Mathspp: mathematics and programming

Pt En What if I told you that right now there is a place on Earth where there is no wind blowing to the sides? None at all! How can I know that? All we need is what is usually called the Hairy Ball Theorem . In less rigorous contexts, one can phrase the Hairy Ball Theorem as such: Theorem (Hairy Ball I): if you have a hairy ball, regardless of the way you comb its hair there will always be a spot where the hair points right up. In this particular image, the hair is pointing up both on the top and on the bottom. More formally, the Hairy Ball Theorem can be formulated like so: Theorem (Hairy Ball II): every continuous vector field over $S^2$ has at least a point where the tangential component is $0$. From this theorem it is actually quite easy to establish our interesting fact! If we think of the wind at the Earth's surface as a continuous vector field, the Hairy Ball Theorem says that there must be a point where the wind isn't blowing to the sid...

Pt En This post will be more theoretical than usual. If you are afraid of mathematics, go away now before it is too late! I will write about a set, the Vitali set, which arises in measure theory , to show that there is no coherent way of assigning a size to every subset of the real line. Because of that I will try to define size for subsets of the real line and then verify that such task is impossible. I will also enforce a couple of (seemingly reasonable) restrictions on my definition. I challenge you to read this post and to calmly follow my reasoning. Whenever something doesn't seem obvious, try to make it clear by yourself with a piece of paper and pen/pencil. If any doubts persist, drop me your question(s) in the comments and I will answer gladly! Let us call $m$ to the function that, given a subset of the real line, returns its size; that is, let us try to define $m: \mathcal{P}(\mathbb{R}) \to [0, \infty]$. Let us also suppose that the function $m$ sat...