Sorting networks Any list can be sorted using Fold[PairSort, list, pairs] by doing a fixed sequence of comparisons of pairs PairSort[a_, p : {_, _}] := Block[{t = a}, t 〚 p 〛 = Sort[t 〚 p 〛 ]; t] (Different comparisons often do not interfere and so can be done in parallel.) … The top two (both with 120 comparisons) have a repetitive structure and correspond to standard sorting algorithms: transposition sort and insertion sort. … (In general all lists will be sorted correctly if lists of just 0's and 1's are sorted correctly; allowing even just one of these 2 n cases to be wrong greatly reduces the number of comparisons needed.)

And indeed my guess is that most of them will actually in the end turn out to depend on all sorts of limits and idealizations in quantum theory—and will emerge just as simple approximations to much more complex underlying behavior. In its development since the early 1900s quantum theory has produced all sorts of elaborate results.

But what the Principle of Computational Equivalence suggests is that abstract descriptions will never ultimately distinguish us from all sorts of other systems in nature and elsewhere. And what this means is that in a sense there can be no abstract basic science of the human condition—only something that involves all sorts of specific details of humans and their history.

For while I believe that the basic science that I develop in this book provides a remarkably powerful new framework, coming up with an actual model requires all sorts of detailed work and analysis. Certainly it would be wonderful if one could just take the ideas and results in this book and somehow immediately use them to create models for all sorts of systems. … And instead—just as in any other framework—there will be no choice but first to learn all sorts of details of a system, and then to use judgement and creativity to see which of them are really essential to a model and which are not.

For any network that has a serious chance of representing actual space—even a supposedly empty part—will no doubt show all sorts of seemingly random activity. … Starting off with a network that is planar—so that it can be drawn flat on a page without any lines crossing—such rules can certainly give all sorts of complex and apparently random behavior.

For it implies that when it comes to computation—or intelligence—we are in the end no more sophisticated than all sorts of simple programs, and all sorts of systems in nature.

Indeed, most often they have in effect been engineered out of all sorts of components that are direct idealizations of various elaborate structures that exist in practical digital electronic computers. … And among other things this means that universality can be expected to occur not only in many kinds of abstract systems but also in all sorts of systems in nature.

So in as much as intelligence is associated with the ability to do sophisticated computations it should in no way require billions of years of biological evolution to produce—and indeed we should see it all over the place, in all sorts of systems, whether biological or otherwise. … Certainly one can identify all sorts of specific features of human intelligence: the ability to understand language, to do mathematics, solve puzzles, and so on.

instructions one can get all sorts of complex behavior is similar to the phenomenon we have seen in cellular automata. … But in the next few chapters [ 3 , 4 , 5 , 6 ] we will see many more examples, both in cellular automata and in all sorts of other systems.

And indeed if one were to talk about how the cellular automaton seems to behave one might well say that it just decides to do this or that—thereby effectively attributing to it some sort of free will. … And it is in this separation, I believe, that the basic origin of the apparent freedom we see in all sorts of systems lies—whether those systems are abstract cellular automata or actual living brains.