Structural realism—in its metaphysical version, championed by the philosopher of science James Ladyman—is the deepest explanation I know, because it serves as a kind of meta-explanation, one that explains the nature of reality and the nature of scientific explanations.
The idea behind structural realism is pretty simple: the world isn't made of things, it's made of mathematical relationships, or structure. A mathematical structure is a set of isomorphic elements, each of which can be perfectly mapped onto the next. To give a trivial example, the numbers 25 and 52 share the same mathematical structure.
When the philosopher John Worrall first introduced structural realism (though he attributes it to physicist Henri Poincaré), he was trying to explain something puzzling: how was it possible that a scientific theory that would later turn out to be wrong could still manage to make accurate predictions? Take Newtonian gravity. Newton said that gravity was a force that masses exert on one another from a distance. That idea was overthrown by Einstein, who showed that gravity was the curvature of spacetime. Given how wrong Newton was about gravity, it seems almost miraculous that he was able to accurately predict the motions of the planets.
Thankfully, we don't have to resort to miracles. Newton may have gotten the physical interpretation of gravity wrong, but he got a piece of the math right. That's why, at weak masses and small velocities, Einstein's equations reduce to Newton's. The problem, Worrall pointed out, was that we mistook an interpretation of the theory for the theory itself. The fact is, in physics, theories are sets of equations, and nothing more. "Quantum field theory" is a group of mathematical structures. "Electrons" are little stories we tell ourselves.
These days, believing in the reality of objects—of physical things like particles, fields, forces, even spacetime geometries—can quickly lead to profound existential crises.
Quantum theory, for instance, strips particles of any sense of "thingness". One electron is not merely similar to another, all electrons are exactly the same. Electrons have no inherent identity—a fact that makes quantum statistics drastically different from the classical kind. Anyone who believes that an electron is a "thing" in its own right is bound to lose big in a quantum casino.
Meanwhile, all of nature's fundamental forces, including electromagnetism and the nuclear forces that operate deep in the cores of atoms, are described by gauge theory, which shows that forces aren't physical things in the world, but discrepancies in different descriptions of the world, in different observers' points of view. Gravity is a gauge force too, which means you can make it blink out of existence just by changing your frame of reference. In fact, that was Einstein's "happiest thought": a person in freefall can't feel their weight. It's often said that you can't disobey the law of gravity, but the truth is you can take it out with a simple coordinate change.
Recent advances in theoretical physics have only made the situation worse. The holographic principle tells us that our four-dimensional spacetime and everything in it is exactly equivalent to physics taking place on the two-dimensional boundary of the universe. Neither description is more "real" than the other—one can be perfectly mapped onto the other with no loss of information. When we try to believe that spacetime is really four-dimensional or really has a particular geometry, the holographic principle pulls the rug out from under us.
The physical nature of reality has been further eroded by M-theory, the theory that many physicists believe can unite general relativity and quantum mechanics. M-theory encompasses five versions of string theory (plus one non-stringy theory known as supergravity) all of which are related by mathematical maps called dualities. What looks like a strong interaction in one theory looks like a weak interaction in another. What look like eleven dimensions in one theory look like ten in another. Big can look like small, strings can look like particles. Virtually any object you can think of will be transformed into something totally different as you move from one theory to the next—and yet, somehow, all of the theories are equally true.
This reality crisis has grown so dire that Stephen Hawking has called for a kind of philosophical surrender, a white flag he terms "model-dependent realism", which basically says that while our theoretical models offer possible descriptions of the world, we'll simply never know the true reality that lies beneath. Perhaps there is no reality at all.
But structural realism offers a way out. An explanation. A reality. The only catch is that it's not made of physical objects. Then again, our theories never said it was. Electrons aren't real, but the mathematical structure of quantum field theory is. Gauge forces aren't real, but the symmetry groups that describe them are. The dimensions, geometries and even strings described by any given string theory aren't real—what's real are the mathematical maps that transform one string theory into another.
Of course, it's only human to want to interpret mathematical structure. There's a reason that "42" is hardly a satisfying answer to life, the universe and everything. We want to know what the world is really like, but we want it in a form that fits our intuitions. A form that means something. And for our narrative-driven brains, meaning comes in the form of stories, stories about things. I doubt we'll ever stop telling stories about how the universe works, and I, for one, am glad. We just have to remember not to mistake the stories for reality.
Structural realism forces us to radically revise the way we think about the universe. But it also provides a powerful explanation for some of the most mystifying aspects of physics. Without it, we'd have to give up on the notion that scientific theories can ever tell us how the world really is. And that, in my humble opinion, makes it a pretty beautiful explanation.