Reasonable Dissent

More (dis)Proof of God

Being somewhat of  a math-head, I tend to like proofs. When done well, they’re inescapable. However, philosophical proofs almost invariably leave me unsatisfied (yes, on both sides of the God-debate). But I still like pointing out where they are lacking, because it keeps people from touting them as actual proof. It’s almost funny when a theist comes with the idea that atheists just want logical proof or evidence, and they must not have heard this perfect argument yet. It’s not that we haven’t heard it; it’s that it isn’t nearly as convincing as you think. So, cue phase two of my commentary on the usual suspects in the ‘Proof for God’.

Last time, I discussed the ‘prime mover‘ argument, specifically the formulation put forth by Aquinas, since his annoys me more than most (it’s the way he asserts the Christian god at the end that does it, I think). But I got an interesting comment that reminds me why mathematical proofs are so satisfying to me. The commenter mentioned the idea, “What’s infinity times zero?” So what does math have to say about this? My response was this:

Strictly math-speaking, you simply cannot do that operation because infinity is not a number. It’s tantamount to saying “what is zero times chair?” Infinity is a tricky thing to get a hold of anyway. For example, there are an infinite number of integers. This is a class of infinity called “countably infinite”. But there’s another class called “uncountably infinite”, for example the real numbers. Take my set of numbers here (1.2, 1.22, 1.222, 1.2222, …) You can see how I’m constructing them. I could continue this series forever, always increasing and never reach 1.3. You could not assign an integer in a 1:1 fashion to the real numbers. So is uncountably infinite greater than countably infinite? They’re both infinity…

But it’s even more fun, because I can illustrate a good (dis)proof and play with infinity at the same time. I’m disproving this claim “Infinity is a number”, and using the common understanding of a number as an element of an ordered set that behaves under the usual arithmetic (+-*/). Also, I’m using the understanding that <inf> + x = <inf>   (ie, adding anything to infinity is still infinity).

So, if <inf> is a number that behaves like a number,

<inf>/<inf> = 1

(<inf>/<inf>) + <inf> = 1 + <inf>

(<inf>/<inf>) + <inf> = <inf>

<inf> / <inf> = 0

1 = 0

See how satisfying that is? I made a claim, if infinity is a number, then the following must be true… 1=0. One very much does not equal zero, so our premise is false. Now let’s take a look at another proof for God. It’s the standard Ontological Argument (meaning a priori, quite a claim), put forth (I believe originally) by St. Anselm. It roughly follows this path:

1. God, by definition is the greatest being imaginable. You cannot imagine anything greater.
2. A being that exists is greater than a being that does not exist
3. Therefore, God must exist

Now see how unstatisfying that is? It kind of ties in with another of Aquinas’ proofs (number 4) . His follows this logic:

1. Things that exist have certain qualities to greater or lesser degrees
2. greater and lesser are relative terms, which relate to the maximum
3. Something must have the maximum possible of every quality
4. That’s God.

One criticism of this I’ve read is that he says nothing that proves that one single object must have the maximum possible of all qualities, but I think this is too easily dismissed. Let’s just add a step, 2.5 that says “One quality you can have is having qualities”.

There, fixed, right? No, they’re both still very flawed. In the first proof, we make no distinction between imagining something and existence when it is actually a very deep divide. How about this?

1. I can imagine no greater proof than the proof against God’s existence.
2. A real proof is greater than an imaginary proof.
3. Lucky me, it’s real and I’m almost done writing it.
4. Therefore, no God.

It’s exactly the same as our infinity proof. I said “If your proof is true, then so is mine, so God exists and God does not exist.” A logical contradiction, therefore our assumptions were wrong. The proof cannot be valid.

The problem that arises from Aquinas’ proof is that he does not prove that because we can determine relative relationships between qualities two things possess, we know that something must have to have the extreme of this quality. He says ‘maximum’ and uses it as ‘infinite’. In his text, he uses concepts like goodness, truth, and nobility. But we are not required to believe in an infinitely good being. His proof merely says that for us to use goodness in a relative way, there must be a ‘most good’ and a ‘least good’. We can conceive of infinite goodness, but it is not required to relate the relative goodness between two real objects. We need only something with some amount of goodness as a reference point, and say that there exists something with more good than anything else that exists (but that my not be infinite goodness).

Again this will tie into our discussion of infinity (see what I did there). There are numbers that are greater and lesser than others. For instance 2 < 3.  15,204 > -10.   We know their relative “greatness”, and we can conceive of the idea that there is a “greatest”. We call this concept infinity. But infinity is not a number, and the fact that we can use the idea of a “greatest number” does not mean that there is one. In fact, I can prove there is not.

For an arbitrarily large number x,

x < x+1

But since x is arbitrarily large, there can be no largest number.

Now if they would just actually define “God” we could find a similar argument and be done with it….

Tags: atheism, logical fallacies, science