goodmath

As an alert reader pointed out, a major mathematical prize was awarded recently. Since 2002, the government of Norway has been awarding a prize modeled on the Nobel, but in mathematics. The prize was originally suggested by Sophus Lie, he of the Lie group, back in 1897, when he heard that Nobel…

On the way to figuring out how to do sign-expanded forms of infinite and infinitesimal numbers, we need to look at yet another way of writing surreals that have infinite or infinitesimal parts. This new notation is called the normal form of a surreal number, and what it does is create a canonical…

When I first read about the sign-expanded form of the surreal numbers, my first thought was "cool, but what about infinity?" After all, one of the amazing things about the surreal numbers is the way that they make infinite and infinitessimal numbers a natural part of the number system in such an…

Orac has posted a really good description of a recent paper discussing how interaction between different antibiotics effects the evolution of antibiotic resistance in bacteria populations. It's a mathematical analysis of experimental results generated by combining drugs which normally interact…

In addition to the classic {L|R} version of the surreal numbers, you can also describe surreals using something called a sign expansion, where they're written as a sequence of "+"s and "-"s - a sort of binary representation of surreal numbers. It's fully equivalent to the {L|R} construction, but…

This post is a quick moderation update, caused by recent crap going on in the comment threads involving George "First Scientific Proof of God" Shollenberger, combined with my recent change of employment. Before changing jobs, my old employer was kind enough to allow me to write this blog, but…

For the basics, I wrote a bunch of stuff about sorting. It seems worth taking a moment to talk about something related: binary search. Binary search is one of the most important and fundamental algorithms, and it shows up in sorts of places. It also has the amazing property that despite being…

The Surreal Reals I was reading Conway's Book, book on the train this morning, and found something I'd heard people talk about, but that I'd never had time to read or consider in detail. You can use a constrained subset of the surreal numbers to define the real numbers. And the resulting…

Coming back from games to numbers, I promised earlier that I would define division. Division in surreal numbers is, unfortunately, ugly. We start with a simple, basic identity: if a=b×c, and a is not zero, then c=a×(1/b). So if we can define how to take the reciprocal of a surreal number, then…

Apparently, Michael Egnor just can't get enough of making himself look like an idiot. His latest screed is an attack on me, for criticizing his dismissal of evolution as a tautology. My observation that "Natural Selection" is a tautology, and therefore useless to modern medicine, seems to have…

A couple of weeks ago, I revisited George Shollenberger, the creator the alleged "First Scientific Proof of God", and commented on his pathetic antics on amazon.com, trying to explain just why no one had bothered to post a single review of his book. (If you'll recall, according to George, it's…

Today we're going to take our first baby-step into the land of surreal games. A surreal number is a pair of sets {L|R} where every value in L is less than every value in R. If we follow the rules of surreal construction, so that the members of L sets are always strictly less than members of R…

Fuck. Ok. Now I'm seriously pissed off. What a miserable way to ruin god-damned april fools day.

In my last post on the surreals, I introduced how the surreal numbers are constructed. It's really fascinating to look back on it - to see the structure of numbers from 0 to infinity and beyond, and realize that ultimately, that it's all built from nothing but the empty set! Today, we're going…

Late last summer, shortly after moving to ScienceBlogs, I wrote a couple of posts about Surreal numbers. I've always meant to write more about them. but never got around to it. But Conway's book actually makes pretty decent train reading, so I've been reading it during my new commute. So it's a…

So over at the DI whiners blog, Egnor is, once again, trying to pretend that he's actually making a case for why evolution is irrelevant to antibiotic resistance. It's really getting silly; he repeats the same nonsense over and over again, desparately doing the rhetorical version of sticking his…

As promised, today, I'm going to show the Kripke semantics model for intuitionistic logic. Remember from yesterday that a Kripke model has three parts: (W, R, :-), where W is a set of worlds; R is a transition relation between worlds; and ":-" is a forces relation that defines what's true in a…

So, over at the DI's media complains department (aka evolutionnews.org), it appears that Casey Luskin has noticed how we SBers have managed to tear apart his buddy Dr. Egnor. Given that we did it so thoroughly, though, there's no legitimate way to defend him. He's repeatedly made incredibly…

To be able to really talk about what a logic (or a calculus) means, you need to define a model of that logic. A model is a way of associating entities in the logic/calculus with some kind of real entity in a way where all statements in the logic about the logical entity will also be true about the…

I'm incredibly busy right now adjusting to my new job and my new commute, which is leaving me less time than usual for blogging. So I'm going to raid the archives, and bring back some interesting things that appeared on the old Blogger blog, but were never posted here. As usual, that will involve…

Ok. So I'm still tweaking syntax, to try to find a way of writing π-calculus in a way that's easy for me to write in my editor, and for you to read in your browser. Here's the latest version: Sequential Composition: Process1.Process2. Send expressions: !channel(tuple).Process Receive…

Today's bit of pathology is a really silly, and really fun language called Clunk, with a downloadable package containing a perl implementation here. I'm not sure that it's Turing compete, but my best guess is that it is. It's another two dimensional language, but it's very different from any of…

As usual, Casey Luskin over at DI's media complaints division is playing games, misrepresenting people's words in order to claim that that they're misrepresenting IDists words. Nothing like the pot calling the kettle black, eh? This time, he's accusing Ken Miller of misrepresenting Dembski in a…

As a refresher for me, and to give some examples to help you guys understand it, I'm going to go through a couple of examples of interesting things you can build with π-calculus. We'll start with a simple way of building mutable storage. Before getting started, I'm going to be annoying and change…

This came up in a question in the post where I started to talk about π-calculus, but I thought it was an interesting enough topic to promote it up to a top-level post. If you listen to anyone talking about computers or software, there are three worlds you'll constantly hear: parallel, concurrent,…

I feel like a bit of a change of pace, and trying a bit of an experiment. Re-reading Backus's old FP work reminds me of what I was doing the last time I read it, which was back in grad school. At the time, I was working on some stuff involving parallel computation, and I discovered Robin Milner's…

In my earlier post about John Backus, I promised to write something about his later work on functional programming languages. While I was in a doctors office getting treated for an awful cough, I re-read his 1977 Turing Award Lecture. Even 30 years later, it remains a fascinating read, and far…

I just heard that John Backus died last saturday. John Backus was one of the most influential people in the development of what we now know as software engineering. In the early days of his career, there was no such thing as a programming language: there was just the raw machine language of the…

I've been getting tons of mail from people in response to the announcement of the mapping of the E8 Lie group, asking what a Lie group is, what E8 is, and why the mapping of E8 is such a big deal? Let me start by saying that this is way outside of my area of expertise. So I fully expect that I'…

So the Discovery Institute's most recent addition has chosen to reply to my post about tautologies. (Once again, I'm not linking to him; I will not willingly be a source of hits for the DI website when they're promoting dangerous ingorance like this.) Typically, he manages to totally miss the…