Introduction

The development of science, from ancient times to the present, has been a series of nearly unbroken steps where one concept after another has moved out of the shadows of doubt and uncertainty and into the light of accepted scientific fact.  The atomic hypothesis [1] , whether matter is made up of atoms, is only one of many atomic hypotheses.  So far every such question, discrete versus continuous, about a property of our world either remains undecided or it has been decided as discrete (atoms, electricity, light, angular momentum, etc.).  It is hard to imagine the proof that some property will never admit to a finite description, no matter how fine grained.  On the other hand, what is interesting is that so many concepts once thought of as continuous are now accepted as discrete.  Finite Nature assumes that that historical process will continue to a logical conclusion where, at the bottom, everything will turn out to be atomic or discrete, including space and time.

The prime implication of Finite Nature is that every volume of space‑time has a finite amount of information in it.  Every small region of space‑time (a cell) must be in one of a small number of states.  If the future state of that cell is always a function of the space‑time neighbors of the that cell, then we are lucky because there is a branch of mathematics, Automata Theory [2] , that deals with such systems.  Automata Theory is concerned with the behavior of all systems that evolve from one state (out of a finite number of states) to another, on the basis of a rule that takes into account: (1) the current state and (2) an input state.  In the case of physics the input state comes from the states of other cells in the space‑time neighborhood.  There are versions of Automata called Cellular Automata [3] (CA) that are just what we're discussing.

Once we understand the rule of some CA, we name the states.  If there are 3 states per cell, we can call them "1, 2 and 3" or "a, b and c"; it doesn't matter.  Given any rule, we can find other rules, with other numbers of states per cell, that evolve in a manner isomorphic to the original system.  Strangely enough, there is a wonderful chain of logic that proves that a system with just 2 states per cell, the Bank's [4] rule, with a 2-D neighborhood (4 neighbors; north, south, east and west), starting from an appropriate initial condition, can evolve in a way isomorphic to any other CA; no matter how many states per cell, no matter how high the dimensionality of the space, no matter how complex the rule, no matter how the neighborhood is defined!  Of course the Bank's rule may take more cells and more time to do the simulation, but it can do it exactly because it's universal.

Automata, Cellular Automata and Finite State Machines are all forms of computers.  A computer is universal if it can be programmed to simulate any other target computer.  The simulating computer must have enough memory for two things: (1) to represent the state of the target computer and (2) to hold a program that can operate on that state to mimic the behavior of the target computer.  Every ordinary computer is universal.  The pioneers of automata theory were interested in what could be done by machines with arbitrarily large memories, so a common but unnecessary interpretation of "universal computer" is a computer with an infinite memory.  We extend the idea to include computers with finite memory.  A Macintosh, can exactly simulate the behavior of the fastest super computer if it has only a little bit more disk memory than the super computer.  We pay a lot more money for a super computer not for what it can do, but for how fast it does it.

Given Finite Nature, what we have at the bottom is a Cellular Automaton of some kind.  The first thing to wonder about is "Is it Universal?"  The normal way to show that a computer is universal is to demonstrate a program that can simulate a computer known to be universal.  The fact that the laws of physics allow us to build ordinary universal computers is proof of what must be a very fundamental law of physics: "The fundamental process of physics is computation universal." 

Uncertainty is at the heart of quantum mechanics.  Finite Nature requires that we rule out true, locally generated randomness because such numbers would not, in this context, be considered finite.  The reason is that there is no way to create, within a computer, a truly random number that is orthogonal to everything in the computer.  On the other hand, another kind of randomness appears in CA where the values of many of the bits are so influenced by distant events as to be essentially orthogonal to any local process.  The deterministic nature of finite digital processes is different in that it is unknowable determinism.  From within the system an observer will never be able to know very much about the true microscopic state of that system.  Every part of space is computing its future as fast possible, while information pours in from every direction.  The result is the same as caused by the apparent randomness of quantum mechanical processes.