Chapter 12: The Principle of Computational Equivalence

Section 4: The Validity of the Principle

Initial conditions [and continuity]

Traditional mathematics tends to assume that real numbers with absolutely any digit sequence can be set up. And if this were the case, then the digits of an initial condition could for example be the table for an oracle of the kind discussed on page 1126—and even a simple shift mapping could then yield output that is computationally more sophisticated than any standard discrete system. But just as in my discussion of chaos theory in Chapter 7, any reasonably complete theory must address how such an initial condition could have been constructed. And presumably the only way is to have another system that already violates the Principle of Computational Equivalence.

From Stephen Wolfram: A New Kind of Science [citation]