And the point is that at each stage, we need think directly only about the scale of structures that we are currently handling—and not for example about all the pieces that make up these structures. … For what usually happens in such systems is that a change made even to a single cell will eventually spread to affect all other cells.

But it also means that it is not clear whether the axiom system actually describes only the objects one wants—or whether for example it also describes all sorts of other quite different objects. … If an axiom system is far from complete—so that a large fraction of statements cannot be proved true or false—then there will typically be many different kinds of objects that are easy to specify and all satisfy the constraints of the system but for which there are fairly obvious properties that differ.

But what the Principle of Computational Equivalence implies is that there are actually a vast range of very different kinds of rules that all lead to exactly the same computational capabilities—and so can all in principle be used as a basis for making computers.

Messages in DNA Science fiction has sometimes suggested that an extraterrestrial source of life might have left some form of message in the DNA sequences of all terrestrial organisms, but to get evidence of this would seem to require extensive other knowledge of the source.

Had I been starting the book now I would likely have authored all of it directly in Mathematica and Publicon . But a decade ago I made the decision to compose all the original source for the book in FrameMaker.

Representations [for symbolic expressions] Among the representations that can be used for expressions are: Typical transformation rules are non-local in all these representations. … If only a single symbol ever appears, then all that matters is the overall structure of an expression, which can be captured as in the main text by the sequence of opening and closing brackets, given by Flatten[Characters[ToString[expr]]/.{"["  1,"]"  0, " ℯ "  {}}]

To cover all possible proofs, however, requires going up to the ordinal ε 0 . … That this sequence terminates for all n is then provable in set theory, but not Peano arithmetic—and in effect Length[g[n]] must grow like  [ ε 0 ][n] .) … But normally these tend to be complicated and not at all typical of what arise in ordinary mathematics.

But if one allows oneself to generate the object in any way at all, this may still be easy, even if P !

Efficient markets In its strong form the so-called Efficient Market Hypothesis states that prices immediately adjust to reflect all possible information, so that knowing a particular piece of information can never be used to make a profit.

. • Different programs for doing all sorts of different things can be set up. • Any given program can be implemented in many ways. • Programs can behave in complicated and seemingly random ways—particularly when they are not working properly. • Debugging a program can be difficult. • It is often difficult to foresee what a program can do by reading its code. • The lower the level of representation of the code for a program the more difficult it tends to be to understand. • Some computational problems are easy to state but hard to solve. • Programs that simulate natural systems are among the most computationally expensive. • It is possible for people to create large programs—at least in pieces. • It is almost always possible to optimize a program more, but the optimized version may be more difficult to understand. • Shorter programs are sometimes more efficient, but optimizations often require many cases to be treated separately, making programs longer. • If programs are patched too much, they typically stop working at all.