1 Introduction

Quantum mechanics and general relativity—both introduced more than a century ago—have delivered many impressive successes in physics. But so far they have not allowed the formulation of a complete, fundamental theory of our universe, and at this point it seems worthwhile to try exploring other foundations from which space, time, general relativity, quantum mechanics and all the other known features of physics could emerge.

The purpose here is to introduce a class of models that could be relevant. The models are set up to be as minimal and structureless as possible, but despite the simplicity of their construction, they can nevertheless exhibit great complexity and structure in their behavior. Even independent of their possible relevance to fundamental physics, the models appear to be of significant interest in their own right, not least as sources of examples amenable to rich analysis by modern methods in mathematics and mathematical physics.

But what is potentially significant for physics is that with exceptionally little input, the models already seem able to reproduce some important and sophisticated features of known fundamental physics—and give suggestive indications of being able to reproduce much more.

Our approach here is to carry out a fairly extensive empirical investigation of the models, then to use the results of this to make connections with known mathematical and other features of physics. We do not know a priori whether any model that we would recognize as simple can completely describe the operation of our universe—although the very existence of physical laws does seem to indicate some simplicity. But it is basically inevitable that if a simple model exists, then almost nothing about the universe as we normally perceive it—including notions like space and time—will fit recognizably into the model.

And given this, the approach we take is to consider models that are as minimal and structureless as possible, so that in effect there is the greatest opportunity for the phenomenon of emergence to operate. The models introduced here have their origins in network-based models studied in the 1990s for [1], but the present models are more minimal and structureless. They can be thought of as abstracted versions of a surprisingly wide range of types of mathematical and computational systems, including combinatorial, functional, categorical, algebraic and axiomatic ones.

In what follows, sections 2 through 7 describe features of our models, without specific reference to physics. Section 8 discusses how the results of the preceding sections can potentially be used to understand known fundamental features of physics.

An informal introduction to the ideas described here is given in [2].