I like symmetry, and it pleases me aesthetically that much (most? all?) of the universe is conceptually symmetrical. For example, addition and multiplication are commutative and associative:
x + y == y + x
x * y == y * x
x + (y + z) == (x + y) + z
x * (y * z) == (x * y) * z
It always bothered me that division and subtraction weren't commutative and associative, until I realized that neither of those operations is truly a fundamental mathematical concept; both are combinations of two other operations. Subtraction is addition with negation, and division is multiplication with inversion. Thus, conceptual symmetry is maintained.
Please understand that the symmetry I'm talking about is very high level. Addition is symmetric, and so is capitalism -- you put work into the system, you get benefits out. Socialism is so awkward and absurd to me because it attempts to break this natural symmetry by disconnecting work from reward, and it fails for just that reason. I hope my meaning of conceptual symmetry is clear from these examples, because I'm not sure I can define it more rigidly at this juncture.
Conceptual symmetry depends a great deal on how we humans connect and relate concepts together. If a concept does not balance symmetrically, then it is generally the case that the concept is not well-formed, and does not represent reality. SDB gives a perfect example of a malformed concept when he writes that:
There's the old saw about the irresistible force and the immovable object and what happens when the irresistible force is applied to the immovable object. (The question turns out to be nonsense. It's logically impossible for both to exist in the same universe, so it's logically impossible for them to ever meet. Therefore it makes no sense to discuss what would happen if they did.) In our universe it turns out that every force is irresistible and no object is immovable. Any object, no matter how massive, will respond to any force, no matter how small. The response may be miniscule, but it isn't zero.It's an interesting mental exercise to consider what would happen if an irresistible force met an immovable object, and the question may appear symmetrical on the surface. The fact that the question is actually nonsensical within our universe, however, demonstrates that the concept behind it is not actually symmetrical. Force and mass are entirely different concepts that cannot be symmetrically related by the "moves" operation.
(Here is an interesting mental exercise: what if there were immovable objects? It would require some sort of universal static friction. Such a universe would not have Newton's three laws of motion. Another, even more difficult exercise: imagine a world in which addition and multiplication were not commutative. (You can't do it.))