Many of my fellow SBers have been mocking the recently unveiled Conservapedia. Conservapedia claims to be a reaction to the liberal bias of Wikipedia. Ed, PZ, Afarensis, Tim, John, and Orac have all piled on already. But why should they get to have all the fun?

Conservapedia has an extensive list of what they claim to be examples of the liberal bias of Wikipedia. My SciBlings have already covered most of the nonsense to be found within, but one point is clearly mine to mock: grievance number 16:

Wikipedia has many entries on mathematical concepts, but lacks any entry on the basic concept of an elementary proof. Elementary proofs require a rigor lacking in many mathematical claims promoted on Wikipedia.

There *is* currently an entry on “Elementary Proof” on Wikipedia, but to be fair, it was created just two weeks ago, most likely in response to this claim by conservapedia.

But that’s trivial. The important thing here is that the concept of “elementary proof” is actually a relatively trivial one. It’s *sometimes* used in number theory, when they’re trying to pare down the number of assumptions required to prove a theorem. An elementary proof is a proof which makes use of the minimum assumptions that describe the basic properties of real numbers. And even in the case of number theory, I don’t think I’ve ever heard anyone seriously argue that an elementary proof is more rigorous than another proof of the same theorem. Elementary proofs *might* be easier to understand – but that’s not a universal statement: many proofs that make use of things like complex numbers are easier to understand than the elementary equivalent. And I have yet to hear of anything provable about real numbers using number theory with complex numbers which can be proven false using number theory without the complex – proofs about real numbers that use complex are valid, rigorous, and correct.

And that’s not even mentioning the minor fact that the vast majority of the math articles on wikipeidia are *not* about simple real number theory. According to Wikipedia’s mathematics portal, wikipedia currently has 16,093 articles on mathematics. Of those, 182 are an number theory. Conservapedia’s complaint thus makes *no sense whatsoever*

for 98.9% of Wikipedia’s math articles.

But in fact, it gets worse than that. Here’s the Conservapedia entry on

“elementary proof”:

The term “elementary proof” or “elementary techniques” in mathematics means use of only real numbers rather than complex numbers, which relies on manipulation of the imaginary square root of (-1). Elementary proofs are preferred because they are do not require additional assumptions inherent in complex analysis, such as that there is a unique square root of (-1) that will yield consistent results.

Mathematicians also consider elementary techniques to include objects, operations, and relations. Sets, sequences and geometry are not included.

The prime number theorem has long been proven using complex analysis (Riemann’s zeta function), but in 1949 and 1950 an elementary proof by Paul Erdos and Atle Selberg earned Selberg the highest prize in math, the Fields medal.

Sets are not part of “elementary proofs” in mathematics, according to Conservapedia. But relations are. What’s a relation? That terrible biased wikipedia says:

Definition 1.ArelationL over the sets X_{1}, …, X_{k}is a subset of their cartesian product, written L ⊆ X_{1}×…;× X_{k}. Under this definition, then, a k-ary relation is simply a set of k-tuples.

Wikipedia’s other definition is similar. According to wikipedia, a relation is a kind of set. But we know that Wikipedia is biased, right? So lets check a different source. Wolfram’s Mathworld is an excellent source of information on mathematical definitions. Let’s see how they define “relation”:

A relation is any subset of a Cartesian product. For instance, a subset of AxB, called a “binary relation from A to B,” is a collection of ordered pairs (a,b) with first components from A and second components from B, and, in particular, a subset of AxA is called a “relation on A.” For a binary relation R, one often writes aRb to mean that (a,b) is in R.

Uh-oh. “Houston, we have a problem.”.

So… The conservapedia can’t even write a consistent definition of elementary proof. It’s not too surprising – when you read their stuff on math, you get the definite idea that they really don’t have a clue of what the heck they’re talking about. Just take another look at their definition of “elementary proof”, where they’re trying to disparage the poor square root of -1.

As I’ve said before, i is *not* imaginary. It’s an important number, and it does exist in mathematics – as I’ve explained in the linked post, there are a lot of very important and very real phenomena that we experience in the real world that mathematically require *i* to be described.

The “uniqueness” of *i* is also not an assumption. It’s *required* by the fact that the complex numbers are a field. To create the two-dimensional complex number space – the one that we know describes real-world phenomena – we require a unique *i* – otherwise, we don’t get that plane.

What does conservapedia have to say about complex numbers, and their use?

Complex numbers is a branch of mathematics based on the assumption that one can manipulate a unique, imaginary number defined as:

i = \sqrt{-1}

In real numbers the square root of negative one does not exist.

That’s it. The whole shebang. Wikipedia is “liberal biased” because it doesn’t use “elementary proofs” in most of its math articles, even though the whole idea of elementary proof is inapplicable in over 98% of the articles. But Conservapedia just blindly rejects things that they don’t like, regardless of whether they’re important, real, or useful.

The conservapedia babble about “i” is just babble. It’s pretty clear that they’re yet another gaggle of bozos who are upset about the idea that math is more than the simple arithmetic they learned in elementary school, and want to find some way of throwing out everything that makes them uncomfortable.

Sorry guys. Math is more than you think it is. And *nobody* cares whether you like that fact.