whenever there is sufficient uniformity to give a stable structure to space one can still think of something like parallel slices at different angles as representing motion at different fixed speeds.

And the crucial point is that whenever the underlying system is causal invariant the exact same underlying rules will account for what one sees in slices at different angles. And what this means is that in effect the same rules will apply regardless of how fast one is going.

And the remarkable point is then that this is also what seems to happen in physics. For everyday experience—together with all sorts of detailed experiments—strongly support the idea that so long as there are no effects from acceleration or external forces, physical systems work exactly the same regardless of how fast they are moving.

At the outset it might not have seemed conceivable that any system which at some level just applies a fixed program to various underlying elements could successfully capture the phenomenon of motion. For certainly a system like a typical cellular automaton does not—since for example its effective rules for evolution at different angles will usually be quite different. But there are two crucial ideas that make motion work in the kinds of systems I am discussing here. First, that causal networks can represent everything that can be observed. And second, that with causal invariance different slices through a causal network can be produced by the same underlying rules.

Historically, the idea that physical processes should always be independent of overall motion goes back at least three hundred years. And from this idea one expects for example that light should always travel at its usual speed with respect to whatever emitted it. But what if one happens to be moving with respect to this emitter? Will the light then appear to be travelling at a different speed? In the case of sound it would. But what was discovered around the end of the 1800s is that in the case of light it does not. And it was essentially to explain this surprising fact that the special theory of relativity was developed.

In the past, however, there seemed to be no obvious underlying mechanism that could account for the validity of this basic theory. But now it turns out that the kinds of discrete causal network models that I have described almost inevitably end up being able to do this.

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From Stephen Wolfram: A New Kind of Science [citation]