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.

there are all sorts of features in the behavior of these rules that could in principle represent a possible purpose. … For certainly our astronomical observations have revealed all sorts of phenomena for which we do not yet have any very satisfactory explanations.

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.

Capabilities [of sequential substitution systems] Even with the single rule {s[1, 0]  s[0, 1]} , a sequential substitution system can sort its initial conditions so that all 0's occur before all 1's.

With this setup, each step then corresponds to LifeStep[list_] := With[{p=Flatten[Array[List, {3, 3}, -1], 1]}, With[{u = Split[Sort[Flatten[Outer[Plus, list, p, 1], 1]]]}, Union[Cases[u, {x_, _, _}  x], Intersection[Cases[u, {x_, _, _, _}  x], list]]]] (A still more efficient implementation is based on finding runs of length 3 and 4 in Sort[u] .)

And in fact, in general the simple cellular automaton shown below seems remarkably successful at reproducing all sorts of obvious features of snowflake growth.

Huffman coding From a list p of probabilities for blocks, the list of codewords can be generated using Map[Drop[Last[#], -1] &, Sort[ Flatten[MapIndexed[Rule, FixedPoint[Replace[Sort[#], {{p0_, i0_}, {p1_, i1_}, pi___}  {{p0 + p1, {i0, i1}}, pi}] & , MapIndexed[List, p]] 〚 1, 2 〛 , {-1}]]]] -1 Given the list of codewords c , the sequence of blocks that occur in encoded data d can be uniquely reconstructed using First[{{}, d} //.

The results of this book indicate however that even programs that are very small—and thus have low algorithmic complexity—can nevertheless perform all sorts of complex computations.

And if one does this, one immediately gets all sorts of fairly complicated patterns that are often not just purely nested—as illustrated in the pictures on the next page .

Properties [of example multiway systems] The second rule shown has the property that black elements always appear before white, so that strings can be specified just by the number of elements of each color that they contain—making the rule one of the sorted type discussed on page 937 , based on the difference vector {{2,-1}, {-1,3}, {-4,-1}} .