Math

Count Number of Maps: First Exercises of Conceptual Mathematics in Clojure

As previously mentioned, I am learning category theory, beginning with Lawvere's Conceptual Mathematics book. This is a very elementry book that assumes almost nothing as a background. However, it is still a math book, which requires doing some exercises. Since the book provides no answer to exercises, I decide to make my own and post them here as I did them. Hopefully someone will find them useful. Continue reading »

Start learning category theory

Perhaps due to my rather small brain (literally), I dislike remembering tedious details. When in elementary school, I hated reciting classic Chinese poems, but liked composing my own :-).  In high school, I hated chemistry but loved physics, because one could do everything based on a few principles in physics, whereas chemistry was all about memorization. Last year, I was chatting with a colleage of mine who had a Ph.D. degree in chemistry from Harvard. He said he's good at it because he could find patterns in all the tedious details and summerized them in his own head, so he didn't have to remember them all. So I said why not write those patterns down so others can benefit, and he didn't seem to like that idea. Anyway, let's go back to the main topic.

Why category thoery is fundamental yet hard to understand

Category theory is mathematical theory about mapping. Some call it the foundation of mathematics.

According to my understanding of math as reproducible mental object, this is not supprising. As all the brain does is doing mapping. Or in Plovian's words: conduction. Signals going from one set of neuros to another.  Understanding mapping is fundamental obviously.

Now category is billed the hardest to understand branch of mathematics, also known as "abstract nonsense". 

Display LaTeX Math on Drupal with MathJax

MathJax seems to be the emerging standard for displaying math on the Web at this moment. It is supported by American Mathematical Society and American Physical Society, and has already been adopted by major math related discussion venues such as Physics Forums and Stack Exchange. MathJax displays math using CSS and Web fonts instead of images, so the quality is very high and is resize-friendly. Below are some examples:

Inline math: the geometric product $\boldsymbol{uv}$ of vectors $\boldsymbol{u}$ and $\boldsymbol{v}$ is $\boldsymbol{u}\cdot\boldsymbol{v} + \boldsymbol{u}\wedge\boldsymbol{v}$, where $\boldsymbol{u}\cdot\boldsymbol{v}$ is the inner product and $\boldsymbol{u}\wedge\boldsymbol{v}$ is the outer product.

显微镜下的数学

显微镜下的数学 Mathematics under microscope (电子书下载)是本数学家写的
从认知角度看数学的书,挺有启发性的。

数学和心理学的联系,应该要比和物理学的联系还要紧密,还要更自然:数学是心理产
物(Mathematics is the study of mental objects with reproducible properties.
p.IX),而心理学是研究心理产物的产生过程的。如果对心理产物本身都不了解,如 Continue reading »

Yaha! A Math Game for Everyone.

Yaha! is a traditional math game popular among Chinese speaking populations. It has been historically given a varieties of names, but the essence of the game is the same: the players are given 4 playing cards, their task is to use the 4 numbers represented by the cards, use them once and only once, to get the number 24. The players are allowed to apply all four arithmetic operations: addition, subtraction, multiplication and division, as well as parenthesis as they see fit. The winner is whoever comes up with a correct solution first.

This program is a simulation of the game. It is at present form designed to be played by only one player. Remember, your goal is to form an arithmetic expression by dragging cards and operators into appropriate slots and let the result of your expression equals to 24. Got it? Click the picture below to load the game. It will start in a new window.

一个关于逻辑和概率的简单科普

人们一般说的逻辑指命题逻辑,它是理性思维的基本规则。比如以罗素为代表的数理哲学家就把逻辑看作数学的基础。命题逻辑的规则并不多,很多人心里都是知道一些大概的的,不过有人没有正式学过,有些地方不太清楚而已。这篇文章就简单扼要地说一说。更重要的是,本文要介绍现在一个新的进展,就是把概率论看作逻辑的扩展,把传统的亚里士多德的逻辑看成是概率论的特例。 Continue reading »

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