But only very special kinds of systems are in fact ergodic, and even in such systems, the time necessary to visit a significant fraction of all possible states is astronomically long. … With the Second Law accepted as a general principle, there is confusion about why systems in nature have not all dissipated into complete randomness. And often the rather absurd claim is made that all the order we see in the universe must just be a fluctuation—leaving little explanatory power for principles such as the Second Law.

For no two clusters whose dangling connections all point inwards can ever overlap, at least so long as neither of these clusters themselves contain subclusters whose dangling connections similarly all point inwards.

Such schemas are taken to stand for all individual axioms that match the patterns and are well-formed formulas. … Note that as mentioned on page 1150 all the axioms given for basic logic should really be thought of as schemas.

The networks shown on the next page capture all effects like this. … Both rules start on step 1 from random initial conditions in which all sequences of black and white cells are allowed.

And indeed, there are a great many different kinds of systems that all exhibit randomness. … And in the third case, there is effectively no random input at all.

Growth of Plants and Animals Looking at all the elaborate forms of plants and animals one might at first assume that the underlying rules for their growth must be highly complex. … It is usually only at the tip of a stem that growth can occur, and much of the time all that ever happens is that the stem just gets progressively longer.

Indeed, often all that one knows is the sequence of prices at which trades are executed. … And as soon as the underlying rule for the cellular automaton is such that information will eventually propagate from one entity to all others—in effect a minimal version of an efficient market hypothesis—it is essentially inevitable that running totals of numbers of cells will exhibit significant randomness.

So what this means is that any pair of paths in the multiway system that diverge must be able to converge again within just one step—so that all the arrows in pictures like the ones below must lie on the edges of quadrilaterals. … All such multiway systems have the property that every pair of paths which diverge at a particular step can converge again on the following step.

Of course, if one goes to an extreme and looks, say, only at how big each picture is, then all three pictures have very short descriptions. … In this chapter I argue that essentially all common forms of perception and analysis correspond to rather simple programs.

But in most modern computer and communications systems there are processes equivalent to perception and analysis that happen all the time when data is compressed for more efficient storage or transmission. … This means, for example, that instead of having to list explicitly all the cells in a run of, say, 53 identical cells, one instead just gives the number "53".