A few simple heuristics have guided the development of Digital Philosophy: simplicity, economy, and Occam’s Razor [1] . In particular, Digital Philosophy is a totally atomic system. Everything fundamental is assumed to be atomic or discrete and, thereby, so is everything else.
In physics, we already know that certain things are discrete, such as charge, angular momentum, and particles. If, in addition, space and time are discrete, then all other measures and quantities of physics must be discrete. This conclusion is a major result of DP (see chapter on Units). In this context, a discrete measure is defined as one where the numerical value of that measure can always be represented exactly by a finite integer; the exact state of any totally discrete system can always be represented by a set of finite integers.
Thus, DP replaces the continuous flow of time in any dynamic system with a sequence of time steps; each time step is associated with an integer. The calculus, while still very useful for modeling many kinds of discrete dynamical systems, must be understood as no more than a good approximation to a fundamentally discrete process. Two mathematical models of systems of integers are:
- Diophantine analysis; the mathematics of the integers
- Automata Theory; the mathematics of discrete processes
Automata Theory and its variants are the mathematics of fundamental processes in DP because they have the property of allowing for exact, dynamic models of discrete temporal processes.
Tremendous progress in the sciences followed the discovery of the calculus (and partial differential equations) as a way of mathematically representing physical-temporal relationships. The normal mathematics of dynamical systems consists of sets of mathematical laws and formulae, often differential equations, which can be transformed into computational algorithms that are approximate models of continuous systems. Even so, mathematical equations can provide only implicit models for temporal processes. DP takes the point of view that we must look to some version of Automata Theory with respect to the most fundamental process-models of physical systems. What must be demanded of Digital Philosophy is the eventual ability to derive, from DP models of fundamental processes, the same mathematical equations that constitute the basis of science today.
Conway’s game of Life [2] is a good example of a simple digital system and the consequent emergent properties. Digital Philosophy represents state by patterns of bits, as is done in ordinary computers. All of the fundamental transformations that can be done with bits in a computer are really a subset of what mathematics can do with the integers. Computer bits are usually thought of as 1 or 0, but in DM (Digital Mechanics is a DP model of physics) we sometimes choose to use 2 symmetric integers, +1 and –1. The bits of DM exist at points in a regular digital space-time, where each point contains 1 bit of information. Space-time is digital when it is made up of points (bits) all of which have integer coordinates, (t, x, y, z all integers). Automata Theory and Computer Science imply that the representation of state by such bits imposes no limitations on the kinds of possible processes beyond the fact that everything is ultimately quantized and finite.
(Last revised 15-Oct-01)