The Structure and Emergent Computation

Integers: irreducible guides in the search for a unified theory

“…In particular, the dynamics of a physical system is considered in order to understand correlations between the parts. For this purpose we describe a physical system by its certain quantities and are interested to know how many of them remain invariant as the system evolves from one state to another [4]. Let I be an integer alphabet and I N = {x = x 1 ...x n , x i ∈ I, i = 1, ..., N } be the set of all sequences of length N ≥ 2 with symbols in I.…”

“…We use the geometric representation to characterize a state x = x 1 ...x N ∈ I N of a physical system in terms of the definite integrals [4] …”

“…In particular, as a physical system makes transition from a state x = x 1 ...x N to a state x = x 1 ...x N we are interested to know how many of the quantities are conserved [4] …”

“…In particular, the theory has the integers and certain integer relations as the basic elements and interprets the relationships between them as a result of their formation processes. The formation processes start with the integers and are completely controlled by arithmetic through a single organizing principle [4]. Consequently, the theory is irreducible as long as the integers are accepted as the most fundamental entities.…”

“…As a result the correlations do not take into account the distances between the parts of a physical systems and are nonlocal, instantaneous and inseparable in nature. Furthermore, it turns out that it is possible to geometrize the integer relations as two-dimensional patterns and characterize the formation processes in terms of their geometric transformations [4]. This makes the theory of formation processes geometrical and quantitative.…”