Analogy and explanation

Analogies intended to help us understand something often break down. When that happens, the question becomes: just how much does the analogy actually explain? By 'explain', I mean that the analogy in some way helps me understand the inner machinery of the matter. But of course analogies breaks down, in which case an analogy only helps me understand part of the inner machinery of the matter.

But how exactly does this 'explaining' happen? How does an analogy A explain some B? In the simplest case, it would seem that analogical explanation occurs when some parts of an analogy A map onto some parts of the inner machinery of B (taking 'parts' in the broadest sense). For example, suppose A and B have parts in the following ways:

A = parts a, b, c, and d
B = parts a, b, j, and k

In this case, if we understand A, we will understand a and b in B but not j and k, since a and b match up for A and B but not the other parts.

So what's the point of using an analogy? If our analogy only explains a and b, why not just explain a and b, and forget the analogy altogether? Well, sometimes we cannot understand parts without the whole. More specifically, sometimes we cannot understand the relations between parts without the whole. It might be the case that I cannot understand how a is related to b individually, but I can understand a and b along with c and d as a whole, while I cannot understand a and b along with j and k as a whole. That is, explaining how a and b are related in A might be much easier than explaining how a and b are related in B. In this case, the analogy serves the purpose of explaining at least some of the parts of the inner machinery of B, or, more specifically, the analogy explains certain relations between parts of the inner machinery of B.

I mentioned that this seems to me the simplest case. It is the simplest case because I have been taking 'parts' to be atomic in the sense that they themselves don't include parts. Of course things can get more complicated if the parts themselves include their own parts. But this makes no difference. A only explains B in so far as parts map onto parts, no matter how far down the tree of parts we go.

For example, suppose A and B have parts like this:

A = parts a (= parts x and y), b, c, and d
B = parts a (= parts x and z), b, j, and k

Here, A's part a has parts x and y, while B's part a has parts x and z. In this case, the first a only maps onto the second a in virtue of its part x. We can subdivide parts into more parts over and over again, but our analogy will only explain the matter insofar as certain of its parts (and the relations between them) map onto the matter in question.

The same applies to the relations between the parts. As I said, I am taking 'parts' in the broadest sense, so any of these parts might be relations. For example, suppose that x and y are certain relations of a to b in A, while x and z are certain relations of a to b in B. In this case, our analogy A explains B in that A helps me to understand that in B, a is related to b by a relation x, but the analogy breaks down when it comes to the relations y and z.

I am assuming that the parts which 'map' are atomic. That is, one part either maps onto another part or it doesn't. A's part x is either the same as B's part x or it's not. If it's not the same, it doesn't map. If it is the same, it does map. My account turns on this claim. In order to compare two things, there must be something which is the same between them.

Thus far I have started from wholes and moved down to their parts to explain how similar the wholes are. But we can explain this picture the other way around. Begin with two atomic things. Either they are the same or they are not. If they are not the same, they are not comparable at all. If they are comparable, then good. Now, add more atomic things to the picture to form some wholes, and the same rules apply. The first whole is comparable to the second whole only in virtue of those parts which are comparable. One whole cannot be compared to another whole by parts which are not the same.

I must confess that I don't see any other way analogies could be explanatory. For simple things, either one thing is the same as the other thing, or it's not. For complexes, either some of the parts are the same or they are not. There must be something the same in order to compare one thing to another. At this point, I can't seem to see another way that analogy could 'explain' something. If none of the parts of an analogy map onto none of the parts of the analogue, then I wouldn't call it an analogy. I would just call it an alternative model.

This is, of course, just Scotus's theory of analogy in a slightly different form. The basic point is that there must be something in the analogy which maps onto the analogue for the analogy to hold. Without some mapping, the analogy is not at all 'like' the analogue.