goodmath

Next stop on our tour of topology is the idea of *connectedness*. It's an important concept that defines a lot of useful and interesting properties of topological spaces. The basic idea of connectedness is very simple and intuitive. If you think of a topology on a metric space like ℜ3, what…

Just for fun, I've been doing a bit of poking around lately in evolutionary algorithms. It's really fascinating to experiment, and see what pops out - the results can be really surprising. There is one fascinating example for which, alas, I've lost the reference, but here's the summary. Several…

This is going to be a short but sweet post on topology. Remember way back when I started writing about category theory? I said that the reason for doing that was because it's such a useful tool for talking about other things. Well, today, I'm going to show you a great example of that. Last friday,…

Now we're going to try something challenging on the abacus: *division*. Like multiplication, abacus division is close to the way you'd do it on paper. But just like doing paper division is trickier than paper multiplication, abacus division is tricker than abacus multiplication. But the technique…

It's friday again, so in addition to a bizzare programming language, you get a random ten. 1. *Transatlantic, "Mystery Train".*: very cool neo-prog rock track. 2. *Darol Anger and the Republic of Strings, "Dzinomwa Muna Save".* Darol Anger is one the most creative artists of our generation. He's a…

I've got a real treat for you pathological programming fans! Today, we're going to take a quick look at the worlds most *useful* pathological programming language: TECO. TECO is one of the most influential pieces of software ever written. If, by chance, you've ever heard of a little editor called "…

Once you can add on an abacus, the next thing to learn is multiplication. Like addition, it follows pretty closely on the old pencil-and-paper method. But it's worth taking the time to look closely and see it step by step, because it's an important subroutine (to use a programming term) that will…

One of the really neat things you can do in topology is play games with dimensions. Topology can give you ways of measuring dimensions, and projecting structures with many dimensions into lower-dimensional spaces. One of the keys to doing this is understanding how to combine different topologies…

Over in my post accepting my victory as the biggest geek on ScienceBlogs, an interesting discussion about beginners learning to program got started in the comments. It was triggered by someone mentioning David Brin's article in Salon about how terrible it is that computers no longer come with basic…

One of the bad arguments that I've frequently seen from creationists is the argument that some biological system is *too good* to be a possible result of an evolutionary process. On its face, this seems like it's not a mathematical argument. But it actually is, and math is key to showing what the…

Yes, it appears that I have won the great ScienceBlogs nerdoff/geekoff. [Janet announced the results yesterday][geekoff], and despite [much][orac-whines] [whining][pz-whines], I'm proud to say that I was the winner. There was some stiff competition, particularly from Orac, but in the end, no one…

If you want to talk about mechanical computing tools, you can't ignore the abacus. It's the oldest computing tool in the world; and it's still very commonly used. It's also about as different from the slide rule as you could imagine. The abacus is really fundamentally an addition device; the slide…

Just like you can define a sub-set of a set, or a sub-object of an object in a category, you can define a sub-*space* of a topological space. It's a pretty easy thing to understand; interestingly, a sub-space of a topological space works in pretty much exactly the same way as a sub-sets and sub-…

Orac sent me a link to some more HIV denialist material, I assume under the assumption that since I'm already being peppered by insults from the denialist crowd, I might as well cover this now. What I'm looking at today is a [paper by Mark Craddock called "HIV: Science by press conference".][…

Expanding on last weeks theme of minimalism, this week, we've got OISC: the One Instruction Set Computer. This is a machine-code level primitive language with exactly one instruction. There are actually a few variants on this, but my favorite is the "subleq" OISC, which I'll describe beneath the…

This is *very* off-topic for this blog; it's really more of a rant on a personal subject which I think it's worth saying publicly. I am mentally ill. I have clinical depression. CD is a thoroughly miserable illness. I'm incredibly lucky to live at a time when CD like mine is easily treated by…

Slides rules are actually astonishingly powerful things. The simple slide rule does multiplication and division using the C and D scales; strictly speaking, you can have a basic rule with nothing but C and D. But you almost never see a rule that simple. (The only one I've ever seen with only the…

Several people in the geekout thread asked me to explain how a sliderule works, and I've been meaning to write a couple of article about manual computing devices. So I thought I'd do it. There's a nice slide-rule simulator at [Derek's Virtual Slide Rule Gallery][sr], which is what I used to…

Orac is refusing to surrender and acknowledge the obvious fact that he simple *is not* as much of a geek as I am. So I am obligated to point out several further facts in my attempt to make him surrender the crown of geekiness. ---------- First: compare our professsions. Orac is a cancer surgeon: a…

When we talk about topology, in general, the way we talk about it is in terms of *shapes*: geometric objects and spaces, surfaces, bodies that enclose things, etc. We talk about the topology of a *torus*, or a *coffee mug*, or a *sphere*. But the topology we've talked about so far doesn't talk…

Haven't done one of these in a while. In light of the "Geek-off" this week, I made a playlist out of what I think of as my "geekier" music, and let ITunes assemble a random list from that playlist. 1. **Elizabeth and the Catapult, "Waiting for the Kill"**. E&tC is a NYC band that plays what…

For todays dose of pathological programming, we're going to hit the worlds simplest language. A Turing-complete programming language with exactly *two* characters, no variables, and no numbers. It's called [Iota][iota]. And rather than bothering with the rather annoying Iota compiler, we'll just…

Draw your pocket protectors, it's a geekout! Janet, our lovely resident ethicist has challenged all of the ScienceBloggers to [a geekout][geekout], to determine who, among us, is the geekiest. How could I, a *math* blogger and computer language geek, pass up such a challenge? (Incidentally, Janet,…

The past couple of posts on continuity and homeomorphism actually glossed over one really important point. I'm actually surprised no one called me on it; either you guys have learned to trust me, or else no one is reading this. What I skimmed past is what a *neighborhood* is. The intuition for a…

A few of my recent posts here appear to have struck some nerves, and I've been getting lots of annoying email containing the same questions, over and over again. So rather than reply individually, I'm going to answer them here in the hope that either (a) people will see the answers before send the…

With continuity under our belts (albeit with some bumps along the way), we can look at something that many people consider *the* central concept of topology: homeomorphisms. A homeomorphism is what defines the topological concept of *equivalence*. Remember the clay mug/torus metaphor from from my…

*(Note: in the original version of this, I made an absolutely **huge** error. One of my faults in discussing topology is scrambling when to use forward functions, and when to use inverse functions. Continuity is dependent on properties defined in terms of the *inverse* of the function; I originally…

While I was on vacation, I got some email from Chris Noble pointing me towards a discussion with some thoroughly innumerate HIV-AIDS denialists. It's really quite shocking what passes for a reasonable argument among true believers. The initial stupid statement is from one of Duesberg's papers, […

While I was away on vacation, my family made a stop in Corning NY to see the Corning Glass Museum. I had to snap this photo for PZ. Alas, all I had was the camera in my cellphone, so the resolution leaves something to be desired, but it's the thought that counts, right? As long as I'm posting…

Lambda calculus started off with the simple, untyped lambda calculus that we've been talking about so far. But one of the great open questions about lambda calculus was: was it sound? Did it have a valid model? Church found that it was easy to produce some strange and non-sensical expressions using…