What the Hell is "Calculus"? << Click on the graph on the left to see the categorization of mathematical fields. I came to the states after 4 years of university studies in Slovakia. I was a transfer student and as a transfer student I planned to continue my education on a different university. Among the usual annoyances of transferring to a different school, foreign students have a it little more difficult. Besides having to pay a full tuition and not having any chance to get any kind of scholarship, to transfer units from my old school, I had to get my previous coursework evaluated and approved as an equivalent to the classes at the university I was transferring to. The university has no interest in helping you to transfer as many units as you can, since doing so means you will spend less time and money at their educational institution.
And here started the leg work. One of the most retarded requests was when I was asked to prove that I do speak other language in order to waive a foreign language class. Capturing More Symmetry using Categories: Groupoids. Today’s entry is short, but sweet. I wanted to write something longer, but I’m very busy at work, so this is what you get. I think it’s worth posting despite its brevity. When we look at groups, one of the problems that we can notice is that there are things that seem to be symmetric, but which don’t work as groups. What that means is that despite the claim that group theory defines symmetry, that’s not really entirely true. The fifteen puzzle is a four-by-four grid filled with 15 tiles, numbered from 1 to 15, and one empty space.
If you look at the 15 puzzle in terms of configurations – that is, assignments of the pieces to different positions in the grid – so that each member of the group describes a single tile-move in a configuration, you can see some very clear symmetries. But it’s not a group. To describe that symmetry, we need a less restrictive version of something like a group. . × must be defined for some pairs of members of G. See what I mean? Categorization of mathematical fields. What makes linear logic linear? Sorry for the lack of posts this week.
I’m traveling for work, and I’m seriously jet-lagged, so I haven’t been able to find enough time or energy to do the studying that I need to do to put together a solid post. Fortunately, someone sent me a question that I can answer relatively easily, even in my jet-lagged state. (Feel free to ping me with more questions that can turn into easy but interesting posts. I appreciate it!) The question was about linear logic: specifically, what makes linear logic linear? For those who haven’t ever seen it before, linear logic is based on the idea of resource consumption. Where the normal propositional or predicate logics that most of us are familiar with are focused around an idea of truth, linear logic is focused on the idea of resource posession and consumption. Think of a conventional logic, like simple propositional logic. If you’re working in normal propositional logic, then you can use those statements to infer some more facts.
Number Theory for Programmers, Part 1. Number Theory for Programmers, Part 2. Before Groups from Categories: a Category Refresher. So far, I’ve spent some time talking about groups and what they mean. I’ve also given a brief look at the structures that can be built by adding properties and operations to groups - specifically rings and fields. Now, I’m going to start over, looking at things using category theory. Today, I’ll start with a very quick refresher on category theory, and then I’ll give you a category theoretic presentation of group theory. I did a whole series of articles about category theory right after I moved GM/BM to ScienceBlogs; if you want to read more about category theory than this brief introduction, you can look at the category theory archives.
Like set theory, category theory is another one of those attempts to form a fundamental abstraction with which you can build essentially any mathematical abstraction. But where sets treat the idea of grouping things together as the fundamental abstraction, category theory makes the idea of mappings between things as the fundamental abstraction.