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In the early 1960s, Abraham Robinson established nonstandard analysis rigorously. Not digging into the controversies around hyperreal numbers and infinitesimals, here I give a handwaving introduction to infinitesimals, following the construction in Lectures on the Hyperreals, an introduction to nonstandard analysis, GTM 188, by Robert Goldblatt, Springer 1998. Assuming we know what natural numbers are,Continue reading “A Handwaving Introduction to Infinitesimals”

上一篇《数学严格性》讲了数学的严格性，基本意思就是说要建立严格的定义和公理体系，让数学的概念和演绎推理毫无争议。我还说严谨和严格有点细微的区别，那我是怎么理解严谨这词呢？

数学严谨性和严格性是两个相关但是不相同的东西。我今天讲一下什么是数学严格性。举个简单的例子，如何定义一个线段的长度？

2021年5月1日，Lexington High School邀请丘成桐通过ZOOM做了一个在线讲座。我听了一下颇有感触。先总结一下这个讲座的主要内容吧。

Here are some high quality textbooks for middle and high school mathematics. They are good in the sense that they include a lot of exercises in different difficulty levels (A, B, and C). Some of the Level C problems are actually pretty hard. Prealgebra: Algebra 1, Geometry, Algebra 2: The more challenging Art of ProblemContinue reading “Textbooks”

There is no truth in mathematics, only unambiguity.

There is no speed limit for talented students The US middle and high school math system consists of Prealgebra, Algebra 1, Geometry, Algebra 2, and Precalculus, followed by AP courses: AP Calculus AB, AP Calculus BC, AP Statistics, AP Computer Science, etc. The names of Prealgebra, Algebra 1, Geometry, and Algebra 2 do not makeContinue reading “Math Pace”

John von Neumann said: “Young man, in mathematics you don’t understand things. You just get used to them.” This quote probably is confusing for many people. Here is my interpretation: It is not that mathematics do not need understanding. It is that if you do not get used to them, you can not understand them.Continue reading “Understanding Math”