Two descriptions of bishops on a chessboard

[jpivarski]

Two ways to describe an empty, toroidal (so that I don't have to talk about the edges) chessboard with a single bishop, who starts on a white square and follows the normal chess rule for bishops.

1. Chess squares come in two colors, red and white, and they each have 8 neighbors: N, NE, E, SE, S, SW, W, NW. A white square has red N, E, W, and S neighbors; all other neighbors are white. Vice-versa for red squares. A bishop can move in the NE, SE, SW, and NW directions; the other directions are forbidden. On our board, the bishop starts on a white square. Through some mathematical proof, we can conclude that the bishop will never visit a red square.
2. Chess squares are all alike. Each square has 4 neighbors, NE, SE, SW, NW. Our bishop starts on an arbitrary square. He can go anywhere.

The first description lays down a background setting and introduces rules that constrain possibilities. With those rules, we could imagine s different world in which the bishop starts on a red square and can never visit white squares.

The second description uses the rules to compress the description of the world so that it has fewer entities. It is like using compression to replace 8-bit booleans (in which 0b00000000 means false and all 255 other possibilities mean true) with 1-bit booleans (which don't have the same kind of wiggle room). It's also like replacing an electromagnetic theory in which the field values have Euclidean metric and a gauge freedom rule with an electromagnetic theory living on a $U(1)$ manifold. Or like replacing very specific infinite series of epicyles with elliptical orbits.

In the second description, the "law of physics" ceases to be a law and it becomes an identity: "whereas you might think you have two different things, you only have one thing" (e.g. time has the same units as space). In this point of view, casting it as a law (the first description) is considered a mistaken perception, in which the same thing from two points of view look like different things.

Another distinction between the two descriptions is that the first describes a cause: "because the bishop started on a white square and has to follow certain moves, it will never reach a red square." In the second description, there is no rule and there is no cause or effect.

[jpivarski]

Or for that matter (responding to my midnight-self), take this in the opposite direction: consider our universe as described by the Standard Model and add one new particle type that doesn't interact with anything. It has no electric charge, no weak charge, no strong charge, and not even any interactions with hidden valley sectors that are visible to us as dark matter or dark energy. This new particle type is superdark matter. (It also has to be "gravitationally neutral," which can't be done with general relativity as-is: massless particles must have kinetic energy, so this requires some modification of gravity to allow the new particle to have zero gravitational interactions.)

This superdark matter has and can have no impact on any past, present, or future sense perceptions, through any kind of scientific instrument, even transitively through other particles that do.

Does superdark matter exist? The evidence to say that it does can never be stronger or weaker than the evidence to say that it does not. You can take it or leave it at will. William of Ockham doesn't like it, but parsimony is not a requirement for nature.

The difference between the Standard Model + gravity and the Standard Model + gravity + superdark matter is like the difference between the chessboard in description 1 and the chessboard in description 2, except for the order in which the simpler and more complex descriptions were presented. Adding superdark matter feels like madness, but removing the red squares because the bishop can never "sense" them doesn't feel obligatory.

Maybe it's for reasons like this that some people feel that Ockham's razor is obligatory, that we must shrink-wrap our descriptions of the universe, but it's not always clear (as it is in these two examples) what "simpler" means and what "more complex" means. Kolmogorov complexity has to be expressed in some language, and some things that are simple in one language are complex in another and vice-versa.

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However, this is tangential from my point last night. This is about simple and complex descriptions of the same physical situation. What I was getting at last night is that one of these two descriptions is described in terms of a cause—the bishop will never reach a red square because of the way bishops move and the topology of the squares of the chessboard[^1]—and the other description has no cause and no effect—the bishop can move to any square that exists, and only white squares exist.

[^1]: I forgot to say in the original formulation that NE for one square is matched to SW of its NE neighbor, and so on, which is necessary for the theorem that the bishop can't move to a square of a different color.