I’m not going to skewer the radical right’s attempt to relativize Wikipedia in full; better bloggers than me have already done so. But looking at Conservapedia’s mathematics entries is a good reminder that polemical hacks don’t usually produce any useful knowledge.

The combined knowledge of Wikipedia’s NPOV editors has produced a page about the prime number theorem that explains in length how the theorem relates to the Riemann zeta function and how the Riemann hypothesis implies a better estimate, and derives some explicit bounds. The first section, comprising only a small part of the article, says,

Let π(*x*) be the prime counting function that gives the number of primes less than or equal to *x*, for any real number *x*. For example, π(10) = 4 because there are four prime numbers (2, 3, 5 and 7) less than or equal to 10. The prime number theorem then states that the limit of the *quotient* of the two functions π(*x*) and *x* / ln(*x*) as *x* approaches infinity is 1. Using Landau notation this result can be written as

.

This does *not* mean that the limit of the *difference* of the two functions as *x* approaches infinity is zero.

Based on the tables by Anton Felkel and Jurij Vega, the theorem was conjectured by Adrien-Marie Legendre in 1796 and proved independently by Hadamard and de la Vallée Poussin in 1896. Both proofs used methods from complex analysis, specifically the properties of the Riemann zeta function and where the function was non-zero.

Meanwhile, the editors of Conservapedia, constrained by the requirements of a radical ideology that displays every radical pathology in the book (for a really egregious example of symbolism, check out the Conservapedia policy on British vs. American spelling), have produced the following article:

The Prime Number Theorem is one of the most famous theorem in mathematics. It states that the number of primes not exceeding n is asymptotic to $latex \frac{n}{\log n}, where log(n) is the logarithm of (n) to the base e.

The number of primes not exceeding n is commonly written as π(*n*), and an asymptotic relationship between a(n) and b(n) is commonly designated as a(n)~b(n). (This does not mean that a(n)-b(n) is small as n increases. It means the ratio of a(n) to b(n) approaches one as n increases.)

The Prime Number Theorem thus states that π(*n*)~*n* / log(*n*) .

In other words, the limit (as n approaches infinity) of the ratio of pi(n) to n/log(n) is one. Put a third way, n/log(n) is a good approximation for π(*n*).

*Section Break*

Gauus [*sic*] conjectured the equivalent statement that π(*x*) was asymptotic to Li(*x*) defined as:

.

In fact, for large x this turns out to be a better approximation than π(*x*).

Now, you might say I’m just picking and choosing, and other articles could be better. In fact, I’m picking and choosing here in Conservapedia’s favor; the prime number theorem is one of the few mathematical entries that even exist on Conservapedia. I could compare the articles on the Langlands program, or local rings, or global fields, or the Riemann hypothesis; on those subjects there is no Conservapedia article. Conservapedia doesn’t even have an article on mathematics.

You might also say that Conservapedia is a young project, so I shouldn’t be comparing it to a 6-year-old encyclopedia. Alright; the news on Conservapedia go back a month, so just compare the math there to the math posts I’ve put up in the month of February. On 2/1, I put up a basic concepts post that could make it to an encyclopedia. That took me maybe an hour net to write; how come the Conservapedia editors can’t come up with something better than a few stubs in a month?

Mark CC’s takedown is a good read; Conservapedia complains that Wikipedia doesn’t use “elementary proofs.” But Mark makes a slight mistake about elementary proofs:

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.

The concept of elementary proof is fairly relative. In number theory, it means no complex analysis, and Mark’s assessment is entirely valid. But in other subjects, it can mean something slightly different. When I took advanced group theory three semesters ago, my professor, an arithmetic geometer/number theorist, told me that to him, “elementary” in a group theoretic context meant no cohomology. There are certainly deeper techniques than just complex analysis; suffice is to say that if someone discovers a proof of Fermat’s Last Theorem that utilizes complex analysis but only at the level of the prime number theorem, his proof will likely be considered more elementary than Wiles’, which uses modular forms, Iwasawa theory, and other state of the art gadgets.