For if the rule for our universe is at all simple, then I expect that to introduce a new phenomenon, however esoteric, will involve modifying some basic part of the rule, which will also affect even common everyday phenomena. … Certainly among all possible rules of a particular kind only a limited number can ever be considered simple, and these rules are by definition somehow special. … For among all possible rules, the overwhelming majority will not be simple; in fact, they will instead tend to be almost infinitely complex.

From the tradition of the existing sciences one might expect that its answer would depend on all sorts of details, and be quite different for different types of physical, biological and other systems. … And indeed in the later parts of this book I will show that even remarkably simple programs seem to capture the essential mechanisms responsible for all sorts of important phenomena that in the past have always seemed far too complex to allow any simple explanation. … Indeed, I even have increasing evidence that thinking in terms of simple programs will make it possible to construct a single truly fundamental theory of physics, from which space, time, quantum mechanics and all the other known features of our universe will emerge.

And I suspect that particles in physics—with all their various properties and interactions—are just yet another example of this very general phenomenon. … In traditional particle physics this is explained by saying that many if not all particles are somehow just manifestations of the same underlying abstract object, related by a simple fixed group of symmetry operations. … But in addition to quantities like electric charge, another important intrinsic property of all particles is mass.

And indeed if one looks at all 4096 Turing machines with 2 states and 2 colors it turns out that the only rates of growth that one ever sees are linear, square and exponential. And of the six examples where exponential growth occurs, all of them are like example (f) above—so that there is another 2-state 2-color Turing machine that computes the same function, but without the maximum number of steps increasing at all with input length.

In fact, all one need do is to encode the axioms of set theory in something like the arithmetic equation system of page 786 . … For given almost any general property that one can pick out in axiom systems like those on pages 773 and 774 there typically seem to be all sorts of operator and multiway systems—often including some rather simple ones—that share the exact same property. So this leads to the conclusion that there is in a sense nothing fundamentally special about the particular axiom systems that have traditionally been used in mathematics—and that in fact there are all sorts of other axiom systems that could perfectly well be used as foundations for what are in effect new fields of mathematics—just as rich as the traditional ones, but without the historical connections.

Yet despite all this, we do not in our everyday experience typically have much difficulty telling living systems from non-living ones. But the reason for this is that all living systems on Earth share an immense number of detailed structural and chemical features—reflecting their long common history of biological evolution. … And I have little doubt that all sorts of such systems can be identified both terrestrially and extraterrestrially—and certainly require nothing like the elaborate history of life on Earth to produce.

But while traditional engineering has usually ended up finding ways to avoid searches for the limited kinds of systems it considers, the phenomenon of computational irreducibility makes it inevitable that if one considers all possible simple programs then finding particular forms of behavior can require doing searches that involve irreducibly large amounts of computational work. … So in practice a better approach will often be in effect just to do basic science—and much as I have done in this book to try to build up a body of abstract knowledge about how all sorts of simple programs behave. In chemistry for example one might start by studying the basic science of how all sorts of different substances behave.

I strongly suspect that all class 4 rules, like rule 110, will turn out to be universal.

Mathematical Constants The last few sections [ 2 , 3 , 4 ] have shown that one can set up all sorts of systems based on numbers in which great complexity can occur. … But it turns out that even though this definition is simple, the digit sequence of π is not simple at all.

The second set of pictures below show what happens, for example, with a rule in which each cell becomes black if just one or all four of its neighbors were black on the previous step, but otherwise stays the same color as it was before. … Steps in the evolution of a two-dimensional cellular automaton whose rule specifies that a particular cell should become black if exactly one or all four of its neighbors were black on the previous step, but should otherwise stay the same color.