Previously I argued that the defining mark of an explanation is that it covers multiple situations. But there is a further distinction we can make, between valuable and value-less explanations. For example, consider the following explanation x ξ y = √(x + 7) * y. Certainly this explains the ξ operator, since the formula given allows us to determine the result of applying it to every pair of numbers. However, the explanation isn’t valuable, since we have no use for the ξ operator. Usefulness is exactly what makes an explanation valuable, I think. And thus it is natural to say that a valuable explanation is one that play a role in good theory.
Good theories, as established previously, are theories that have content, and thus made falsifiable predictions. Which is fine when we are considering science (the yardstick we naturally judge most talk about theories by), but it seems to invalidate mathematics. After all, mathematics, strictly speaking, doesn’t make falsifiable predictions. And thus no mathematical explanation could be valuable. In this case I think we should bite the bullet and acknowledge that math by itself really doesn’t provide valuable explanations, because math by itself is totally unconnected to reality. Fortunately for mathematicians, math doesn’t exist by itself. For example, arithmetic is naturally applied to the counting of certain kinds of objects. And arithmetic plus a theory that matches numbers with some way of counting things is a good theory because it does make falsifiable predictions about the world. Given that arithmetic is part of such a theory explanations of the ideas of arithmetic are thus valuable explanations. Of course not all mathematics can be tied directly to some kind of theory about the physical world. So what can we say about the rest of it? Well many branches of mathematics look like they could be used in such explanations, or are simply waiting to be used, and thus their explanations are probably valuable because they can be incorporated into various theories that are good (in the sense of having content), but which at the same time might be false (they make bad predictions). Of course there are branches of mathematics that are extremely abstract, which seem to have no possibility whatsoever of being part of a theory about the world. However, such abstract math is usually about math itself. Because much of math provides valuable explanations by being incorporated, at least possibly, into good theories then math about math is thus providing valuable explanations, by being usefully about something that is useful, and thus being useful itself, transitively. So most of the math that mathematicians actually engage in does result in valuable explanations. But of course not all math necessarily does, mathematicians just restrict themselves to problems that have some usefulness. As shown via the example of the ξ operator it is perfectly possible to create arbitrary and thus useless mathematics.
So let us now turn our attention to philosophy. Clearly philosophy does attempt to provide explanations. For example, attempting to define what justice is is an attempt to provide a kind of explanation, because attempting to define justice is to attempt to create a rule by which we would be able to tell which situations are and aren’t just. Thus it is a kind of explanation because it says whether a property (being just) holds for multiple situations. But are such explanations valuable, or are they empty word play? It comes down to whether the philosophical theory it is part of is a good theory. Now, done correctly, many branches of philosophy automatically provide good theories. For example, epistemology deals with what knowledge is and what justification is, and thus it makes predictions about what kind of reasoning is most likely to lead to true beliefs. Ethics, by being normative, is also a predictive theory, because it says that by acting in some way described as ethical the outcome will be best (or at least desirable) in some way. This is of course falsifiable because we can check whether the recommendations actually work. But not all areas of philosophy deal with normative questions. Metaphysics notably deals more with defining things, like what a property is, than in making predictions. And even in ethics we occasionally address questions like “what makes an outcome desirable?” that seem more definitional and less predictive. But such philosophy is like mathematics. By itself it doesn’t provide valuable explanations. However, such philosophy does not actually stand by itself, but is incorporated into normative philosophy, just as mathematics is incorporated into scientific theories. For example, philosophy about what properties are contributes to how we describe situations via predicate logic. And such decisions have implications as to what further facts follow by logical deduction, and thus how truth works in such situations. Clearly what facts are true is a claim about the world (given the assumption that saying something is true is saying that the world actually is a certain way). And thus philosophy as abstract as debates about the nature of universals can be seen to play a role in good (predictive) theories, and thus be a valuable explanation.
As a final example we can turn this understanding of valuable explanations on itself. Is the definition of what a valuable explanation is here itself a valuable explanation? I would hold that it is. Firstly it is an explanation because it classifies every explanation (multiple situations) as good or bad. And although it doesn’t directly make any predictions it is part of a good theory that does. By labeling an explanation as valuable or value-less we are obviously making a recommendation as to which explanations we should care about, think about, and reason with, and which explanations we can just ignore. Or, in other words, it can be incorporated into an epistemological position that does make predictions about what process of theorizing is best. Of course I haven’t actually gone into detail as to this connection, or why such theorizing really is best; I assume the connection is relatively obvious, and since this isn’t for a journal there is no need to beat you over the head with more tedious detail than I already have.