goodmath

So in the last few posts, I've been building up the bits and pieces that turn lambda calculus into a useful system. We've got numbers, booleans, and choice operators. The only thing we're lacking is some kind of repetition or iteration. In lambda calculus, all iteration is done by recursion. In…

I'm on vacation this week, so I'm posting reruns of some of the better articles from when Goodmath/Badmath was on Blogger. Todays is a combination of two short posts on numbers and control booleans in λ calculus. So, now, time to move on to doing interesting stuff with lambda calculus. To start…

I thought that for a followup to yesterday's repost of my takedown of Berlinksi, that today I'd show you a digested version of the debate that ensued when Berlinksi showed up to defend himself. You can see the original post and the subsequent discussion here. It's interesting, because it…

I'm on vacation this week, so I'm recycling some posts that I thought were particularly interesting to give you something to read. ------------ In computer science, especially in the field of programming languages, we tend to use one particular calculus a lot: the Lambda calculus. Lambda calculus…

I'm away on vacation this week, so this is a repost of one of early GM/BM entries when it was on Blogger. As usual, I've revised it slightly. Berlinksi actually showed up and responded; a digest of the discussion back and forth is scheduled to appear here later this week…

I'm leaving on vacation today. I'll be away for a week, with intermittent internet access. And even when I have access, I doubt I'll have much time to do blog-related stuff. I've scheduled a bunch of reposts of some of my favorite posts from the early days of _Goodmath, Badmath_ back when it lived…

About 10 days ago, I wrote about [Grigory Perelman and his proof of the Poincare conjecture][poincare]. This is a quick followup. There's a more detailed story over on [Seed][seed]. The Fields medal was supposed to be presented this past week, and they planned on presenting it to Perelman. He…

Over the six months tÃ¥hat I've been writing this blog, I've gotten a bunch of email from people asking about what it's like working as a researcher in industry vs working in academia. It's a good question, one which I've spent a lot of time thinking about. So I thought it was worth turning into…

Todays dose of programming pathology is a monstrosity called *Linguine*. Linguine is a language that (obviously) carries spaghetti code to an extreme. You can see the [language specification][linguine-spec], or download an [implementation with examples][linguine-impl]. It's really a remarkably…

Yesterday, I introduced the idea of a *metric space*, and then used it to define *open* and *closed* sets in the space. (And of course, being a bozo, I managed to include a typo that made the definition of open sets equivalent to the definition of closed sets. It's been corrected, but if you're not…

Topology usually starts with the idea of a *metric space*. A metric space is a set of values with some concept of *distance*. We need to define that first, before we can get into anything really interesting. Metric Spaces and Distance ------------------------------ What does *distance* mean? Let'…

Back when GM/BM first moved to ScienceBlogs, we were in the middle of a poll about the next goodmath topic for me to write about. At the time, the vote was narrowly in favor of topology, with graph theory as a very close second. We're pretty much done with category theory, so it's topology time!…

Everyone is scientific circles is abuzz with the big news: there's proof that dark matter exists! The paper from the scientists who made the discovered is [here][dark-matter-paper]; and a Sean Carroll (no relation) has [a very good explanation on his blog, Cosmic Variance][cv]. This discovery…

Since we talked about the surreals, I thought it would be interesting to take a *very* brief look at an alternative system that also provides a way of looking at infinites and infinitessimals: the *hyperreal* numbers. The hyperreal numbers are not a construction like the surreals; instead they're…

Last thursday, I introduced the construction of John Conway's beautiful surreal numbers. Today I'm going to show you how to do arithmetic using surreals. It's really quite amazing how it all works! If you haven't read the original post introducing surreals, you'll definitely want to [go back and…

I figured it was time I did the latest random thing to be wandering its way around Scienceblogs. [Janet has introduced the "random quotes" meme.][janet], in which we're supposed to go wandering through the [quotes here][quotes], and pick the first five that reflect "who you are or what you believe…

Remember my post several weeks ago about ["The First Scientific Proof of God"?][georgie] The author, Georgie-boy Shollenberger popped up [in the comments yesterday][georgie-comments], and posted [a response][georgie-responds] on his blog. This is how he describes this blog: >This website is an…

Todays programming language insanity is a real winner. It's a language called SNUSP. You can find the language specification [here][snuspspec], a [compiler][snuspcomp], and [an interpreter embedded in a web page][snuspinterp]. It's sort of like a cross between [Befunge][befunge] and [Brainfuck][…

Surreal numbers are a beautiful, simple, set-based construction that allows you to create and represent all real numbers, so that they behave properly; *and* in addition, it allows you to create infinitely large and infinitely small values, and have *them* behave and interact in a consistent way…

I've always been perplexed by roman numerals. First of all, they're just *weird*. Why would anyone come up with something so strange as a way of writing numbers? And second, given that they're so damned weird, hard to read, hard to work with, why do we still use them for so many things today? The…

The Poincarė conjecture has been in the news lately, with an article in the Science Times today. So I've been getting lots of mail from people asking me to explain what the Poincarė conjecture is, and why it's a big deal lately? I'm definitely not the best person to ask; the reason for the recent…

After yesterdays post about the sloppy probability from ann coulter's chat site, I thought it would be good to bring back one of the earliest posts on Good Math/Bad Math back when it was on blogger. As usual with reposts, I've revised it somewhat, but the basic meat of it is still the same…

A reader sent me a copy of an article posted to "chat.anncoulter.com". I can't see the original article; anncoulter.com is a subscriber-only site, and I'll be damned before I *register* with that site. Fortunately, the reader sent me the entire article. It's another one of those stupid attempts by…

How can you talk about interesting numbers without bringing up π? History --------- The oldest value we know for π comes from the Babylonians. (Man, but those guys were impressive mathematicians; almost any time you look at the history of fundamental numbers and math, you find the Babylonians in…

The stupidity and innumeracy of Americans, and in particular American fundamentalists, never ceases to astound me. Recently on Yahoo, some bozo posted [something claiming that the bible was all correct][yahoo], and that genetics would show that bats were actually birds. But that's not the real…

Remember back in the end of june, when I [talked about these insane bozos][code] who were [predicting that a nuclear bomb would be blown up in the UN plaza?][firsttime] And they were on their fourth prediction about when this would happen? And how each time they blew it, [they came up with an…

In the comments onmy post about φ, Polymath, (whose [blog][polymath] is well worth checking out) provided a really spectacular [link involving φ][desert]. It's an excerpt from a book called "[Mathematical Gems 2][gems]", showing a problem that came from John Conway, called the "Sending Scouts…

Todays tasty treat: a variable-free nearly-pure functional programming language: Unlambda. Unlambda is based on the SKI combinator calculus. The [SKI calculus][ski] is a way of writing lambda calculus without variables, and without lambda. It's based on three combinators: S, K, and I: 1. S = λ x y…

This is one of the last posts in my series on category theory; and it's a two parter. What I'm going to do in these two posts is show the correspondence between lambda calculus and the [cartesian closed categories][ccc]. If you're not familiar with lambda calculus, you can take a look at my series…

In email, someone pointed me at an automated proof system called [Metamath][metamath]. Metamath generates proofs of mathematical statements using ZF set theory. The proofs are actually relatively easy to follow, which is quite unusual for an automated theorem prover. I'll definitely write more…