goodmath
goodmath | June 17, 2006Just saw a nice post at another math blog called Polymathematics about something that bugs me too... The way that people don't understand what repeating decimals mean. In particular, the way that people will insist that 0.9999999... != 1. As a CS geek, I tend to see this as an issue of how people screw up syntax and semantics.
And it has some really funny stupidity in the comments. 0.9999999... = 1.
One quick quote from the post, just because it's a nifty demonstration of the fact which I've not seen before: (I replaced a GIF image in the original post with a text transcription.)
Let x =…
goodmath | June 12, 2006 Let's talk a bit about functors. Functors are fun!
What's a functor? I already gave the short definition: a structure-preserving mapping between categories. Let's be a bit more formal. What does the structure-preserving property mean?
A functor F from category C to category D is a mapping from C to D that:
Maps each member m ∈ Obj(C) to an object F(m) ∈ Obj(D).
Maps each arrow a : x → y ∈ Mor(C) to an arrow F(a) : F(x) → F(y), where:
(∀ o ∈ Obj(C)) F(1o) = 1F(o). (Identity is preserved by the functor mapping of morphisms.)
(&forall m,n ∈ Mor(C)) F(n º o) = F(o) º F(n). (…
goodmath | June 9, 2006 For me, the frustrating thing about learning category theory was that
it seemed to be full of definitions, but that I couldn't see why I should care.
What were these category things, and what could I really talk about using this
strange new mathematical language of categories?
To avoid that in my presentation, I'm going to show you a couple of examples up front of things we can talk about using the language of category theory: sets, partially ordered sets, and groups.
Sets as a Category
We can talk about sets using category theory. The objects in the category of sets are, obviously, sets…
goodmath | June 8, 2006One of the things that I find niftiest about category theory is category diagrams. A lot of things that normally turn into complex equations or long-winded logical statements can be expressed in diagrams by capturing the things that you're talking about in a category, and then using category diagrams to express the idea that you want to get accross.
A category diagram is a directed graph, where the nodes are objects from a category, and the edges are morphisms. Category theorists say that a graph commutes if, for any two paths through arrows in the diagram from node A to node B, the…
goodmath | June 7, 2006Back at my old digs last week, I put up a post about programming languages and types. It produced an interesting discussion, which ended up shifting topics a bit, and leading to a very interesting question from one of the posters, and since the answer actually really does involve math, I'm going to pop it up to the front page here.
In the discussion, I argued that programmers should know and use many different programming languages; and that that's not just a statement about todays programming languages, but something that I think will always be true: that there will always be good reasons…
goodmath | June 7, 2006To get started, what is category theory?
Back in grad school, I spent some time working with a thoroughly insane guy named John Case who was the new department chair. When he came to the university, he brought a couple of people with him, to take temporary positions. One of them was a category theorist whose name I have unfortunately forgotten. That was the first I'd ever heard of cat theory. So I asked John what the heck this category theory stuff was. His response was "abstract nonsense". I was astonished; a guy as wacky and out of touch with reality as John called something abstract…