And it could be that even the structures shown above can be combined to produce all the richness that is needed for universality.

In most kinds of mathematics there are all sorts of additional details, particularly about how to determine which parts of one or more previous expressions actually get used at each step in a proof.

If continued forever the network would eventually include all possible true statements (tautologies) of logic (see also page 818 ).

Orthogonality is then the property that a 〚 i 〛 . a 〚 j 〛  0 for all i ≠ j .

Ever since the 1960s all sorts of schemes for nonlinear processing of images have been discussed and used in particular communities.

But there is absolutely no reason to think that the specific concepts that have arisen so far in the history of mathematics should cover all of science, and indeed in this book I give extensive evidence that they do not.

Things appear somewhat simpler with boiling points, and as noticed by Harry Wiener in 1947 (and increasingly discussed since the 1970s) these tend to be well fit as being linearly proportional to the so-called topological index given by the sum of the smallest numbers of connections visited in getting between all pairs of carbon atoms in an alkane molecule.

But the Principle of Computational Equivalence suggests at some level a remarkable uniformity among systems, that allows all sorts of general scientific statements to be made without dependence on context.

With a list s of possible symbols, c[s, n] gives all possible expressions with LeafCount[expr]  n : c[s_, 1] = s; c[s_, n_] := Flatten[ Table[Outer[#1[#2] &, c[s, n - m], c[s, m]], {m, n - 1}]] There are a total of Binomial[2n - 2, n - 1] Length[s] n /n such expressions.

Recurrence relations The rules for the sequences given here all have the form of linear recurrence relations.