The Cellular Automaton

In this paper, we take the position that Finite Nature implies that the basic substrate of physics operates in a manner similar to the workings of certain specialized computers called cellular automata[4][5][6]. The logic behind this assumption is that a cellular automaton is a system of cells where each cell is in one of a finite number of states. Each cell transitions from one state to the next according to a rule where the outcome for a particular cell depends on the states of cells in the neighborhood of that cell. The definition of cellular automata seems so broad as to encompass every kind of discrete cellular process.

Some cellular automata are universal; they have the property that they can be made to do any computation for which they have enough volume and time. If Finite Nature is true it means that a volume of space has a certain amount of computational capability. If Finite Nature is not true, then it seems necessary to assume that infinite computational resources are required to model physics exactly. Conversely it is reasonable to equate the order of computational power of any system with the order of computational power necessary to exactly model that system. If Finite Nature is false, then any volume of space time, no matter how small, would represent an infinite capacity for computation. The author believes that computational capability is a quantitative resource, like area, or energy. It should be possible to relate the physical units of computation to ordinary physical units (e.g. Mass, Length, Time...). This would make it untenable for a finite volume of space time to require an infinite amount of computation in order to model physics exactly; for example, if it requires infinite resources that are equivalent to mass.

In this paper we will be working with large finite numbers, but any kind of infinity dwarfs them all. It is very hard to imagine what the purpose or necessity could be for any sort of infinity, since simple to express finite resources can clearly dwarf the needs of a universe like the one we live in. Consider the Ackerman function:

A(a, n, j);
A(a, n, 1) = a n
A(a, n ,2) = a * n,
A(a, n, 3) = a^n,
A(a, n, 4) = a^a^a^a (n times)

In Mathematica, A can be defined as follows:

A[a_, n_, j_]:= If [ j==1, a n, If [ n=1, a, A[a, A[a, n-l, j], j-1] ] ]

Assume a = A(9, 9, 9). C is a large but finite number that easily exceeds the value and the precision of any number that might be encountered in any calculation about our universe. General recursion allows us to define much larger finite numbers as necessary. Nothing to do with anything we can learn from experimental physics will ever have any need whatsoever for such large or precise numbers. The ideas of infinite and infinitesimal numbers and of continuous variables are handy for allowing the use of calculus and for creating approximate physical formulas, but we shouldn't confuse those ideas with what is happening in the real world.

 

[4] S. Ulam, Random Processes and Transformations, Proceedings of the International Congress on Mathematics, 1950, Vol. 2 (1952) 264-275

[5] J. von Neumann, Theory of Self-Reproducing Automata, edited and completed by A. Burks (University of Illinois Press, Champaign, IL, 1966).

[6] T. Toffoli and N Margolus, Invertible Cellular Automata: A Review, Physica D 45 (1990) 229-253