Nevertheless, if a network is going to correspond to space as it seems to exist in our universe, such phenomena must not be too important—and in the end there must to a good approximation be the kind of straightforward locality that exists for example in the simple causal network of page 518 .

reached—corresponding to the fact that statements must exist that cannot be proved either true or false from a given set of axioms. … And essentially the same turns out to be the case for pure predicate logic, in which one just formally adds "for all" and "there exists" constructs.

Yet at least in principle there should be examples that can be constructed based just on statements that no solutions exist to particular integer equations. … If in general one was able to tell whether such a solution exists then it would mean that one could always answer the question of

But in a sense such a thing cannot exist. … So even if we cannot find an abstract way to give evidence of purpose or intelligence, what about using the practical fact that both the sender and receiver of a signal exist in the same physical universe?

And assuming that one is able to look at a multiway system from the outside, one will immediately see that different paths exist corresponding to different histories. … Yet for time the belief still exists that we—and our universe—somehow have a unique history.

The idea that even a vacuum without particles will have a complicated and in some ways random form also exists in standard quantum field theory in traditional physics. … But the basic notion is that for each possible type of particle there is some kind of continuous field that exists throughout space—with the presence of a particle corresponding to a simple type of structure in this field.

In general one can think of technology as trying to take systems that exist in nature or elsewhere and harness them to achieve human purposes. … And indeed a vast quantity of current technology is in the end based on trying to set up our own systems to emulate features that we have noticed exist in ordinary biological or physical systems.

Undecidability [for 2D constraints] The general problem of whether an infinite pattern exists that satisfies a particular constraint is formally undecidable (see page 1139 ).

History [of recursive sequences] The idea of sequences in which later terms are deduced from earlier ones existed in antiquity, notably in the method of induction and in various approximation schemes (compare page 918 ). … A fairly clear idea of integer recurrence relations has existed since about the 1600s, but until very recently mainstream mathematics has almost never investigated them.

Conserved Quantities and Continuum Phenomena Reversibility is one general feature that appears to exist in the basic laws of physics.