I’ve been getting a lot of mail from people asking for my take on
the news about the Washington GOP primary. Most have wanted me to
debunk rumours about vote fixing there, the way that I tried to debunk the
rumours about the Democratic votes back in New Hampshire.
Well, sorry to disappoint those of you who were hoping for a nice debunking
of the idea of fraud, but to me, something sure looks fishy.
In the last couple of posts, I showed how we can start looking at group
theory from a categorical perspective. The categorical approach gives us a
different view of symmetry that we get from the traditional algebraic
approach: in category theory, we see symmetry from the viewpoint of
groupoids – where a group, the exemplar of symmetry, is seen as an
expression of the symmetries of a simpler structure.
We can see similar things as we climb up the stack of abstract algebraic
constructions. If we start looking for the next step up in algebraic
constructions, the rings, we can see a very different view of
what a ring is.
Before we can understand the categorical construction of rings, we need
to take a look at some simpler constructions. Rings are expressed in
categories via monoids. Monoids are wonderful things in their own right, not
just as a stepping stone to rings in abstract algebra.
What makes them so interesting? Well, first, they’re a solid bridge
between the categorical and algebraic views of things. We saw how the
category theoretic construction of groupoids put group theory on a nice
footing in category theory. Monoids can do the same in the other direction:
they’re in some sense the abstract algebraic equivalent of categories.
Beyond that, monoids actually have down-to-earth practical applications –
you can use monoids to describe computation, and in fact, many of the
fundamental automatons that we use in computer science are, semantically,
monoids.
This post started out as a response to a question in the comments of my last post on groupoids. Answering those questions, and thinking more about the answers while sitting on the train during my commute, I realized that I left out some important things that were clear to me from thinking about this stuff as I did the research to write the article, but which I never made clear in my explanations. I’ll try to remedy that with this post.
In my introduction to groupoids, I mentioned that if you have a groupoid, you can find
groups within it. Given a groupoid in categorical form, if you take any object in the
groupoid, and collect up the paths through morphisms from that object back to itself, then
that collection will form a group. Today, I’m going to explore a bit more of the relationship
between groupoids and groups.
Before I get into it, I’d like to do two things. First, a mea culpa: this stuff is out on the edge of what I really understand. My category-theory-foo isn’t great, and I’m definitely
on thin ice here. I think that I’ve worked things out enough to get this right, but I’m
not sure. So category-savvy commenters, please let me know if you see any major problems, and I’ll do my best to fix them quickly; other folks, be warned that I might have blown some of the details.
Second, I’d like to point you at Wikipedia’s page on groupoids as a
reference. That article is quite good. I often look at the articles in Wikipedia and
MathWorld when I’m writing posts, and while wikipedia’s articles are rarely bad, they’re also
often not particularly good. That is, they cover the material, but often in a
somewhat disorganized, hard-to-follow fashion. In the case of groupoids, I think Wikipedia’s
article is the best general explanation of groupoids that I’ve seen – better than most
textbooks, and better than any other web-source that I’ve found. So if you’re interested in
finding out more than I’m going to write about here, that’s a good starting point.
A bunch of people have been sending me links to a USA Today article about a math professor who wants to change math education. Specifically, he wants to stop teaching fractions, and de-emphasize manual computation like multiplication and long division.
Frankly, reading about it, I’m pissed off by both sides of the argument.
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?
One of the staples of chinese cooking is fried rice. Unfortunately, what we get in
American restaurants when we order fried rice is dreadful stuff. The real thing is
absolutely wonderful – and very different from the American version.
The trick to getting the texture of the dish right is to use leftover rice. Freshly cooked rice won’t work; you need it to dry out bit. So cook some other chinese dish one night, make an extra 2 cups of rice, and then leave it in the fridge overnight. If you can, take it out of the fridge a couple of hours before you’re going to cook, to get it to room temperature. Then when you’re ready to start preparing all the ingredients, use your hands to crumble the rice – that is, break up the clumps so that the grains aren’t sticking together.
What I found surprising about real fried rice was that you don’t put any soy sauce into the rice. The rice is the heart of the dish, and you don’t want anything as strongly flavored as soy sauce to disturb the fine, delicate flavor of good rice.
I generally start with around two cups of uncooked rice to make a large portion for four people.
Ingredients
Instructions
This is a recipe that you should feel free to fool around with. It’s pretty versatile. After thanksgiving, I make leftover turkey fried rice; if we buy a roast duck and have leftovers, I make duck fried rice. It also supposedly comes out very well with smoked ham as the meat. One chinese chef even suggested adding finely diced tomatoes to it, which
surprising worked extremely well!
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. My favorite example of this is the fifteen puzzle.
The fifteen puzzle is a four-by-four grid filled with 15 tiles, numbered from 1 to 15, and one empty space. You can make a move in the puzzle by sliding a tile adjacent to the empty
space into the empty. In the puzzle, you scramble up the tiles, and then try to move them back so that they’re in numerical order. The puzzle, in its initial configuration, is shown to the right.
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. For example, the moves that are possible when the empty is in any corner are equivalent to the moves that are possible when the empty is in any other corner. The possible moves when the space is in any given position are the same except for the labeling of the tiles around them. There’s definitely a kind of symmetry there. There are also loops – sequences of moves which end in exactly the same state as the one in which they began. Those are clearly symmetries.
But it’s not a group. In a group, the group operation most be total – given any pair of values x and y in the group, it must be possible to combine x and y via x+y. But with the 15 puzzle, there moves that can’t be combined with other moves. If x = “move the ‘3’ tile from square 2 to square 6”, and y = “move the ‘7’ tile from square 10 to square 11”, then there’s no meaningful value for “x+y”; the two moves can’t be combined.