数学2019arxiv.org
数学中的一些基本定理
Some Fundamental Theorems in Mathematics
进阶 · 约 90 分钟读完前置知识：线性代数、概率论基础、数理逻辑基础
导读
这篇论文（Keith Conrad 等人的 arXiv 综述）以鸟瞰视角罗列并简要讲解数学各分支中的一些基本定理，如广义勾股定理、中心极限定理、遍历定理、哥德尔不完备定理等。HN 读者认为这是一份难得的全景式数学概览，但讨论中也指出文中若干表述存在疏漏。
HN 关注点：数学家平时大多局限于自己的细分领域，这种跨领域的基础定理总览很少见，读者既欣赏其价值，也热衷于挑错和讨论表述的严谨性。
HN 评论精选
这几条评论里：支持 1 质疑 3
质疑认为广义勾股定理的讲解应保留内积空间形式，并展示抽象与初等形式的关系
第二个定理（基于内积空间的广义勾股定理）的讲法让我觉得不完整。既然第一页就出现内积空间，读者显然学过线性代数，那么结尾不该停留在 a²+b²=c² 这种初等形式，而应给出 ||x+y||² = ||x||² + ||y||² + 2⟨x,y⟩，并说明抽象形式如何蕴含我们熟悉的简单形式。既然这份综述本来就不自足，不如更完整地展示优雅的抽象如何漂亮地推出已知结论。
throwawaymath查看英文原文
The second theorem (the generalized Pythagorean theorem, which uses inner product spaces) feels incomplete to me. The exposition is fine, but I don't think the explanation should have concluded with the regular a^2 + b^2 = c^2 equality. You wouldn't use that in the context of an inner product space. The generalized Pythagorean theorem actually looks like ||x + y||^2 = ||x||^2 + ||y||^2 + 2<x, y>. I think it would be better if there's less editorializing of the equalities and more relating of their abstract forms to their "simple", familiar forms. Otherwise I don't really understand the exercise - if the first page has a theorem involving inner product spaces, this is clearly targeted at people with at least a full course of linear algebra under their belt. Unless the audience is already aware of how that second equality implies the first equality, this explanation doesn't capture the heart of it. As another commenter said, this is not a self-contained exposition (and I don't think it realistically could be). But if it's not going to be self-contained, I think it could be improved by more completely showing how the elegant abstractions imply the things we're already familiar with in a neat way.
质疑指出哥德尔不完备定理被误述：不完备性只针对递归可枚举的算术公理系统
第 20 条（逻辑部分）的表述不对，这种错误很常见。完备的公理系统其实很多——把某个模型中所有真命题都取为公理，就得到一个完备系统。真正不完备的是包含标准整数算术的递归可枚举公理系统。总体而言这份清单不错。
skh查看英文原文
Number 20 in logic is incorrect. It’s surprising how often this is misstated. There are lots of complete axiom systems. Indeed, take as your axioms all true statements in a given model. Voila, you now have a complete axiom system. It’s not just a useful one but it is complete. What isn’t complete is a recursively enumerable axiom system that contains arithmetic of the standard model of integers. Overall it is a nice list and well done. EDIT: Deleted a sentence that was pointed out to me that was wrong in the context of the assumptions made in the paper.
支持赞赏这份综述提供了数学家少有的跨领域全景视角
我本来是准备进来挑刺的（这个毛病我确实该反省一下），但这其实是一份难得的领域鸟瞰图。数学家可能比其他学科的人更容易困在自己的专门子领域里。
thearn4查看英文原文
I came into the link ready to nitpick the choices (for some reason which I really should examine), but it's actually a nice birds-eye-view of the field that is rare to see. I feel like mathematicians, perhaps more than most, tend to stay siloed within our specialized subtopics.
质疑逐条列举文中多处数学表述的不严谨之处
这份清单有趣但有点粗糙：中心极限定理依赖均值和方差存在，但有些随机变量没有；遍历定理中'保测'的定义漏了测度；dim(ran A)^⊥ = dim ker A^T 可以加强为直接的空间相等；第 20 条有问题；Haar 测度的定义需固定某个非零测度集的测度，否则'唯一'应改为'差一个缩放倍数下唯一'；还有'Sounders Mac Lane'拼写错误、牛顿法收敛域可能极小等多处问题。
tmyklebu查看英文原文
This list is fun, but it's a little bit sloppy: - The central limit theorem relies on the mean and variance of your random variable existing; there are random variables for which mean and variance don't exist. - The definition of "measure-preserving" in the ergodic theorem statement is missing the measure. - dim (ran A)^perp = dim ker A^T can be strengthened by dropping the dim's. If v in ker A^T, then A^T v = 0. Pick any Ax in ran A; then <v, Ax> = <A^T v, x> = <0, x> = 0, so v is in (ran A)^perp. - Others pointed out that 20 looks busted. - The definition of Haar measure needs to fix the measure of some nonzero-measure set, or "unique" needs to be replaced with "unique up to scaling" otherwise the theorem isn't true. - "Sounders Mac Lane" - 25: x_0 came out of nowhere; it's unclear from the text which "invertible" is meant; the basin of convergence for Newton's method for finding a local continuation can be very small indeed. - 36: Why is the adjoint of A named T^*? - 46: You need some assumptions about f.
历次讨论
- 2019Hacker News 讨论帖419 分 · 71 评
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