But unlike in the space networks that we discussed two sections ago , the connections in the causal networks we consider here always go only one way: each connection corresponds to a causal relationship in which one event leads to another, but not the other way around. … For the progress of time can be defined by saying that only those events that occur later in time than a particular event can be affected by that event.

But the very complexity of this behavior was usually taken to show that it could not be relevant for real mathematical work—and could only be of recreational interest. … But as I will discuss on page 149 , the traditional mathematics that was used to analyze such systems ended up concentrating only on certain specific features, and completely missed the main phenomenon discovered in this chapter .

But if we can observe only the causal network for the universe then our information about space and time must inevitably be deduced indirectly from looking at slices of causal networks. And indeed only some causal networks even yield a reasonable notion of space at all.

So one possibility is to define randomness so that something is considered random only if no short description whatsoever exists of it. … So to say that something is random only if no short description whatsoever exists of it turns out to be a highly restrictive definition of randomness.

For the model does not say that such sequences are impossible—it merely says that they should occur only about 1% of the time. … For in such a case one can always just make up an extreme model in which only one very long block is allowed, with this block being precisely the sequence that is observed in the data.

all its development over the past few thousand years mathematics itself has continued to concentrate only on rather specific types of abstract systems—most often ones somehow derived from arithmetic or geometry. … The threshold for such universality has however generally been assumed to be high, and to be reached only by elaborate and special systems like typical electronic computers.

So what has mostly been done is to try to construct systems that perform only rather specific tasks. … So what this means is that most likely it will at some level be much easier to reproduce general human-like thinking than to set up some special version of human-like thinking only for specific tasks.

But my uniform experience has been that if one wants to specify processes of any significant complexity in a fashion that can reasonably be understood then the only realistic way to do this is to use a language—like Mathematica—that has essentially an ordinary sequential grammatical structure. … Or perhaps it is a consequence of our apparent ability to pay attention only to one thing at a time.

The only new structure not already seen in elementary rules is the one in code 420—but this occurs only quite rarely.

And in a class 4 cellular automaton such as rule 110 one can readily shortcut the process of evolution for at least a limited number of steps in places where there happen to be only a few well-separated localized structures present. … But what makes there be computational reducibility is when only a short computation is needed to find from the compressed description any feature of the actual behavior.