Friday, February 29, 2008

Specious Explanations

There are two kinds of bad explanations I've blogged about in the past.

The first kind is the restatement of data. If we look at the data points on a graph, and our explanation is equivalent to drawing dots over the data points, then we're not explaining the points, merely restating them.

The second kind of specious explanation is the reference to an explanation we do not have. "God" is an example of this. In theory, if God were visible and we knew the mind of God and had his omniscience, God's actions would be explained by his choosing the best solution in each case. However, we don't have any of those things, and so God refers to an explanation we don't have.

In both cases, the way to avoid the trap is to have in one's hands a predictive model. It may not actually be the correct model, but at least it is explanatory. In the first case, points on the graph are explained by a particular curve (or a limited class of curves) passing through those data points. In the second case, God is only explanatory when we know what actions were God's, why he acted as he did, and what his actions are likely to be in the future.

In describing these specious cases to a friend, it occurred to me that these two varieties of fallacy are two sides of the same coin.

Suppose that we have a graph with data points on it, and we propose that they are explained by a curve passing through the points. So far so good. Under normal circumstances, we would fit N data points to an Mth-order polynomial, where M < N. That way, we can fit the polynomial to the data and make a prediction (by interpolation or extrapolation).

However, if I only have two data points and I propose to fit the points to a 2nd-order (quadratic) polynomial which requires three parameters, then my theory is under-determined. I need another data point to know what the polynomial looks like. Nonetheless, this is acceptable as an explanation because it makes the prediction that I can measure another data point, then make predictions from there.

Now, if we were to describe the space of all possible explanations, what would it look like? It would look like an unknown-order polynomial. For such a polynomial, we have no idea how many data points need to be accumulated before a predictive pattern emerges. Moreover, it is impossible to make any predictions from the unknown-order polynomial. A Kth-order polynomial (where K is finite and specified) would make a definite prediction, but an unknown-order polynomial can't do that much.

Hence, the first variety of specious explanation, the restatement, is equivalent to the second variety, a reference to an explanation we don't have.

As a clarification, I am talking about potential explanations here. Explanatory power. The explanatory power of a theory I do not yet have is zero! I'm not just saying that "God" is an unconfirmed theory, I'm saying that it's a non-explanatory theory (if it even merits the title "theory").

To add insult to injury, supernaturalists will argue that predictions about supernatural causes are not just an unknown distance away, but are fundamentally impossible. They don't merely invoke an unknown-order polynomial, but an infinite-order one.