May 2007 – Good Math Bad Math

Behe's Dreadful New Book: A Review of "The Edge of Evolution"

MarkCC May 31, 2007 Intelligent Design

I’ve gotten my hands on a review copy of Michael Behe’s new book, “The Edge of Evolution”. The shortest version of a review is: Bad science, bad math, and bad theology, all wrapped up in a pretty little package.
As people who’ve followed his writings, lectures, and court appearances know, Behe is pretty much a perfect example of the ignoramus who makes a bad argument, and then puts his fingers in his ears and shouts “La la la, I can’t hear you” whenever anyone refutes it. He *still* harps on his “irreducible complexity” nonsense, despite the fact that pretty much *every aspect* of it has been thoroughly refuted. (The entire concept of IC is a pile of rubbish; the entire argument about IC is based on the idea that evolution is a strictly additive process, which is not true; there are numerous examples of observed evolution of IC systems, both in biology and in evolutionary algorithms. But none of these facts makes a bit of difference: like the energizer bunny of ignorance, he just keeps going, and going…)
Anyway, the new book is based on what comes down to a mathematical argument – a mathematical argument that I’ve specifically refuted on this blog numerous times. I’m not mentioning that because I expect Behe to read GM/BM and consider it as a serious source for his research; even if I were an expert in the subject (which I’m *not*), a blog is *not* a citable source for real research. But I mention it because the error is so simple, so fundamental, and so bleeding *obvious* that even a non-expert can explain what’s wrong with it in a spare five minutes – but Behe, who apparently spent several *years* writing this book still can’t see the problem. (In fact, one of the papers that he cites as *support* for this ridiculous theory contains the refutation!)

(more…)

The Disgrace of Memorial Day

MarkCC May 28, 2007 Politics

Today is Memorial Day, and I feel compelled to say something about it. We're in the middle of a horrible and pointless war. A war that we started, based on…

The Axiom of Choice

MarkCC May 27, 2007 Set Theory

The Axiom of Choice
The axiom of choice is a fascinating bugger. It’s probably the most controversial statement in mathematics in the last century – which is pretty serious, considering the kinds of things that have gone on in math during the last century.

The axiom itself is quite simple, and reading an informal description of it, it’s difficult to understand how it managed to cause so much trouble. For example, wikipedia has a rather nice informal statement of it:

given a collection of bins each containing at least one object, exactly one object from each bin can be picked and gathered in another bin

(more…)

The Axiom of Infinity

MarkCC May 24, 2007 Set Theory

The axiom of infinity is a bundle of tricks. As I said originally, it does two things. First, it gives us our first infinite set; and second, it sets the stage for representing arithmetic in terms of sets. With the axiom of infinity, we get the natural numbers; with the natural numbers, we can get the integers; with the integers, we can get the rationals. Once we have the rationals, things get a bit harder – but we can get the reals via Dedekind cuts; and by transfinite induction, we can get the transfinite numbers. But before we can get to any of that, we need a sound representation of the naturals in terms of sets.

(more…)

Popups

MarkCC May 24, 2007 Meta

We've been having some problem with popups showing up on scienceblogs. It's not deliberate: Seed does not accept popup ads. But it appears that one snuck in somehow, but no…

The Axiom of Extensionality

MarkCC May 23, 2007 Set Theory

Some of the basic axioms of ZFC set theory can seem a bit uninteresting on their own. But when you take them together, and reason your way around them, you…

The Axiom of Pairing

MarkCC May 23, 2007 Set Theory

The axiom of pairing is an interesting beast. It looks simple, and in fact, it
is simple. But it opens up a range of interesting things that we’d like to be able
to do. For example, without the axiom of pairing, we wouldn’t be able to formulate the
cartesian products of sets – and without cartesian product, huge ranges of interesting and
important areas of mathematics would be inaccessible to us. (Note that I’m saying that
pairing is necessary, not that it’s sufficient. You also need replacement
to get the projection functions that are part of the usual definition of the cartesian
product.)

(more…)

The Man I called Fink

MarkCC May 22, 2007 Chatter

My father died on sunday. To some degree, I'm still in shock. Even though we knew it was coming, when something like this happens, no amount of preparation really helps.…

The Axioms of Set Theory

MarkCC May 20, 2007 Set Theory

Axiomatic set theory builds up set theory from a set of fundamental initial rules. The most common axiomatization, which we’ll be used, is the ZFC system: Zermelo-Fraenkel with choice set theory. The ZFC axiomatization consists of 8 basic rules which are pretty much universally accepted, and two rules that are somewhat controversial – most particularly the last rule, called the axiom of choice.

(more…)

Why Axiomatize Set Theory?

MarkCC May 19, 2007 Set Theory

Naive set theory is fun, and as we saw with Cantor’s diagonalization, it can produce some incredibly beautiful results. But as we’ve seen before, in the simple world of naive set theory, it’s easy to run into trouble, in the form of Russell’s paradox and a variety of related problems.

For the sake of completeness, I’ll remind you that Russell’s paradox concerns the set R={ s | s ∉ s}. Is R∈R? If R∈R, then by the definition of R∉R. But by definition, if R∉R, then R∈R. So R is clearly not a well-defined set. But there’s nothing about the form of its definition which is prohibited by naive set theory!

Mathematicians, being the annoying buggers that they are, weren’t willing to just give up on set theory over Russell’s paradox. It’s too beautiful, too useful an abstraction, to just give up on it over the self-reference problems. So they went searching for a way of building up set theory axiomatically in a way that would avoid problems by making it impossible to even formulate the problematic statements.

(more…)