数学2009dpmms.cam.ac.uk编辑精选
数学的两种文化
The Two Cultures of Mathematics (2000)
入门 · 约 35 分钟读完前置知识：基础概率论、组合数学常识、线性代数
导读
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“理论构建者”，致力于建立抽象概念框架、追求整体理解。文章讨论了两类工作的价值标准、彼此的张力与互补，并借“测度集中”等现象说明好结果往往能被简洁地组织与传达给后人。
HN 关注点：是因为文中“解题者 vs 理论构建者”的划分可以几乎原样套用到编程领域（hacker 型 vs 理论型程序员），引发了关于两类思维方式如何互相借鉴的讨论。
HN 评论精选
这几条评论里：支持 1 补充 3
补充把 Gowers 的测度集中概念联系到 LASSO 和 Johnson-Lindenstrauss 等 ML/统计应用。
Gowers 文中提到的“测度集中”现象在机器学习/统计中有很多应用，可能比文中的概率组合数学更让 HN 读者感兴趣。例如 LASSO 估计量（最小化 |Y-βX|₂+λ|β|₁）用于变量选择，其合理性来自 Candes 与 Tao 2005 年的结果：在相当一般的条件下该估计能精确恢复真实的活跃变量集合，证明的关键一步正是次高斯随机矩阵的测度集中结果；一个推论是我们可以高概率地通过求解凸松弛 |Y-βX|₂+λ|β|₁ 来精确解决非凸的 |Y-βX|₂+λ|β|₀ 问题，并保证松弛解最优。Johnson-Lindenstrauss 降维引理也基本可由同一原理（随机投影“大概率”近似等距）证明。
ajtulloch查看英文原文
One technique Prof Gower's mentions (the phenomenon of the concentration of measure) has many applications in machine learning/statistics (which may be of more interest to HN readers than some of the probabilistic combinatorics mentioned in the paper). A few (non-rigourous) examples in ML/statistics are: - The use of the LASSO estimator [0] (minimizer of |Y - β X|₂ + λ |β|₁) for variable selection is justified by Candes & Tao's 2005 results [1, 2] that show, under fairly general conditions, that this estimate exactly recovers the true set of active variables. A key step in the proof is finding a concentration of measure result for sub-Gaussian random matrices. A corollary is that (with high probability) we can exactly solve the non-convex problem |Y - β X|₂ + λ |β|₀ by passing to the convex relation |Y - β X|₂ + λ |β|₁ and have strong guarantees that our relaxed solution is optimal. - The Johnson-Lindenstrauss lemma for dimensionality reduction, etc [3] can be proven by essentially the same principle (random projections are "probably" nearly isometries) that Prof Gowers describes in this note. [0]: http://en.wikipedia.org/wiki/Lasso_regression#Lasso_method [1]: http://authors.library.caltech.edu/10092/1/CANieeespm08.pdf [2]: http://arxiv.org/pdf/math/0410542.pdf [3]: http://en.wikipedia.org/wiki/Johnson%E2%80%93Lindenstrauss_lemma
支持赞同“好结果=可压缩、可传达”，并吐槽数学记号难以解析、计算机科学靠源码提供语境。
这篇论文有很多精彩观点，都让我深有共鸣：* 能长久流传的结果，是那些能被连贯组织、经济地向后代解释清楚的结果（对，就是有效压缩！）* 一个结果能被多有效地传达给另一位数学家（或许甚至是任意一个人）。我“学数学”的时间有 90% 花在解析记号上：这些符号到底什么意思？我甚至没法把它复制粘贴到 Google 里得到任何有意义的结果！你注意到我们在计算机科学里没有这个问题吗？因为源代码给了你解释语法含义的上下文！同像性（homoiconicity）是形式语言的万灵药。
ukj查看英文原文
This paper makes a ton of brilliant points all of which strongly resonate with me: * The results that will last are the ones that can be organized coherently and explained economically to future generations (yes! effective compression!) * How effectively a result can be communicated to another mathematician (and perhaps even s/mathematician/person/) 90% of my time spend 'studying mathematics' is spent lexing the notation. What do those symbols even mean? I can't even copy-paste this into Google to get any meaningful results!!!!!!!!!!!!@#!$@#$@#$@#$#@!$@#$ Have you noticed how we don't have this problem in Computer Science? Because the source code gives you the context in which to interpret the meaning of the grammar! Homoiconicity is the panacea of formal languages.
补充把数学的两种文化类比为三类程序员，并强调跨圈子传播好想法（如不可变性进入前端）的价值。
几年前我写过一篇博客，对计算机科学提出了类似的主张：程序员大致有三个阵营——因为编程在数学上很美而喜欢编程的人（比如 Haskell 程序员）；因为喜欢推理机器、有机械同理心而喜欢编程的人（比如 C 程序员）；以及因为编程能为用户解决实际问题而喜欢编程的人。我在 HN 和其他地方遭到反驳，说很多人同时属于多个阵营——这有道理，也许更好的描述是“价值观的生态”。但我仍认为这里有真实的东西。几年前我在一家咨询公司工作，老板从一个前端开发大会回来，兴奋地让我一定要看一个演讲：那是用通俗语言讲函数式编程的不可变性如何应用到 React。我老板从没听说过不可变性，因为它在 Web 前端圈子里从未出现过。通过让各个领域互相“交叉感染”最好的想法，我们的软件有很多改进机会。
josephg查看英文原文
I wrote a blog post a few years ago[1] making a similar claim about computer science. I said there’s 3 main camps of programmers: - People who enjoy programming because it’s mathematically beautiful (eg Haskell programmers) - People who enjoy programming because they like reasoning about machines, and like mechanical sympathy (eg C programmers) - And people who like programming because it can solve real problems for their users. I got pushback here and elsewhere that lots of people fit into multiple camps - which makes sense. Maybe they’re better described as an ecology of values. But I also think there’s something real here. I worked in a consulting shop a few years ago and my boss came back from some front end development conference gushing about a talk he’d seen that I had to watch. The talk was a layman’s description of FP’s immutability approach applied to react. My boss had never heard of immutability before because it had never come up in front end web development circles. There’s lots of opportunities for our software to improve by cross contaminating all our best ideas. [1] https://josephg.com/blog/3-tribes/
补充用“解题者 vs 理论构建者”对照 hacker 与理论型程序员，指出两边各有盲点。
这个论证几乎可以原封不动地套用到 hacker 型与理论型程序员身上：一方属于解题者阵营，另一方属于追求根本理解与理论构建的阵营。硅谷由 hacker 主导，因为缺乏理论理解而不断重复造轮子（比如 NoSQL）；而理论构建者研究的问题，几乎注定会在大学图书馆某个阴暗角落积灰。
_qc3o查看英文原文
The argument can be adapted almost verbatim for hacker type vs theory type programmers. One is in the problem solver camp the other in the fundamental understanding and theory building camp. Silicon Valley is dominated by hackers and they are constantly re-inventing wheels because of lack of theoretical understanding, e.g. NoSQL, and the theory builders are working on problems that will almost surely collect dust in some dark corner of a university library.
相关资源
从这篇论文所有 HN 讨论帖的全部评论里整理出来的链接。
历次讨论
- 2009Hacker News 讨论帖24 分 · 5 评
- 2014Hacker News 讨论帖138 分 · 28 评
- 2021Hacker News 讨论帖145 分 · 69 评
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