goodmath

Many of my fellow SBers have been mocking the recently unveiled Conservapedia. Conservapedia claims to be a reaction to the liberal bias of Wikipedia. Ed, PZ, Afarensis, Tim, John, and Orac have all piled on already. But why should they get to have all the fun? Conservapedia has an extensive…

So, after our last installment, describing the theory of monads, and the previous posts, which focused on representing things like state and I/O, I thought it was worth taking a moment to look at a different kind of thing that can be done with monads. We so often think of them as being state…

So as many other folks have been pointing out, the Koufax awards have come out with their "Best Expert Blog" nominations, and I'm incredibly pleased to say that GM/BM was nominated! In case you're not familiar, the Koufaxes are one of the really serious, prestigious web-awards, aimed primarily at…

It's that time again - yes, we have yet another wacko reinvention of physics that pretends to have math on its side. This time, it's "The Electro-Magnetic Radiation Pressure Gravity Theory", by "Engineer Xavier Borg". (Yes, he signs all of his papers that way - it's always with the title "Engineer…

Many of my fellow ScienceBloggers have recently declared their membership in Order of the Science Scouts of Exemplary Repute and Above Average Physique. I've been busy, so I haven't been able to get around to signing up until now. That's a shame, since some of the badges appear to have been…

Just a quick reminder: the second Carnival of Mathematics is coming up this friday, to be hosted here at GM/BM. If you've written any math related articles, get me a link by thursday at the latest. You can either send it to me here at markcc at gmail.com, or via the carnival submission form.

One of the more advanced topics in topology that I'd like to get to is homology. Homology is a major topic that goes beyond just algebraic topology, and it's really very interesting. But to understand it, it's useful to have some understandings of some basics that I've never written about. In…

Yet another term that we frequently hear, but which is often not properly understood, is the concept of optimization. What is optimization? And how does it work? The idea of optimization is quite simple. You have some complex situation, where some variable of interest (called the target) is based…

Frameshift, "Walking through Genetic Space": a track from an album inspired by the writings of Steven Jay Gould about genetics and evolution. The leader of the project is the lead singer of Dream Theater; the end result has a very DT like feeling to it. The album overall is quite good; bit this…

Insane Stacking Todays pathology is playing with stacks. Lots of lots of stacks. Stacks for data. Stacks for control. Stacks out the wazoo. It's called Kipple for no particularly good reason that I know of. Kipple happens to be one of the pathological languages that I highly recommend trying to…

In yesterdays basics post, I alluded to the second kind of calculus - the thing that computer scientists like me call a calculus. Multiple people have asked me to explain what our kind of calculus is. In the worlds of computer science and logic, calculus isn't a particular thing: it's a kind of…

Calculus is one of the things that's considered terrifying by most people. In fact, I'm sure a lot of people will consider me insane for trying to write a "basics" post about something like calculus. But I'm not going to try to teach you calculus - I'm just going to try to explain very roughly…

One of the fundamental branches of modern math - differential and integral calculus - is based on the concept of limits. In some ways, limits are a very intuitive concept - but the formalism of limits can be extremely confusing to many people. Limits are basically a tool that allows us to get a…

Basics: Algebra While I was writing the vectors post, when I commented about how math geeks always build algebras around things, I realized that I hadn't yet written a basics post explaining what we mean by algebra. And since it isn't really what most people think it is, it's definitely worth…

There's another way of working with number-like things that have multiple dimensions in math, which is very different from the complex number family: vectors. Vectors are much more intuitive to most people than the the complex numbers, which are built using the problematic number i. A vector is…

In light of the recent posts and discussions about multidimensional numbers,today's pathological language is Recurse, a two-dimensional language - like Befunge, sort of. But I find it more interesting in its own peculiar little way. It's actually a function-oriented two-dimensional language where…

When we think of numbers, our intuitive sense is to think of them in terms of quantity: counting, measuring, or comparing quantities. And that's a good intuition for real numbers. But when you start working with more advanced math, you find out that those numbers - the real numbers - are just a…

Many people would probably say that things like computability and the halting program aren't basics. But I disagree: many of our basic intuitions about numbers and the things that we can do with them are actually deeply connected with the limits of computation. This connection of intuition with…

I haven't taken a look at Uncommon Descent in a while; seeing the same nonsense get endlessly rehashed, seeing anyone who dares to express disagreement with the moderators get banned, well, it gets old. But then... Last week, DaveScott (which is, incidentally, a psueudonym!) decided to retaliate…

What are the real numbers? Before I go into detail, I need to say up front that I hate the term real number. It implies that other kinds of numbers are not real, which is silly, annoying, and frustrating. But we're pretty much stuck with it. There are a couple of ways of describing the real…

As long as I'm doing all of these basics posts, I thought it would be worth explaining just what a Turing machine is. I frequently talk about things being Turing equivalent, and about effective computing systems, and similar things, which all assume you have some clue of what a Turing machine is.…

Marillion, "Ocean Cloud": Long, wonderful piece of neo-prog rock from my favorite prog band. Mogwai, "Acid Food": Mogwai is a brilliant post-rock group, leaning more towards the rock than the classical. This is a slow track with vocals, with a very dark sound to it. Very cool. Trey Gunn…

I've had a long, difficult week, so I've decided to pick something pointlessly pathological for today. It's a remarkably goofy language called href="http://www.p-nand-q.com/humor/programming_languages/gplz/gplz_chef.html">"Chef", designed by David Morgan-Mar, in which programs are recipes.…

Finally, the math geeks of the blogosphere are going to have a carnival of our own! Alon Levy of Abstract Nonsense (which I really need to add to my blogroll!) has taken the initiative and started the Carnival of Mathematics. The very first edition will appear on February 9th. If you've got…

Sets are truly amazing things. In the history of mathematics, they're a remarkably recent invention - and yet, they're now considered to be the fundamental basis on which virtually all of mathematics is built. From simple things (like the natural numbers), to the most abstract and esoteric things…

As promised, I'm finally going to get to the theory behind monads. As a quick review, the basic idea of the monad in Haskell is a hidden transition function - a monad is, basically, a state transition function. The theory of monads comes from category theory. I'm going to assume you know a little…

Another great basics topic, which came up in the comments from last fridays "logic" post, is the difference between syntax and semantics. This is an important distinction, made in logic, math, and computer science. The short version of it is: syntax is what a language looks like; semantics is what…

Time for more monads. In this article, I'm going to show you how to implement a very simple state monad - it's a trivial monad which allows you to use a mutable state consisting of a single integer. Then we'll expand it to allow a more interesting notion of state. Let's get the trivial stuff out…

It's been a while since I've written a topology post. Rest assured - there's plenty more topology to come. For instance, today, I'm going to talk about something called a fiber bundle. I like to say that a fiber bundle is a cross between a product and a manifold. (There's a bit of a geeky pun in…

This is another great basics topic, and it's also one of my pet peeves. In general, I'm a big science fiction fan, and I grew up in a house where every saturday at 6pm, we all gathered in front of the TV to watch Star Trek. But one thing which Star Trek contributed to our vocabulary, for which I…