Non-theistic unificationist accounts of explanation

Here is a problem for non-theistic unificationist accounts of explanation. Suppose that in world w₁, I arrange a large sequence of n perfectly round stones (where n is large enough that the unification of the data on these n stones calls invites a unifying explanation) according to the rule that I place the kth stone at a distance f(k) from the Washington Monument, where f is some moderately complex mathematical function that I have picked for the occasion. Suppose there are no other perfectly round stones in the world. Then we can neatly unify the data: there are n perfectly round stones, where the kth is placed by me at distances f(k) from the Washington Monument.

Suppose, further, that as it happens there is another mathematical function, g, with the properties that (a) g is simple, and in fact significantly simpler than f, (b) g(k)=f(k) for k from 1 to n, and (c) g(n+1) is not equal to f(n+1). Then we don't want to say that the fact that the kth round stone is placed at distance g(k) for k between 1 and n explains the placement of the stones. For it is the function f, not the function g, that I am using. (If we want to go in for counterfactuals, we might say that if there were an (n+1)st stone, it would be at distance f(n+1), not g(n+1).) It is a mere coincidence that f equals g in this limited range.

The unificationist either can or cannot accept the claim that it is f, not g, that explanatorily unifies the stone placement. If she cannot, then unificationism is false (unless perhaps we have here a theistic unificationism on which God causes me to arrange things g-wise, even though I think of the arrangement as f-wise). Suppose, then, that she can. Presumably, this is because she notes that the f-based unification unifies not only the placement of the stones, but unifies my activity with it. Now, suppose a world w₂ where the stones have the same arrangement as in w₁, but where I don't exist and the stones causelessly appeared where they did. Then, in w₂, g, because it is significantly simpler than f, provides the better unification. Thus, in w₂, the fact that the distances of the stones obey g explains the stone placement.

But this is weird. For by moving from w₁ to w₂, we took away that which explained the placement of the stones—my activity governed by f—and, if theism is false, we put nothing in the place of the explanation. (If theism is true, then God had to have some reasons for arranging the stones as he did. More on that later.) The g-based arrangement of the stones in w₂ is also present in w₁, and is not explanatory there. It should not become explanatory in w₂ just because a cause was taken away. Therefore, unificationism says that in w₂ the stones' placement is explained by g, while the correct thing to do is that the one and only explanation that was present in w₁ is removed in w₂ if theism is false, and hence the stones' placement is unexplained. Thus, if theism is false, unificationism is false.

The theistic unificationist can say, along the lines of my previous post, that in w₂, the stones are placed by God, and not by me. And since God always acts on all the (unexcluded) good reasons that favor his action, and since g-arrangement exemplifies the good of orderliness (as g is a simple function) and hence provides a good reason for God to arrange the stones g-wise (and maybe a weaker reason for him to arrange the stones f-wise), we can say that God arranged the stones in w₂ in a g-wise way (and under that description, as well as to a lesser degree under the f-wise description), and thus the g-arrangement is, indeed, explanatory.