goodmath
goodmath | December 30, 2006By 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 of the sun at the same time every day, instead of being in exactly the same place
every day, it will traverse a figure eight, like in this image. This is an effect caused by a combination of the eccentricity of the earth's orbit, and the tilt of the earth's axis. It can be a bit hard to visualize just where the figure-eight shape comes from; the analemma site uses a combination…
goodmath | December 5, 2006Yesterday at Pharyngula, PZ posted a description of his favorite signaling pathway in developmental biology, the [Notch system.][notch] Notch is
a cellular system for selecting one cell from a collection of essentially indistinguishable cells, so that that one cell can take on some different role.
What I found striking was that the problem that Notch solves is extremely similar to one of the classic problems of distributed systems that we study in computer science, called the leader-election problem; and that
the mechanism used by Notch is remarkably similar to one of the classic
leader-…
goodmath | November 21, 2006One of my fellow ScienceBloggers, [Karmen at Chaotic Utopia](http://scienceblogs.com/chaoticutopia/2006/11/puzzling_at_a_simpleminde…) pointed out a spectacularly stupid statement in [Casey Luskin's critique of Carl Zimmer][lutkin] (*another* fellow SBer) at the Discovery Institutes "Center for Science and Culture". Now normally, I might not pile on to tear-down of Casey (not because he doesn't deserve it, but because often my SciBlings do such a good job that I have nothing to add); but this time, he's crossed much too far into *my* territory, and I can't let that pass without at least a…
goodmath | November 14, 2006A 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 mathematics. He invented a technique for computing areas that's the closest thing to calculus before Newton and Leibniz. Modern mathematicians call Archimedes technique "the method of exhaustion".
The basic idea of the method of exhaustion is to take the figure whose area you want to compute, and to divide it into pieces whose area you already know…
goodmath | October 9, 2006There'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 Andrew Wiles first showed his proof of Fermat's last theorem, he discovered a very serious error. After that, it took him a couple of years, and some help from a colleague, but he *did* eventually fix the problem and complete the proof.
Whatever develops, it remains true that Professor Smith has made *huge* strides in her work on Navier-Stokes, and if she hasn'…
goodmath | September 4, 2006*(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 terms of the *inverse* of the function; I originally wrote it in the other direction. Thanks to commenter Dave Glasser for pointing out my error. I'll try to be more careful in the future!)*
Since I'm back, it's time to get back to topology!
I'm going to spend a bit more time talking about what continuity means; it's a really important concept in…
goodmath | August 30, 2006I'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 with, for convenience, I'll introduce a bit of syntactic sugar to let us
name functions. This will make things easier to read as we get to complicated
stuff.
To introduce a *global* function (that is a function that we'll use throughout our lambda calculus introduction without including its declaration…
goodmath | August 29, 2006I'm on vacation this week, so I'm recycling some posts that I thought were particularly interesting to give you something to read.
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In computer science, especially in the field of programming languages, we tend to use one particular calculus a lot: the Lambda calculus. Lambda calculus is also extensively used by logicians studying the nature of computation and the structure of discrete mathematics. Lambda calculus is great for a lot of reasons, among them:
1. It's very simple.
2. It's Turing complete.
3. It's easy to read and write.
4. It's semantics are strong enough that we can…
goodmath | August 24, 2006Yesterday, 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 familiar with this stuff, you might want to go back and take a look at the corrected version. It's just replacing a ≤ with a <, but that makes a *big* difference in meaning!)
Today I'm going to explain what a *topological space* is, and what *continuity* means in topology. (For another take on…
goodmath | August 22, 2006Back 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!
So what's topology about? In some sense, it's about the fundamental abstraction of *continuity*: if I have a bunch of points that form a continuous line or surface, what does that really mean? In particular, what does it mean *from within* the continuous surface?
Another way of looking at is as the…
goodmath | August 22, 2006Everyone 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 happens to work as a great example of just why good science needs good math.
As I always say, one of the ways to recognize a crackpot theory in physics is by the lack of math. For an example, you can look at the [electric universe][electric] folks. They have a theory, and they make predictions: but…
goodmath | August 21, 2006Last 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 read that][surreals] before looking at this post!
Transfinite Induction and ≤
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I'm going to start off by working through the way that the recursive
definition of the surreals and the "≤" operator work. It's based on
something called *transfinite induction*.
Transfinite…
goodmath | August 12, 2006In 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 into the Desert" problem.
The problem is:
Suppose you're given a checkerboard with all of the squares on the bottom filled. You're allowed to do standard checks jumps (jump over a man and remove it), but you can't jump diagonally, only up, left, or right. How far *up* can you get a man? How many men…
goodmath | August 10, 2006This 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 of articles on it [here][lambda]. You might also want to follow the CCC link above, to remind yourself about the CCCs. (The short refresher is: a CCC is a category that has an exponent and a product, and is closed over both. The product is an abstract version of cartesian set product; the exponent…
goodmath | August 10, 2006In 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 about Metamath some other time, but I thought it would be interesting today to point to [metamaths proof of the fifth axiom of Peano arithmetic][meta-peano], the principle of induction. Here's a screenshot of the first 15 steps; following the link to see the whole thing.
[metamath]: http://us.metamath…
goodmath | August 7, 2006Quaternions
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 things.
The basic idea behind quaterions is: we see some amazing things happen when we expand the dimensionality of numbers from 1 (the real numbers) to 2 (the complex numbers). What if we add *more* dimensions?
It doesn't work for three dimensions But you *can* create a system of numbers in *four* dimensions. As with complex numbers, you need a…
goodmath | August 7, 2006Ω 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 computation, that is important, and that expresses meaningful fundamental mathematical properties. It's also deeply non-computable; meaning that not only is it non-computable, but even computing meta-information about it is non-computable. And yet, it's *almost* computable. It's just all around awfully cool.
So. What is it Ω?
It's sometimes called the *halting…
goodmath | August 4, 2006If 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 leads us into todays strange numbers: irrational and transcendental numbers. Both of them were huge disappointments to the mathematicians who discovered them.
So what are they? We'll start with the irrationals. They're numbers that aren't integers, and that aren't a ratio of any two integers. So you can't write them as a normal fraction. If you write them as a continued…
goodmath | August 3, 2006One 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 what I want you to ask, or else there's no point in my writing the rest of this!)
It's annoying because both fractions and decimals can both only describe
*rational* numbers - that is, numbers that are a perfect ratio of two integers. And *most* numbers aren't rational.
But it's even more annoying than that: if you use decimals, then there are lots of rational numbers that you can't…
goodmath | August 1, 2006After 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 (not in the sense of real numbers, but in the sense of representing something *real* and meaningful)?
History
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The number *i* has its earliest roots in some of the work of early arabic mathematicians; the same people who really first understood the number 0. But they weren't quite as good with *i* as they were with 0: they didn't really get it. They had some concept of…