# goodmath

As regular readers have no doubt noticed by now, posting on the blog
has been slow lately. I've been trying to come back up to speed, but so
far, that's been mainly in the form of bad math posts. I'd like to get
back to the good stuff. Unfortunately, the chaos theory stuff that I was
posting just isn't good for my schedule right now. Once you get past
the definitions of chaos, and understanding what it means, actually
analyzing chaotic systems is something that doesn't come easily to me - which
means that it takes a lot of time to put together a post. And
my work schedule…

One of the things that's endlessly fascinating to me about math and
science is the way that, no matter how much we know, we're constantly
discovering more things that we don't know. Even in simple, fundamental
areas, there's always a surprise waiting just around the corner.
A great example of this is something called the Ulam spiral,
named after Stanislaw Ulam, who first noticed it. Take a sheet of graph paper.
Put "1" in some square. Then, spiral out from there, putting one number in
each square. Then circle each of the prime numbers. Like the following:
If you do that for a while - and…

As I've mentioned before, I've been spending a lot of time working on a book.
Initially, I was working on a book made up of a collection of material from blog posts;
along the way, I got diverted, and ended up writing a book about cloud computing using
Google's AppEngine tools. The book isn't finished, but my publisher, the Pragmatic Programmers,
have a program that they call beta books. Once a book is roughly 60% done, you
can buy it at a discount, and download drafts electronically immediately. As more sections
get done, you can download each new version. And when the book is finally…

I've got a bunch of stuff queued up to be posted over the next couple of days. It's
been the sort of week where I've gotten lots of interesting links from
readers, but I haven't had time to finish anything!
I thought I'd start off with something short but positive. A reader sent
me a link to a post on Reddit, with the following question:
Throughout elementary and high school, I got awful marks in math. I always
assumed I was just stupid in that way, which is perfectly possible. I also
hated my teacher, so that didn't help. A friend of mine got his PhD in math
from Harvard before he was 25…

As you've no doubt heard by now, there's been a new recommendation issues
which proposes changing the breast-cancer screening protocol for women under
50, by eliminating mammograms for women who don't have significant risk
factos. While Orac has done a terrific job of covering this here and
here, I wanted to throw
in a couple of notes and a personal perspective.
To begin with, there's a bit of math which has been bandied about, and
I thought I'd just quickly walk through it.
When you look at things like screening programs, what you're doing is
performing some kind of test on a very large…

So I'm trying to ease back into the chaos theory posts. I thought that one good
way of doing that was to take a look at one of the class chaos examples, which
demonstrates just how simple a chaotic system can be. It really doesn't take much
at all to push a system from being nice and smoothly predictable to being completely
crazy.
This example comes from mathematical biology, and it generates a
graph commonly known as the logistical map. The question behind
the graph is, how can I predict what the stable population of a particular species will be over time?
If there was an unlimited…

So, why math?
The short version of the answer is remarkably simple: math provides
a tool where you can, without ambiguity, prove that something is true or false.
I'll get back to that - but first, I'm going to make a quick diversion, to help you understand my basic viewpoint on things.
This blog actually started in response to something specific. I was reading
Orac's blog "Respectful Insolence", and
he was fisking a study published by the Geiers, purporting to show a change in the trend in autism diagnoses. Orac was attacking it on multiple bases, but it struck me
that the most obvious…

Today the 2008 Nobel Prize winners were announced for physics. It was given to three physicists who described something called symmetry breaking. Since most people don't know what symmetry breaking is, but people remember me writing about group theory and symmetry, I've been getting questions about what it means.
I don't pretend to completely understand it; or even to mostly understand it. But I mostly understand the very basic idea behind it, and I'll try to pass that understanding on to you.
We'll start with the idea of symmetry. Intuitively, we think of symmetry as a situation where…

Every year at ScienceBlogs, we do a charity drive for
DonorsChoose.org. If you haven't heard of them, DonorsChoose is a charity that takes proposals from schoolteachers, and lets people pick specific proposals to donate money to. We run our charity challenge through the month of October.
For personal reasons, I couldn't participate last year. The year before that, Good Math/Bad Math readers donated just over two thousand dollars to support math education in impoverished New York area schools.
This year, I'm still focusing on the NYC area, because with where I live and work, I get to…

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 in the 1970s. The question concerns one
claim in the article. Bertie was paid Â£10,000 for his part in a robbery in 1972. The article alleges that Â£10,000 in 1972 is equivalent to Â£200,000 today.
Lots of people think that that looks fishy, and have been sending me mail asking
if that makes any sense.
Full disclosure. I'm not an economist - I've never studied…

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 at recent editions. In fact, I completely forgot that I was hosting it again until I started receiving
submissions.
Much to my disappointment, it appears that spam has managed to invade even the carnivals. Close to half of the submissions that I received were blatant spam, including one for a penis-enlargement pill. But hey, when a theme hits me in the face, I run with it. So, welcome to the…

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 the article demonstrates the existence of free will, which they argue is inconsistent with naturalism and darwinism.
The thing is, the paper says nothing of the sort.
The paper did a very interesting study on the behavior of fruit-flies. They basically tethered fruit flies inside of a small cylindrical apparatus, which basically amounts to
a little tiny…

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 analysis of the effect, and their prediction and verification of the effect.
The Kaye effect is an incredibly bizarre phenomenon. Basically, if you take a substance like liquid shampoo, and allow a thin stream of it to pour down from a height onto a smooth surface, the stream will periodically "bounce", producing a stream leaping up from the point of contact.…

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 involve Gödel style problems - any self-referential cycle is a strange loop.
Reading this book reminded me of my favorite strange-loop story. It's actually
a story about software security, and the kinds of stunts you can play with
software if you're clever and subtle. It's the story of the Unix C compiler, and the virtually invisible back-door security hole inserted…

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 was setting up his awards, and was not including
mathematics. The prize is named after Niels Abel, the Norwegian mathematician who
discovered the class of functions that are now known as Abelian functions; the same person
that Abelian groups are named after, etc.
Anyway, this year, the Abel
prize was…

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 poorly with one another, and analyzing the distribution of resistance in the resulting populations. It turns out that under the right conditions, you can create a situation in which the selective pressure of the combination of drugs - which are less effective when combined - can select in favor of…

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 formulation of the reals is arguably superior to the more traditional formulations of the reals via Dedekind cuts or Cauchy sequences.
First, let's look at how we can create a set of just the real numbers using the
surreal construction. What we want to do is get a notion of the simplest surreal number that…

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 messy office. And I've got a problem. The Math Carnival is coming to town. All those geeks, and the chaos that they always cause. Oy.
But I'm stuck. After all, I have student loans that need to be payed off early, and the kind of work I do doesn't exactly bring in as much cash as running an oil company. Besides, maybe I can talk someone who's been confused by all the math geekery into…

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. So it's worth taking the time to look at it, and see what it means and how it works.
The cleverest definition that I've seen of recursion comes from the Hackers dictionary. In there, it has:
recursion
n. See {recursion}.
Recursion is about defining things in terms of themselves. For what is probably the canonical example, think about the…

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 system, and a Turing complete computation. It's true
that in informal use, we often tend to muddy the line between these two related but distinct concepts. But in fact, they are distinct, and the difference between them can be extremely important. In some sense, it's the difference between "capable of" and "requires"; another way of looking at it is
"sufficient" versus "…