goodmath:
I've been getting a ton of questions about an article from the Independent about a guy named Bertie Smalls. Bertie was a british thief who died quite recently, who was famous for testifying against his organized crime employers back...
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Posted on February 12, 2008 9:56 AM • 24 Comments •
To be honest, I haven't been following the Carnival of Math much since it's inception; my new job keeps me busy enough that I barely have time to keep the blog going, and so I haven't really looked much...
Posted on October 19, 2007 11:43 AM • 12 Comments •
I've been seeing articles popping up all over the place about a recent PLOS article called Order in Spontaneous Behavior. The majority of the articles seem to have been following the lead of the Discovery Institute, which claims that...
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Posted on May 17, 2007 1:24 PM • 49 Comments •
I came across this while looking through the referrals to GM/BM. This is an incredibly cool video of a strange phenomenon called the Kaye effect. It includes high speed video of the effect, and a demonstration of their mathematical...
Posted on April 17, 2007 8:55 PM • 16 Comments •
I'm currently reading "I am a Strange Loop" by Douglas Hofstadter. I'll be posting a review of it after I finish it. A "strange loop" is Hofstadter's term for a Gödel-esque self-referential cycle. A strange loop doesn't have to...
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Posted on April 15, 2007 12:34 PM • 40 Comments •
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,...
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Posted on April 12, 2007 8:05 PM • 14 Comments •
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...
Posted on April 9, 2007 11:52 AM • 10 Comments •
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...
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Posted on April 5, 2007 10:09 PM • 13 Comments •
Please make sure you read to the end. A couple of late submissions didn't get worked into the main text, and a complete list of articles is included at the end. Oy. So I find myself sitting in my disgustingly...
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Posted on February 23, 2007 8:18 AM • 9 Comments •
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....
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Posted on January 23, 2007 9:05 PM • 8 Comments •
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...
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Posted on January 5, 2007 1:08 PM • 40 Comments •
By way of the astronomy picture of the day, I encountered a really fantastic site about the analemma. The analemma is the apparent path that the sun takes in the sky during the year. If you record the precise position...
Posted on December 30, 2006 10:57 AM • 2 Comments •
Yesterday at Pharyngula, PZ posted a description of his favorite signaling pathway in developmental biology, the Notch system. Notch is a cellular system for selecting one cell from a collection of essentially indistinguishable cells, so that that one cell can...
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Posted on December 5, 2006 12:21 PM • 22 Comments •
One of my fellow ScienceBloggers, Karmen at Chaotic Utopia pointed out a spectacularly stupid statement in Casey Luskin's critique of Carl Zimmer (another fellow SBer) at the Discovery Institutes "Center for Science and Culture". Now normally, I might not pile...
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Posted on November 21, 2006 1:45 PM • 35 Comments •
A lot of people have asked me to write something about "Archimedes Integration", and I'm finally getting around to fulfilling that request. As most of you already know, Archimedes was a philosopher in ancient Greece who, among other things, studied...
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Posted on November 14, 2006 2:54 PM • 12 Comments •
There's bad news on the math front. Penny Smith has withdrawn her Navier Stokes paper, because of the discovery of a serious error. But to be optimistic for a moment, this doesn't mean that there's nothing there. Remember that when...
Posted on October 9, 2006 9:48 PM • 17 Comments •
(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...
Posted on September 4, 2006 4:22 PM • 19 Comments •
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...
Posted on August 30, 2006 8:33 AM • 0 Comments •
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...
Posted on August 29, 2006 8:30 AM • 5 Comments •
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...
Posted on August 24, 2006 8:20 PM • 7 Comments •
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...
Posted on August 22, 2006 7:11 PM • 34 Comments •
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; and a Sean Carroll (no relation) has a very good explanation on his...
Posted on August 22, 2006 10:03 AM • 47 Comments •
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...
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Posted on August 21, 2006 8:30 AM • 11 Comments •
In the comments onmy post about φ, Polymath, (whose blog is well worth checking out) provided a really spectacular link involving φ. It's an excerpt from a book called "Mathematical Gems 2", showing a problem that came from John Conway,...
Posted on August 12, 2006 10:10 AM • 1 Comments •
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. If...
Posted on August 10, 2006 2:33 PM • 5 Comments •
In email, someone pointed me at an automated proof system called 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...
Posted on August 10, 2006 10:53 AM • 3 Comments •
Quaternions Last week, after I wrote about complex numbers, a bunch of folks wrote and said "Do quaternions next!" My basic reaction was "Huh?" I somehow managed to get by without ever being exposed to quaternions before. They're quite interesting...
Posted on August 7, 2006 7:53 PM • 34 Comments •
Ω is my own personal favorite transcendental number. Ω isn't really a specific number, but rather a family of related numbers with bizzare properties. It's the one real transcendental number that I know of that comes from the theory of...
Posted on August 7, 2006 9:00 AM • 54 Comments •
If you look at the history of math, there've been a lot of disappointments for mathematicians. They always start off with an idea of math as a clean, beautiful, elegant thing. And they seem to often wind up disappointed. Which...
Posted on August 4, 2006 11:00 AM • 18 Comments •
One of the annoying things about how we write numbers is the fact that we generally write things one of two ways: as fractions, or as decimals. You might want to ask, "Why is that annoying?" (And in fact, that's...
Posted on August 3, 2006 12:00 PM • 43 Comments •
After the amazing response to my post about zero, I thought I'd do one about something that's fascinated me for a long time: the number i, the square root of -1. Where'd this strange thing come from? Is it real...
Posted on August 1, 2006 2:06 PM • 63 Comments •
Today we'll finally get to building the categories that provide the model for the multiplicative linear logic. Before we jump into that, I want to explain why it is that we separate out the multiplicative part. Remember from the simply...
Posted on July 31, 2006 2:25 PM • 0 Comments •
In the comments on my post mocking Granville Sewell's dreadful article, one of the commenters asked me to write something about why evolution is frequently modeled as a search process: since there is no goal or objective in evolution, search...
Posted on July 26, 2006 3:28 PM • 26 Comments •
Things are a bit busy at work on my real job lately, and I don't have time to put together as detailed a post for today as I'd like. Frankly, looking at it, my cat theory post yesterday was half-baked...
Posted on July 26, 2006 1:57 PM • 3 Comments •
So, we're still working towards showing the relationship between linear logic and category theory. As I've already hinted, linear logic has something to do with certain monoidal categories. So today, we'll get one step closer, by talking about just what...
Posted on July 25, 2006 3:42 PM • 3 Comments •
Back during the DonorsChoose fundraiser, I promised a donor that I'd write an article about the math of zero. I haven't done it yet, because zero is actually a suprisingly deep subject, and I haven't really had time to do...
Posted on July 21, 2006 2:57 PM • 64 Comments • 1 TrackBacks
Time to come back to category theory from out side-trip. Category theory provides a good framework for defining linear logic - and for building a Curry-Howard style type system for describing computations with state that evolves over time. Linear logic...
Posted on July 19, 2006 4:13 PM • 3 Comments •
(This post has been modified to correct some errors and add some clarifications in response to comments from alert readers. Thanks for the corrections!) Today, we're going to take a brief diversion from category theory to play with some logic....
Posted on July 17, 2006 4:52 PM • 8 Comments •
Via Migrations, I've found out about a really beautiful computational biology paper that very elegantly demonstrates how, contrary to the assertions of bozos like Dembski, an evolutionary process can adapt to a fitness landscape. The paper was published in the...
Posted on July 14, 2006 11:27 AM • 3 Comments •
One of the questions that a ton of people sent me when I said I was going to write about category theory was "Oh, good, can you please explain what the heck a monad is?" The short version is: a...
Posted on July 12, 2006 10:55 AM • 4 Comments •
So, at last, we can get to Yoneda's lemma, as I promised earlier. What Yoneda's lemma does is show us how for many categories (in fact, most of the ones that are interesting) we can take the category C, and...
Posted on July 11, 2006 11:43 AM • 4 Comments •
Today's contribution on category theory is going to be short and sweet. It's an example of why we really care about natural transformations. Remember the trouble we went through working up to define cartesian categories and cartesian closed categories? As...
Posted on July 10, 2006 3:10 PM • 0 Comments •
We're almost at the end of this run of category definitions. We need to get to the point of talking about something called a pullback. A pullback is a way of describing a kind of equivalence of arrows, which gets...
Posted on July 6, 2006 7:26 PM • 4 Comments •
What's a subset? That's easy: if we have two sets A and B, A is a subset of B if every member of A is also a member of B. What's a subgroup? If we have two groups A and...
Posted on July 4, 2006 11:30 AM • 9 Comments •
Before I dive into the depths of todays post, I want to clarify something. Last time, I defined categorical products. Alas, I neglected to mention one important point, which led to a bit of confusion in the comments, so I'll...
Posted on June 27, 2006 4:19 PM • 6 Comments •
On my way home from picking up my kids from school, I heard a story on NPR that included a line from one of the editors of the New England Journal of Medicine, which I thought was worth repeating here....
Posted on June 26, 2006 6:14 PM • 8 Comments •
In the comments to one of my earlier Demsbki posts, I was responding to a comment by a poster, and realized that what we were discussing was probably interesting enough to promote up to a front-page post. A poster named...
Posted on June 25, 2006 11:59 AM • 23 Comments •
Back when I first started this blog on blogspot, one of the first things I did was write an introduction to information theory. It's not super deep, but it was a decent overview - enough, I thought, to give people a good idea of what it's about, and to help understand why the common citations of it are almost always misusing its terms to create bizzare conclusions, like some ["law of conservation of information",][conservation]
[conservation]: http://en.wikipedia.org/wiki/Law_of_conservation_of_information "wikipedia on Dembski's law of CoI"
This is a slight revision of that introduction. For the most part, I'm just going to clean up the formatting, but once I'm going through it, I'll probably end up doing some other polishing.
Posted on June 24, 2006 5:06 PM • 28 Comments •
Sorry, but I actually jumped the gun a bit on Yoneda's lemma. As I've mentioned, one of the things that I don't like about category theory is how definition-heavy it is. So I've been trying to minimize the number...
Posted on June 22, 2006 3:31 PM • 3 Comments •
Harald Hanche-Olsen, in the comments on my earlier post about the Principia Mathematica, has pointed out that this months issue of the Notices of the American Mathematical Society is a special issue in honor of the 100th anniversary of Kurt...
Posted on June 21, 2006 10:40 AM • 2 Comments •
The thing that I think is most interesting about category theory is that what it's really fundamentally about is structure. The abstractions of category theory let you talk about structures in an elegant way; and category diagrams let you illustrate...
Posted on June 19, 2006 10:49 AM • 1 Comments •
Just saw a nice post at another math blog called Polymathematics about something that bugs me too... The way that people don't understand what repeating decimals mean. In particular, the way that people will insist that 0.9999999... != 1. As...
Posted on June 17, 2006 2:01 PM • 49 Comments • 1 TrackBacks
Let's talk a bit about functors. Functors are fun! What's a functor? I already gave the short definition: a structure-preserving mapping between categories. Let's be a bit more formal. What does the structure-preserving property mean? A functor F from...
Posted on June 12, 2006 8:40 AM • 7 Comments •
For me, the frustrating thing about learning category theory was that it seemed to be full of definitions, but that I couldn't see why I should care. What were these category things, and what could I really talk about...
Posted on June 9, 2006 3:59 PM • 9 Comments •
One of the things that I find niftiest about category theory is category diagrams. A lot of things that normally turn into complex equations or long-winded logical statements can be expressed in diagrams by capturing the things that you're talking...
Posted on June 8, 2006 11:25 AM • 17 Comments • 1 TrackBacks
Back at my old digs last week, I put up a post about programming languages and types. It produced an interesting discussion, which ended up shifting topics a bit, and leading to a very interesting question from one of the...
Posted on June 7, 2006 5:47 PM • 26 Comments •
To get started, what is category theory? Back in grad school, I spent some time working with a thoroughly insane guy named John Case who was the new department chair. When he came to the university, he brought a couple...
Posted on June 7, 2006 10:02 AM • 19 Comments •
