 12 continents. [...] A goal of this project is also to get back up to speed up to the level of a first year grad student (one forgets a lot of things over the years) and maybe pass the quals (with some luck). via http://people.math.harvard.edu/~knill/graphgeometry/papers/fundamental.pdf#page=127
质疑文档整体有价值但需谨慎使用，评论者指出了第 7 页关于幺半群扩张的一个具体数学错误。
看到有人整理自己喜爱的定理很好，但如果你要使用这些内容，务必谨慎。除了拼写错误（如 'Perseval' 应为 'Parseval'）和一些奇怪之处外，我注意到第 7 页有一个错误陈述：把'将幺半群扩张为群'理解为把它实现为某个更大群中的子幺半群，这在一般情况下是做不到的。例如'最小值幺半群'（对象是自然数，两个数的乘积是它们的较小值）不满足消去律，因此无法嵌入群中。正确的说法是：对每个交换幺半群，存在一个到群的泛同态，所有其他此类同态都通过它分解（在我举的例子里，这个泛同态就是到零群的常值 0 同态）。
JadeNB查看英文原文
It's great to see someone assemble some of their favourite theorems, but, if you have occasion to use these, make sure that you do so with caution. Aside from some typos ('Perseval' for 'Parseval') and some curiosities (stating on p. 3 that the cardinalities of a set and its power set are different, which is true; but why not say that the latter is larger?), I noticed a wrong statement on p.7: assuming that to 'extend' a monoid to a group is to realise it as a submonoid of a larger monoid that is also a group, this cannot be done in general. (For example, the 'min-monoid' whose objects are the natural numbers, and in which the product of two natural numbers is their minimum, does not embed in a group because it obeys no cancellative law.) What is true is that, for every commutative monoid, there is a universal homomorphism from that monoid to a group, in the sense that all other such homomorphisms factor through it. (In my example, this universal homomorphism is the constant-0 homomorphism to the 0 group.)
补充以幽默方式对比：应用数学中默认一切函数和级数都'足够好'，与纯数学的严格条件形成反差。
我想起应用数学的基本定理，粗略的说法是'在应用数学里，看起来对的它就是对的'，更精确的表述是：在应用数学中，所有合理的级数都收敛，所有函数几乎处处连续、可微且光滑，所有极限（和积分）都存在，所有求和与积分都可交换次序，所有泰勒级数都是好的近似，你想用的定理的那些'良好性态'条件都自动满足，而且每个方程都有唯一解。
dan-robertson查看英文原文
I recall learning the fundamental theorem of applied maths. The approximate phrasing is “in applied maths, if it looks right then it is” and a more precise phrasing is that “in applied maths, all reasonable series converge, all functions are continuous, differentiable and smooth almost everywhere, all limits (and integrals) exist, all sums or integrals commute, all Taylor series are good approximations, all of the silly well-behaved ness conditions of the theorems you want apply, and every equation has one solution”.
支持这篇文档名不虚传，内容有深度而非泛泛罗列。
太棒了。通常标题带'一些基本的'的东西都是简短肤浅的清单，但这篇内容相当有深度。
jmount查看英文原文
That was fantastic. Often things titled "some fundamental" are a short superficial list, but this was some great deep stuff.
历次讨论
- 2020Hacker News 讨论帖234 分 · 91 评
同领域还值得读
一种新的相关系数
A New Coefficient of Correlation
提出了一种新的相关性系数（由 Sourav Chatterjee 提出），能够检测变量之间任意形式（不限于线性）的相关关系，且计算简单高效，主要基于排序和距离计算。
数学2021 · 682 分 · 158 评 · 引用 1进阶 · 约 60 分钟编辑精选数学的两种文化
The Two Cultures of Mathematics (2000)
这是数学家 Timothy Gowers 于 2000 年发表的一篇随笔，他把数学研究大致分成两种文化：一类是“解题者”，热衷于攻克具体难题、发展技巧（如组合数学）；另一类是“理…
数学2009 · 307 分 · 102 评入门 · 约 35 分钟编辑精选重新定义统计显著性
Redefine statistical significance
提议将统计显著性阈值从常用的 p<0.05 收紧到 p<0.005，以降低假阳性率、缓解可重复性危机。
数学2017 · 303 分 · 114 评入门 · 约 45 分钟编辑精选统计建模：两种文化（2001）
Statistical Modeling: The Two Cultures (2001)
讨论统计学中两种建模文化：一种假设数据由某种随机模型生成，侧重推断与系数解释；另一种把数据生成机制视为未知，侧重预测精度，主张使用算法建模。
数学2015 · 254 分 · 36 评进阶 · 约 60 分钟编辑精选通信的数学理论 (1948)
A Mathematical Theory of Communication (1948)
这是香农 1948 年的经典论文《通信的数学理论》，奠定了信息论的基础。
数学2016 · 169 分 · 53 评入门 · 约 90 分钟编辑精选