Not too long ago, I was listening to a radio interview with Seth MacFarlane and Adam Carolla. I find both of these comedians hilarious, and I don't intend any antagonism in critiquing something that was said. They were talking about their atheism and the host brought up the point that (and this is a paraphrase), "there has to be something behind the universe." He was referring specifically to the apparent order within the universe. MacFarlane responded briefly by noting all of the chaos in the universe. I want to make two quick, general points about this before moving on to the conceptualist argument (CA).
First of all, the existence of chaos does not cancel out the reality of order. No matter how strange, paradoxical, and unintuitive some things can be, there's still quite a bit of order to be found. Always, or for the most part, whenever I throw something into the air, it comes back down. Secondly, and more to the point, even chaotic events are intelligible. And, since intelligibility presupposes order, it follows that there is order even behind elements of chaos. It is true at all times and in all places that A=A; that if A=B and B=C, then A=C; and that 2+2=4. No instance of chaos negates any of that.
As a result, I agree with the host's conclusion that there must be some kind of ordering principle, what Heraclitus called the "Logos," that makes the universe intelligible to our experiences. However, many people might reasonably ask: why does the Logos have to be God. More specifically, why does the Logos have to be the personal God that classical theists believe in? This is where I believe the CA can be used to supplement the argument from uniformity detailed above.
Chad McIntosh has already ably defended the CA on a number of occasions. One of his latest contributions can be found here: http://www.doxazotheos.com/?p=101. In this post, I'd like to simply offer my own take on the argument.
First of all, we have already noted the necessity and indispensability of abstract propositions, such as 2+2=4. Instead of using my time to tackle this issue in depth, I will turn instead to the conceptual nature of abstracta and what this implies for theism. So far, the argument can be summarized like this:
1. Abstract objects are either contingent, necessary and mind-independent, or necessary concepts of a mind.
2. Abstract objects are not contingent or mind-independent.
3. Therefore, abstract objects are necessary concepts of a mind.
We will focus, then, on the premise that abstracta are not mind-independent. What reason do we have for coming to this conclusion? Here's how I would outline the proof:
4. There is a causal relationship between a subject and the external object that is known.
5. Abstract objects do not stand in causal relations.
6. Hence, if abstract objects are external to the mind, then they cannot be known.
7. Abstract objects are known.
8. Therefore, abstract objects are not external to the mind (i.e. they are not mind-independent).
The truth of this syllogism depends upon (4) and (5). In defense of (4), imagine some object external to the mind, like your computer screen. As you read this, your eyes act as a bridge in the causal relationship between your mind and the screen in order for you to have knowledge of the screen, and of what is on the screen. Now, if there is no such causal relationship, then you wouldn't know that the computer screen is in front of you. This becomes highly problematic unless one thinks of abstracta as mental concepts, since abstract objects do not stand in causal relations, per (5). Abstract objects just don't do anything; they are causally inert. The transitive axiom has no weight or measurability, and it certainly cannot mow my lawn or file my taxes. This would mean that if the transitive axiom were mind-independent, then we would have no knowledge of it. Yet, we clearly do have knowledge of it, which means that it and other abstracta must be conceptual in nature.
This is the really interesting part. Take the union of all true propositions; we can call it U. U is itself an abstract object, and is therefore the concept of a mind. However, it cannot be the concept of just any mind, since only an omniscient mind would know all true propositions. Therefore, it logically follows that an omniscient mind exists. So, not only do we have rational justification for believing in a Logos, but we are reasonably brought to the conclusion that this Logos is a personal mind. And, to borrow a phrase from Thomas Aquinas, this everyone understands to be God.