As I will discuss in the last section of this chapter , quantum mechanics tends to make one think of particles with higher momenta as being somehow progressively less spread out in space. … The Phenomenon of Gravity At an opposite extreme from elementary particles one can ask how the universe behaves on the largest possible scales. … And it is this that allows one to think of gravity as a general feature of space—rather than for example as some type of force that acts specifically on different objects.

Defining Complexity Much of what I have done in this book has been concerned in one way or another with phenomena associated with complexity. But just as one does not need a formal definition of life in order to study biology, so also it has not turned out to be necessary so far in this book to have a formal definition of complexity. … But if one chooses to look

And the reason that this happens so quickly when we look at an image is no doubt that the procedure for picking out such features is a very simple one that can readily be carried out in parallel by large numbers of separate cells in our eyes and brains. … Note that in a pattern generated by repeating one particular block, there will normally be other blocks that occur with other alignments. Page 215 shows patterns obtained in systems based on constraints in which one effectively requires that only certain blocks or sets of blocks occur.

One way to see that this must be true is to note that any particular computer system or computer language can always be set up by appropriate programming to emulate any other one. … For when one enters a command such as Log[15] , what actually happens is that the program which implements the Mathematica language interprets this command

And if such complication was indeed necessary in order to achieve universality, then one would not expect that universality would be common, for example, in the systems we see in nature. … With their simple and rather specific underlying structure one might think that cellular automata would never be capable of emulating a very wide range of other systems. … The main difference between a mobile automaton and a cellular automaton is that in a mobile automaton there is a special active cell that moves around from one step to the next, while in a cellular

One can always in principle find out how a particular system will behave just by running an experiment and watching what happens. But the great historical successes of theoretical science have typically revolved around finding mathematical formulas that instead directly allow one to predict the outcome. … One feature of many of the most important advances in science throughout history is that they show new ways in which we as humans are not special.

and especially a simple one—then it can be extremely difficult to determine whether or not the rule is universal. … For what one needs to show is that no possible scheme can be devised that will allow the system to emulate any other universal system. … But as we saw in Chapter 10 , in almost no other case do standard methods of perception and analysis allow one to make much progress at all.

Yet while this means that there is in effect less new information in each theorem that is proved, it turns out that in most areas of mathematics these theorems are usually the ones that are considered interesting. … Yet in a sense this is no different from what has happened, say, in physics, where the phenomena that have traditionally been studied are mostly just those ones that show enough computational reducibility to allow analysis by traditional methods of theoretical physics. But whereas in physics one has only to look at the natural world to see that other more complex phenomena exist, the usual approaches to mathematics provide almost no hint of anything analogous.

The pattern required to satisfy this constraint corresponds to a shifted version of the one generated by the evolution of the rule 30 elementary one-dimensional cellular automaton.

And in fact I expect that there are some significantly simpler ones. … If one looks at the 4096 Turing machines with 2 states and 2 colors it is fairly easy to see that their behavior is in all cases too simple to support universality.