goodmath

Todays pathological language is actually in the form of a challenge for you. (Isn't that exciting?) It's a very clever numerical programming language in the vein of Conway's href="http://scienceblogs.com/goodmath/2006/10/prime_number_pathology_fractra.php">Fractran, called NULL. The author of…

Correlation and Causation Yet another of the most abused mathematical concepts is the concept of correlation, along with the related (but different) concept of causation. Correlation is actually a remarkably simple concept, which makes it all the more frustrating to see the nonsense constantly…

Time for another sort-of advanced basic. I used some recursive definitions in my explanation of natural numbers and integers. Recursion is a very fundamental concept, but one which many people have a very hard time wrapping their head around. So it's worth taking the time to look at it, and see…

The biggest nightmare for most people learning Haskell is monads. Monads are the key to how you can implement IO, state, parallelism, and sequencing (among numerous other things) in Haskell. The trick is wrapping your head around them. On the most basic level, they're actually pretty easy to…

One of the interestingly odd things about how people understand math is numbers. It's astonishing to see how many people don't really understand what numbers are, or what different kinds of numbers there are. It's particularly amazing to listen to people arguing vehemently about whether certain…

The margin of error is the most widely misunderstood and misleading concept in statistics. It's positively frightening to people who actually understand what it means to see how it's commonly used in the media, in conversation, sometimes even by other scientists! The basic idea of it is very…

Yet another reader forwarded me a link to a rather dreadful article. This one seems to be by someone who knows better, but prefers to stick with his political beliefs rather than an honest exploration of the facts. He's trying to help provide cover for the anti-global warming cranks. Now, in light…

I decided that for today, I'd show the most thoroughly evil programming language ever devised. This is a language so thoroughly evil that it's named Malbolge after a circle of hell. It's so evil that it's own designer was not able to write a hello world program! In fact, the only way that anyone…

Yet another reader sent me a great bad math link. (Keep 'em coming guys!) This one is an astonishingly nasty slight of hand, and a great example of how people misuse statistics to support a political agenda. It's by someone named "Dr. Deborah Schurman-Kauflin", and it's an attempt to paint illegal…

I've gotten complaints from a bunch of commenters about problems with comments getting thrown into the moderation queue by the spam filter. Things with too many links, or with certain text properties, were getting caught even though they are clearly not spam. In order to get around this, I've re-…

When we look at a the data for a population+ often the first thing we do is look at the mean. But even if we know that the distribution is perfectly normal, the mean isn't enough to tell us what we know to understand what the mean is telling us about the population. We also need to know something…

Another piece of junk that I received: "The Invisible Link Between Mathematics and Theology", by a guy named "Ladislav Kvasz", published in a rag called "Perspectives on Science and Christian Faith". (I'm not going to quote much from this, because the way that the PDF is formatted, it requires a…

A reader sent me a link to this, thinking that it would be of interest to me, and he was absolutely right. I actually needed to let it sit overnight before writing anything because it made me so angry. I've come to realize that probably one reason I struggled with algebra, geometry et.al., was…

One thing that we've seen already in Haskell programs is type classes. Today, we're going to try to take our first look real look at them in detail - both how to use them, and how to define them. This still isn't the entire picture around type-classes; we'll come back for another look at them…

What started me on this whole complexity theory series was a question in the comments about the difference between quantum computers and classical computers. Taking the broadest possible view, in theory, a quantum computer is a kind of non-deterministic machine - so in the best possible case, a…

In general, when we gather data, we expect to see a particular pattern to the data, called a normal distribution. A normal distribution is one where the data is evenly distributed around the mean in a very regular way, which when plotted as a histogram will result in a bell curve. There are a lot…

Statistics is something that surrounds us every day - we're constantly bombarded with statistics, in the form of polls, tests, ratings, etc. Understanding those statistics can be an important thing, but unfortunately, most people have never been taught just what statistics really mean, how they're…

Here at ScienceBlogs, we've got our own back-channel forums for the bloggers to chat with each other. An idea that came up, which a bunch of us are interested in, is doing some posts about basic definitions and basic concepts. There are many people who read various blogs around here who've had…

Shelley started it, then PZ joined in. Who am I to stop it?

I'm hitting on something deeply twisted this week. It's called homespring. Homespring is interesting for two reasons. First, it's got a sort of reverse flavor to it: it consists of active agents in a passive structure. So, for example, you can't do anything like control flow - that's a dynamic…

There's another classic example of sheaves; this one is restricted to manifolds, rather than general topological spaces. But it provides the key to why we can do calculus on a manifold. For any manifold, there is a sheaf of vector fields over the manifold. Let's start by explaining what a vector…

As I've mentioned in the past, complexity theory isn't really one of my favorite topics. As a professional computer scientist, knowing and loving complexity up to the level of NP-completeness is practically a requirement. But once you start to get beyond P and NP, there are thousands of complexity…

Now that we've gone through a very basic introduction to computational complexity, we're ready to take a high-level glimpse at some of the more interesting things that arise from it. The one that you'll hear about most often is "P vs NP", which is what I'm going to write about today. So, what are…

In the comments thread of the post on Turing Equivalent vs Turing Complete, there've been a flurry of questions and comments about some stuff involving computational complexity. So I thought it would be interesting to write a couple of posts about computational complexity. I'm not going to get…

As an experiment, I decided to try making a iMix of the items in my FRT that are available via iTunes. Please let me know if you like this; it's a bit of extra work for me which I don't mind doing, as long as people use it... but if no one wants it, then I'd rather not spend the time setting it up…

Today, I have something really fun and twisted for you. It's my favorite variation on BrainF**k, called "BFM", which is short for "BrainFunct-Macro". It's a doubly-Turing-equivalent language - it's got two complete and distinct Turing equivalent computing systems in it: it's got regular BF on the…

In my discussion with Sal Cordova in this post, one point came up which I thought was interesting, and worth taking the time to flesh out as a separate post. It's about the distinction between a Turing equivalent computing system, and a Turing complete computation. It's true that in informal use,…

Over at [Dispatches][dispatches], Ed Brayton has been shredding my old friend Sal Cordova. Ed does a great job arguing that intelligent design is a PR campaign, and not a field of scientific research. Ed does a fine job with the argument; you should definitely click on over to take a look. But Sal…

Since the posts of sheaves have been more than a bit confusing, I'm going to take the time to go through a couple of examples of real sheaves that are used in algebraic topology and related fields. Todays example will be the most canonical one: a sheaf of continuous functions over a topological…

The big news over the last couple of days has been the execution of Saddam Hussein. I want to put in my own two cents about it. It's not math, but it does at least involve a bit of logic. I wish I could remember who first said this, but I really don't know. But the important thing, in a moral sense…