The Principle of Computational Equivalence has implications for many issues long discussed in the field of philosophy. Most important are probably those in epistemology (theory of knowledge). In the past, it has usually been assumed that if we could only build up in our minds an adequate model of the world, then we would immediately know whatever we want about the world. But the Principle of Computational Equivalence now implies that even given a model it may be irreducibly difficult to work out its consequences. In effect, computational irreducibility introduces a new kind of limit to knowledge. And it implies that one needs a criterion more sophisticated than immediate predictability to assess a scientific theory—since when computational irreducibility is present this will inevitably be limited. In the past, it has sometimes been assumed that truths that can be deduced purely by operations like those in logic must somehow always be trivial. But computational irreducibility implies that in general they are not. Indeed it implies that even once the basic laws are known there are still an endless series of questions that are worth investigating in science. It is often assumed that one cannot learn much about the world just by studying purely formal systems—and that one has to rely on empirical input. But the Principle of Computational Equivalence implies that at some level there are inevitably common features across both abstract and natural systems. In ontology (theory of being) the Principle of Computational Equivalence implies that special components are vastly less necessary than might have been thought. For it shows that all sorts of sophisticated characteristics can emerge from the very same kinds of simple components. (My discussion of fundamental physics in Chapter 9 also suggests that no separate entities beyond simple rules are needed to capture space, time or matter.) Arguments in several areas of philosophy involve in effect considering fundamentally different intelligences. But the Principle of Computational Equivalence implies that in fact above a certain threshold there is an ultimate equivalence between possible intelligences. In addition, the Principle of Computational Equivalence implies that all sorts of systems in nature and elsewhere will inevitably exhibit features that in the past have been considered unique to intelligence—and this has consequences for the mind-body problem, the question of free will, and recognition of other minds. It has often been thought that traditional logic—and to some extent mathematics—are somehow fundamentally special and provide in a sense unique foundations. But the Principle of Computational Equivalence implies that in fact there are a huge range of other formal systems, equivalent in their ultimate richness, but different in their details, and in the questions to which they naturally lead. In philosophy of science the Principle of Computational Equivalence forces a new methodology based on formal experiments—that is ultimately the foundation for the whole new kind of science that I describe in this book.