The new kind of science in this book connects in all sorts of ways with mathematics and the existing sciences—and it can be used at an educational level to place some of the fundamental ideas in these areas in a clearer context.

Even though it is not inevitable from lattice symmetry, one might think that if there is some kind of effective randomness in the underlying rules then sufficiently large patterns would still often show some sort of average isotropy.

The idea that space might be defined by some sort of causal network of discrete elementary quantum events arose in various forms in work by Carl von Weizsäcker (ur-theory), John Wheeler (pregeometry), David Finkelstein (spacetime code), David Bohm (topochronology) and Roger Penrose (spin networks; see page 1055 ).

But in giving the specific axiom systems that have been used in traditional mathematics one needs to take account of all sorts of fairly complicated details.

Note however that this result is extremely specific to looking only at what is considered output from the system, and that inside the system there are all sorts of components that are definitely universal.

= {}, AllNet[k], q = ISets[b = Map[Table[ Position[d, NetStep[net, #, a]] 〚 1, 1 〛 , {a, 0, k - 1}]&, d]]; DeleteCases[MapIndexed[#2 〚 2 〛 - 1  #1 &, Rest[ Map[Position[q, #] 〚 1, 1 〛 &, Transpose[Map[Part[#, Map[ First, q]]&, Transpose[b]]], {2}]] - 1, {2}], _  0, {2}]]] DSets[net_, k_:2] := FixedPoint[Union[Flatten[Map[Table[NetStep[net, #, a], {a, 0, k - 1}]&, #], 1]]&, {Range[Length[net]]}] ISets[list_] := FixedPoint[Function[g, Flatten[Map[ Map[Last, Split[Sort[Part[Transpose[{Map[Position[g, #] 〚 1, 1 〛 &, list, {2}], Range[Length[list]]}], #]], First[#1]  First[#2]&], {2}]&, g], 1]], {{1}, Range[2, Length[list]]}] If net has q nodes, then in general MinNet[net] can have as many as 2 q -1 nodes.

All these parts seem to depend almost completely on detailed common conventions—and I suspect that without all sorts of human context their meaning would be essentially impossible to recognize.

Even in antiquity it was nevertheless recognized that all sorts of aspects of language are purely matters of convention, so that shared conventions are necessary for verbal communication to be possible.

In the 1700s and 1800s all sorts of celestial mechanics was done on the basis of this—with occasional observational anomalies being resolved for example by the discovery of new planets.

In recent years there has started to be increasing use of the language component of Mathematica for all sorts of applications outside the area of technical computing where Mathematica as a whole has traditionally been most widely used.