In his book The Beginning of Infinity (BoI), physicist David Deutsch says explanations are "[s]tatement[s] about what is there, what it does, and how and why" (chapter 1 glossary). Elsewhere in the same chapter, he says explanations are "assertions about what is out there and how it behaves". He also writes that "creating [explanatory] knowledge means understanding better what is really there, and how it really behaves and why [...]". While these quotes provide a good defense of realism, I believe they are insufficient when it comes to explaining what explanations are.

First of all, we can explain things that aren't "out there". For example, we can explain fiction, which doesn't exist "out there" by definition. When you explain the plot of a movie to someone, that usually involves more than what you see on screen.

At first I thought maybe Deutsch's explanation of explanations could be limited to science: as something about the physical world only. After all, anything to be explained about the physical world really exists "out there" somewhere – or so I thought. For example, you can explain how solar systems work. But then I realized that this approach isn't satisfactory either: sometimes our explanations, even those of the physical world, involve things that don't exist but could, or that don't happen but could: counterfactuals. I'm no expert on Deutsch's Constructor Theory but I believe counterfactuals play an important role in it, and it is a theory of physics.

In addition, consider what happens to an explanation once we learn it is false. We do not suddenly stop considering it an explanation. For example, Einstein's theory of general relativity superseded Newton's theory of gravity and negated Newton's force of gravity. But Newton's theory didn't suddenly stop being an explanation even though it couldn't possibly be about something "out there" anymore since what it proposed existed "out there" – a force of gravity – does not, to the best of our knowledge, exist. His theory is a false explanation, but an explanation nonetheless. As fallibilists, we expect even our best theories to contain errors, but we still consider them explanations.

Consider the conflict between quantum theory and general relativity: from what I understand, we know that at least one of them must be false because they conflict. Maybe both are false, but let's say it's only one of them. Then it does not, in fact, describe what's "out there". At best it's only partly right about that. But that doesn't stop us from considering both theories explanations – excellent explanations, I'm told – and we don't hold their status as explanations as something tentative or uncertain, to be changed when we change our minds about their truth status, which we do hold tentatively. If we are to take Deutsch's concept of explanation seriously, we need to consider at least one of these two theories to not be an explanation at all – but which one?

Then there's innovation. Any explanation that allows you to do or create something new can't possibly be about something "out there" and how it behaves: the new thing doesn't exist yet. Our very ability to create depends on this fact. The inventor of the airplane first understood (i.e., explained to himself) how it would work, and then built it. Deutsch writes (BoI chapter 1): "For millennia people dreamed about flying, but they experienced only falling. Then they discovered good explanatory theories about flying, and then they flew – in that order." Initially, there were no airplanes, and to build them, people first had to explain how they would work. Airplanes exist today because that explanation came first. It had to come first.

There are also historical explanations. They're about things that happened in the past but do not necessarily happen anymore. For example, you can explain the causes of World War 1 and its course – and just like explaining the plot of a movie, this usually involves more than just a list of what happened. It involves decisions of key actors, why they made those decisions, previous events and their influence on the war, and so on. But that war is over now. It doesn't exist "out there" anymore.

Analyzing Deutsch's quotes in the first paragraph above, to be an explanation, a statement must answer the following questions:

• What is "out there"?
• What does it do?
• How does it do that?
• Why does it do that?

These are Deutsch's requirements. Now, consider the following modification of Bertrand Russell's teapot. (He proposed his teapot for different epistemological reasons, none of which interest me here.) Let's say there's a teapot orbiting the sun in a roughly elliptical orbit because of gravity. This checks all of Deutsch's boxes. What is "out there"? A teapot. What does it do? It orbits the sun. How? In a roughly elliptical orbit. Why? Because of gravity. We have checked all the boxes, but is this statement really an explanation? To be sure, it contains an explanation – gravity as the cause of the teapot's orbit – but that's only part of the entire statement. The statement as a whole is not an explanation. Why not? Because it doesn't solve a problem.

An explanation is a statement about how to solve a problem. Sometimes such a statement is about what is really "out there"; sometimes it involves counterfactuals, fiction, or creations. This explanation of explanations covers all the use cases I mentioned above, and it itself solves the problem of answering what an explanation is.

Also consider this quote from Karl Popper:

The problem of explanation itself. It has often been said that scientific explanation is reduction of the unknown to the known. If pure science is meant, nothing could be further from the truth. It can be said without paradox that scientific explanation is, on the contrary, the reduction of the known to the unknown. In pure science, as opposed to an applied science which takes pure science as 'given' or 'known', explanation is always the logical reduction of hypotheses to others which are of a higher level of universality; of 'known' facts and 'known' theories to assumptions of which we know very little as yet, and which have still to be tested. The analysis of degrees of explanatory power, and of the relationship between genuine and sham explanation and between explanation and prediction, are examples of problems which are of great interest in this context.

And:

In seeking pure knowledge our aim is, quite simply, to understand, to answer how-questions and why-questions. These are questions which are answered by giving an explanation. Thus all problems of pure knowledge are problems of explanation.

I think Deutsch's conception of explanation breaks significantly and unnecesarily with Popper's. In addition, Deutsch limits it to the behavior of things. But one can instead ask what something is like and why. And it doesn't have to be about a thing at all: we can ask why someone did something and why; we can ask about abstract 'things', like why explanations work the way they do and how. So Popper's phrasing is more general and better. But I also think it isn't quite good enough – again, we can 'explain' how the teapot moves and why, but that doesn't mean the teapot itself explains anything. Phrasing it in terms of statements that solve problems seems best: if some statement doesn't (help) solve whatever problem you're working on, you might not consider that an explanation at all, even if it answers how and why questions in form.

On a more technical level, I believe explanations are functions in the sense of the lambda calculus. I say more about that in my book. Once you view explanations in this way, it's easier to make them explicit and analyze their structure. It also becomes really easy to distinguish "between genuine and sham explanation and between explanation and prediction […]". Programmers do so routinely without being aware of this equivalence. On the flip side, it's more difficult to understand the structure of explanations without knowing how to code.

Thanks to Aaron Stupple for reading a draft of this post.