## Friday, September 03, 2010

### Much needed closure.

Closure is an important concept in mathematics, and is deceptively simple. If you have a set of things and some operation acting on those things, then the closure of your set is the smallest set that contains your original set along with everything that operation gives you.

The words get in the way, though, so let's consider an example. If you have the numbers zero and one, then their closure under addition would be all positive integers. Why? Because you can get to any positive integer by adding one to itself over and over. For instance, 2 is in the closure, since addition produces 2 from our set: 1 + 1 = 2. By the same argument, 3 is in the closure since 1 + 2 = 3, and since 2 must be in our closure.

We say that this set is the closure of our original set since it is the smallest set which is, well, closed. If, in our previous example, we omitted the number 2, our set wouldn't be closed any more, since addition could take us outside of the set.

As of late, however, the way that mathematicians use the word closure has started to be seen well outside of mathematics. Witness the rise of "epistemic closure" (closely related to deductive closure) as a useful term in political science. The word finds much use even outside of mathematics, as it gets to the heart of a very powerful technique in rational thinking: asking what, given some tool, one can produce. In epistemic closure, the tool is reasoning itself, while in our more pedestrian example, our tool was basic addition. In both cases, however, what remains is the use of closure as a mechanism for understanding and characterizing an operation.

In the spirit, then, of exploring closure, I'd like to bring some much needed closure forward. Specifically, I'd like to consider a kind of causal closure. If we consider some set of events which may or may not be causally related, we can for any specific event ask what events may be caused and what other events may cause it. Both of these are a kind of operation; extrapolating both directions in time to understand the causal structure of your set of events. The causal closure, then, of a set of events is the full set of events which caused the original set, along with the full set of effects caused by these events.

Luckily, we already have a term for this kind of causal closure. What we mean when we say that two events are causally related is that they lie within the same universe, so that the universe can be thought of as the set of all events which are causally related to an event representing our powers of observation. Under this realization, if A causes B, then A cannot be in a different universe than B.

All this is well and good, but why do I bring it up now? In a recent post, I asserted that religious claims were of a material nature, and thus amenable to the methods of science. That this is the case can be easily seen by invoking a principle of causal closure; if religious claims include any causal relation to the material universe, then they must, by closure, be entirely about the material universe. Of course, that alone does not mean that such claims are subject to scientific understanding, but that is an argument I have made before and don't wish to repeat here. Rather, my intent was simply to bring some much needed closure to bear on an argument that has gone on too long.