Don’t you dare use the number 271277229129081016424883074559900780951 under any circumstances. It’s mine, mine I tell you, and if you use it, or copy it, I can have you arrested and sent to do hard time in prison. And it doesn’t matter whether you use it in decimal, like I used above, or it’s hexidecimal form, “CC16180895F94705F667F1BB6DB20997”, or any other way of encoding it. It’s my number, and you’re not allowed to use it. In fact, I don’t think I want to allow you to look at it – so I’m going to sue all of you for having read this post!
This is something that came up in some of the comments on the recent “nimbers” post, and I thought it was worth promoting to the front, and getting up under an easy-to-find title in the “basics” series.
In a lot of discussions in all different areas of math, you encounter talk about sets and classes, and you’ll find people worried about whether they’re talking about sets or classes. What’s the difference? I mentioned this once before, but it’s buried in a discussion of the concept of “meta”, which is why I thought it was worth moving it to its own top-level post: if you don’t know the difference, you’re not going to look in the body of a discussion about the concept of going meta to find the explanation!
I’ll start with just the definitions, and then I’ll dive into the discussion of why we make the distinction.
- A class is any collection of things which have some common property that defines them: the class of logical statements, the class of numbers.
- A set is a class which is a member of a class.
- A proper class is a class which is not a set.
- Rachel’s, “Even/Odd”: Rachel’s is a very classically-oriented post-rock
ensemble – violin, bass, woodwinds. They’re absolutely brilliant. “Even/Odd” is a short, extremely rhythmic track with an interesting pulse with an almost siren-like string lead played over it. Very, very cool.
- Rush, “Spindrift”: Rush is back! They released a new album this week. It’s a
much better work than their last effort (which wasn’t bad, mind you, but it wasn’t as
as good as it could have been). It actually sounds a lot more like older Rush than
most of their other recent work. Really good. Not spectacular or anything, but definitely quite good. This track has a nice edge to it, dark chords, very classic
Lifeson guitars, a strong Geddy Lee bass lead driving things. Peart is rather non-descript on this track – he’s got that almost inhumanly perfect timing as always, but he’s not
doing a lot that catches my attention.
- Marillion, “The Invisible Man”. The opening track off of Marillion’s last
album. It’s an amazing song, one of the best they’ve ever done. I can’t listen to
it without getting chills. Just the first 30 seconds of it is enough to start me
- Porcupine Tree, “Fear of a Blank Planet”. Porcupine Tree also has a new album out in the last couple of weeks. It’s one of their best, which is really saying an awful
lot when you realize how good their catalog is. This is the opening track from the album. Very typically PT: interesting rhythms, dark chords, interesting transitions, and a great contrast between very smooth soft vocals and hard-edged instrumental lines.
- Lunasa, “Mean Fomhair”: would you believe, a bagpipe solo? Ok, so it’s not the awful scottish greatpipes that you probably think of when I say bagpipes – it’s the Irish Uillean pipes which have a much less grating sound to them. And it’s played by one of the worlds greatest Uillean piper’s, Cillean Vallely. But it is a bagpipe solo. And it’s great.
- Edgar Meyer, “Concerto in D, 2nd Movement”. Edgar Meyer is one of those musicians that just make me sick. He plays this incredibly awkward instrument (the double-bass), and makes it look like it’s easier to play than a basic violin. And he can play anything on it – anything from Bluegrass to Rock to Jazz to Classical. He can play the Bach cello suites on his bass better than any cellist I’ve ever heard can play it on a cello. And he’s an amazing composer, who’s practically redefined the repertoire for
the bass. This track is him performing the part of the Concerto for double bass that he wrote.
- Moxy Fruvois, “Spiderman”. Interesting timing, given that the new Spiderman Movie is just coming out. This is Moxie Fruvous doing their incredibly silly version of the old theme from the Spiderman cartoon show. “Spidermans master plan, build his own little spider clan. In the woods, now they’re troops, fighting for special interest groups”. How can you not love a song making fun of low-budget superhero cartoon with lyrics like that?
- Tony Trishka Band, “Woodpecker”. Tony is one of the most talented musicians
I’ve ever gotten to know personally. He’s a banjo player who pioneered the Banjo
as a serious Jazz instrument. (Bela Fleck is one of his students.) This is a track off of his first album with his own Jazz fusion band. It’s very typical of Tony’s playing. As much as I love Bela Fleck’s playing, he’s still got a lot to learn from Tony in terms of how to make Jazz really work on the banjo. Tony just pulls out all the stops – using some of the pentatonic rolling tricks Bela is known for, as well as some fitting some more traditional bluegrass rolls, and some single-string work.
- Explosions in the Sky, “Catastrophe and the Cure”. Great rock-oriented post-rock.
- Shirim Klezmer Orchestra, “Nokh A Gleyzl Vayn”. Klezmer/Jazz fusion. Yeah, really, I’m not joking. Shirim is an amazing bunch of players who can move seamlessly back and forth between very traditional Klezmer and Bebop – but they’re most at home playing something in between. This is a traditional Klezmer tune played Shirim style.
- Miles Davis, “Deception”. What can I say about Miles Davis? One of the most
amazing, influential musicians of the 20th century. One of the creators of an
entirely new genre of Jazz. Every note he plays is virtually perfect. You could spend
hours just listening to one little track by him, and still not absorb everything that
he did to make it so perfect.
I thought that it would be fun to stick with the “stack-based” theme of last week’s pathological post, but this time, to pick an utterly pointlessly twisted stack based language, but one that would be appreciated by the mascot of one of my fellow ScienceBlogs. Orac, this one’s for you! 🙂 Our target these week is the language “Enema”.
So, today we’re going to play a bit more with nimbers – in particular, we’re
going to take the basic nimbers and operations over nimbers that we defined last time, and
take a look at their formal properties. This can lead to some simpler definitions, and
it can make clear some of the stranger properties that nimbers have.
An astute reader pointed me towards a monstrosity of pompous bogus math. It’s an oldie, but I hadn’t seen it before, and it was just referenced by my old buddy Sal Cordova in a thread on one of the DI blogs. It’s a “debate” posted online by Lee Spetner, in which he rehashes the typical bogus arguments against evolution. I’m going to ignore most of it; this kind of stuff has been refuted more than enough times. But in the course
of this train wreck, he pretends to be making a mathematical argument about search spaces and optimization processes. It’s a completely invalid argument – but it’s one which is constantly rehashed by creationists, and Spetner’s version of it is a perfect demonstration of exactly what’s wrong with the argument.
Before diving off into real content, the name of my little π-calculus monstrosity has been chosen. As several people recommended in the comments, I’m going to go with Pica.
So, today, we’re going to take the dive, and start looking at some interesting semantics of the language. The goal of this is still to work on Pica’s type system: I want to work towards a first draft of what a process type will look like. But to figure out what a process type should look like, we need to think about how process abstractions will be defined and used in Pica, what control structures are going to look like. So that’s what I’m going to do in this post: ramble on a bit about what processes are really going to end up looking like. Just a warning: this post is definitely on the half-baked side – it’s much more of a train-of-thought/brainstorming thing than the things I usually post.