This post won't be everyone's cup of tea.
Each year, the online salon "Edge.org" asks the smartest people it can find a different Big Question. Earlier years' questions have included "What have you changed your mind about," "What are you optimistic about," and "What do you believe is true even though you cannot prove it?" This year's question asked them to name their "favorite deep, elegant or beautiful explanation." The responses are profound, trivial, thoughtful, silly and incomprehensible. Most are worth mulling over.
One person chose the Scientific Method, calling it "an explanation for explanations." I like that. A few chose Einstein’s explanation of gravity as the curvature of spacetime. More biologically inclined respondents picked the DNA double-helix. Other answers included behavioral economics, sexual conflict theory, "Like Attracts Like," Pascal's Wager, plate tectonics, the germ theory of disease, or how the experience of time changes when one stops wearing a wristwatch.
I'm not an especially deep thinker, but I did major in Physics and try to keep up with the popular scientific literature, and in that context I've encountered a few ideas that brought me up short and made me say, "Whoa." (Told you I'm not deep.)
For example, I'm amazed how often trigonometric functions like sine and cosine appear in unexpected places. It's not surprising when they crop up in systems of a cyclic nature; for example, the motion of any spot on a rolling wheel describes a sinusoidal curve. You'd expect to see them when calculating a planet's orbit or electron's path. But they also appear in weird places that have nothing to do with circles or angles, hinting that they are deeply woven into the fabric of the universe.
In any event, why should math--an internally consistent system of analysis and calculation invented by humans--describe reality at all? Yet it does, to as fine a point as we can measure. Moreso, it sometimes seems that no result from mathematics is so wacky or abstract that some scientist can't find an unimaginably small, large, fast, dense, early or late chunk of the universe that it describes perfectly.
I'm amazed by Euler's Identity: e^iπ + 1 = 0. In this equation, "e" is Euler's Number, (1 + 1/n)^n as n approaches infinity, equal to 2.71828... ; π is the ratio of a circle's circumference to its radius equal to 3.14159... ; and both are transcendental numbers that never end. They have nothing in common. Yet if you take "e" and raise it to the power of imaginary π (that's what the "i" indicates) then add 1, you get 0. To my ear, that's like saying "if you stack crushed glass and Pop-Tarts on top of a Stephen King novel, you get fried chicken." It makes no sense but there it is, again hinting at a deeper truth I can't begin to comprehend (not least of which is the notion that imaginary numbers--multiples of the square root of -1, which makes no sense in real life--actually mean something).
I think Newton's insight that Force = Mass x Acceleration (F = ma) is THE pinnacle of human intellectual achievement, real Promethean napalm. It's not intuitive or obvious, and yet the most profound insights and results flow from it. First, it implies that objects only feel a force when they accelerate or decelerate (which includes changing direction, and was stated more colloquially by Newton as "an object in motion tends to stay in motion, an object at rest tends to stay at rest"). To put it another way, it's not jumping off the cliff that kills you, it's the sudden stop at the bottom. F = ma makes the tides rise and the planets spin. It took me a couple of years to really grasp the profundity of F = ma and I can't recapture that learning in a brief blog post. Suffice it to say that F = ma is nearly all you need to know to shoot a rocket into space and land a man on the Moon (although it wouldn't let you build a GPS satellite network, which requires the refinement of relativity). Studying that equation changed the way I regard the world.
Also deep, elegant or beautiful: the idea that a profound insight into the universe can be expressed at all, let alone economically with only four or five symbols. F = ma. E = mc^2. Maxwell's Equations, which in the late 19th Century explained the entire field of electromagnetism (and, incidentally, could be solved to prove E = mc^2 although no one realized it until Einstein achieved it a different way decades later), are four expressions that can be written with 30 or so symbols. One of the great quests in science is for a Grand Unified Theory (GUT) that would unite General Relativity and Quantum Mechanics, which govern different realms (the very big/fast and very small, respectively) and are currently incompatible. No one quite knows what a GUT will look like, but I remember one researcher saying he'd know it when he saw it because it'd fit on a t-shirt.
That seems right, doesn't it? That when we eventually find the universe's owners' manual, it'll comprise one page with a single line of type (or, per Douglas Adams's Hitchhiker's Guide to the Galaxy, the number "42")? There's no reason our aesthetic sense of elegance should coincide with how the universe actually operates--the universe doesn't owe us simple answers--but so far that's how it seems to work. That's deep.
However, none of these are really "explanations" in the sense the question asks. They may be helpful tools or Mysterious Mysteries of the Unknown, but they don't really explain anything. So for my favorite deep, elegant or beautiful explanation I'm going outside Physics to Darwin's Theory of Natural Selection. The reason it's my favorite is that when I first learned about Evolution it seemed so self-evidently obvious to me that I couldn't believe it hadn't always been common knowledge. No math required.
Lifeforms with characterstics that offer advantages in a particular environment live to reproduce more successfully than those without them. Different environments reward different characteristics. That's it. Repeat for a couple billion years and it explains everything from the oldest fossil strata to the latest flu vaccine (we wouldn't need a new flu shot every year if the bugs didn't keep evolving). That's just common sense! Why didn't everybody know that all along? We needed Darwin to tell us that? But of course we did.
(I had the same reaction after finishing Jared Diamond's Guns, Germs and Steel, which attributed the developmental differences among cultures to their environments, including whether they happened to have handy domesticatable plants and animals (the Old World had many, the New World few) or mountain ranges that ran west-to-east (Old World) or north-to-south (New World). I understand Diamond's work has its critics and I'm not qualified to judge, but for me the broad sweep of his thesis was so obvious I couldn't believe I hadn't known it all along. Yet I hadn't even considered it until I read his book.)
Physicist Richard Feynman once posed the question, "If all of scientific knowledge were to be destroyed, and only one sentence passed on to the next generation of creatures, which statement would contain the most information in the fewest words?" Feynman's answer: "All things are made of atoms." That's a pretty deep, elegant and beautiful answer.