In case you've ever wondered whether God exists, a simple answer can be found in mathematics. Here it is: God is defined as the perfect being. One of the properties contained in the definition of "perfect" is existence. Therefore, to avoid logical contradiction, God must exist. This argument, known as an 'ontological argument' due to its attempt to use logic and reason alone to prove God's existence, was originally proposed by Anselm of Canterbury in the 11th century, and repeated in various forms by later philosphers, including famous mathematician Rene Descartes.
So, what do you think of this argument? Something about it seems a bit unsatisfying, though it can be a little tricky to put your finger on it at first. It's pretty easy to spoof it though. Have you ever thought about a perfect island, with light breezes, coconut trees, and beautiful beaches on which cute seals frolic? Well, by this argument, it must exist too. The world must in fact be full of perfect stuff that all exists, by this argument. Some perfect podcaster must be recording the perfect August 1st 2010 math podcast sometime today, in fact. I'm pretty sure this isn't it though; sorry to disappoint you!
Extending this argument to the mathematical domain further exposes its absurdity: let's talk about the perfect 32-sided die, which we will define as a 3-d object which has 32 regular polygons for its faces, and in which all angles and faces are congruent. Since we have defined this as perfect, it must exist. But we know, as discussed in episode 18, that such an object cannot exist, as there are only five regular polyhedra, and a 32-sider is not one of them! What's going on?
You have probably spotted the key problem by now, pointed out most clearly by Immanuel Kant in 1781. When we define a theoretical object and state a number of properties it has, this definition is necessarily conditional: we are stating that IF the object exists, it has all the stated properties. We cannot cause an object to exist by just adding existence to its definition: this is a circular argument. And if other aspects of its definition prevent an object from existing, these cannot be undone merely by adding existence to its definition. All we are saying is that IF the object exists, then it exists, which doesn't really add much to the discussion.
If you look at the wikipedia page on ontological arguments, you will find a number of convoluted reformulations by modern philosophers attempting to resurrect Anselm's ontological argument, but they all look to me like fundamentally the same argument in more complicated words. Ultimately, I don't care how fancy the philsophers get, we can't cause something to exist by adding the existence attribute to its definition. When I can sit on my perfect island beach rolling 32-sided dice as I play the perfect edition of Dungeons & Dragons with frollicking seals, maybe I'll start to be convinced by this argument. Until then, I'm afraid the question of the existence of divine beings is probably beyond the domain of mathematics.
And this has been your math mutation for today.