For whether one does calculations by hand, by mechanical calculator or by electronic computer, one always needs an explicit representation for numbers, typically in terms of a sequence of digits of a certain length.

And when one goes to higher orders progressively more account is taken of viscosity, but the chaos phenomenon becomes progressively weaker. … Even within the Lorenz equations, however, one can see evidence of intrinsic randomness generation, in which randomness is produced without any need for randomness in initial conditions.

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. … As discussed in the main text (see also page 1155 ) one can think of axioms as giving rules for transforming symbolic expressions—much like rules in Mathematica. … And one way to represent this process is just to have the pattern a_  a_ ∧ (b_ ∨ ¬ b_) and then to say that any actual rule that can be used must match this pattern.

The pictures below show one example.

Growth of Plants and Animals Looking at all the elaborate forms of plants and animals one might at first assume that the underlying rules for their growth must be highly complex.

And by making simple algebraic changes to the way that time enters a differential equation one can often arrange, as in the pictures below, that processes that would normally take an infinite time will actually always occur over only a finite time.

Looking at the progress of science over the course of history one might assume that it would only be a matter of time before everything would somehow be predicted by science.

Instructions that are shown as light gray boxes refer to the first register; those shown as dark gray boxes refer to the second one.

Note that particularly in computer languages higher redundancy is found if one takes account of grammatical structure.

Self-reproduction That one can for example make a mold that will produce copies of a shape has been known since antiquity (see note above ).