The only way to discover the limits of the possible
is to go beyond them into the impossible.

Arthur C. Clarke (1917 -2008)

Hypotheses about The Nature of Arithmetic

With the discovery of Pre-Arithmetic (Prearithmetic), figuratively speak- ing, the Queen of mathematics – Arithmetic, reveals his true face, which allows deeper understanding of its essence and the world around us.
In Pre-Arithmetic the operation of addition and subtraction is characterized by the result of the operation – n-bits of the base variable G and its non-linear complement P. Unlike the arithmetic, the formation of a series of elements that differ by unity may be accompanied by the presence of a transient non-linear segment (attractor) of length формула, dependent on the initial conditions, after the passage of which, thanks to the phenomenal self-regulation of the variable G and its complement P, the basic variable reaches its maximum period формула, and then everywhere, within every subsequent period, behaves stationary and nonrepetedly.
In fact, the complementary pair формула of elements формула and формула of this series sets the operation of addition (subtraction) in Pre-Arithmetic. At the same time, the non-linear complement P does not degenerate into a constant and does not vanish, as is the case of arithmetic following from them.

Mutual equivalence (isomorphism) of physics and mathematics allows one to extend these results to physical phenomena and processes occurring in nature. From this, if we use the hypothesis about a potentially finite velocity of propagation of interactions, it inevitably follows that the operation of addition (subtraction) and derivatives from their arithmetic operations, being deployed in time, cannot be realized immediately, are accompanied by non-linear transient processes that are caused by changes in the induction component P*, reflected by the complement P introduced by Pre-Arithmetic.
From these facts follows the hypothesis «The Nature of Arithmetic» (Arithmetics), reported in the article of the same title. According to the hypothesis, real systems in the course of development and transition from non-stationary to stationary behavior, or formally, as they pass from Pre-Arithmetic through Incomplete Arithmetic to complete one, become self-synchronized. As the induction component is stabilized P* -> const (disappears), the systems reach their equ- ilibrium (stationary) state, and develop further according to the experimentally observed and known physical laws.

Pre-arithmetic gives birth to its other varieties.

Pre-Arithmetic forms the basis for constructing (represented by the Stochastic Technologies) the Regular and Non-Regular Random Method (Randomization Method, Stochastic Method), which gives birth to Stochastic Cryptography (Minimalistic Cryptography, Light-Weight Cryptography), Binary Nonlinear Functions, One-sided Functions, One-way Functions and Hash Functions.
The Regular Random Method (Stochastic Method) is used to construct generators and to form n-bit binary sequences, endowed with a special structure, similar to that of a binary tree, called Dichotomic Sequence, with a repetition period of формула. In turn, Dichotomic Generators can be linked and form networks and compositions of any structural and functional complexity, as well as provide the basis for constructing high-quality Random Number Generators (RNG, PRNG, TRNG), Stream Ciphers and Stochastic discrete-time systems for various applications.
The Non-Regular Random Method (Stochastic Method) is wider, inherits structural and functional properties of the regular method, lays the ground for constructing Block Ciphers and, in its characteristics and statistical parameters, is directly associated with the ideal Chaos and truly random processes.

Although this trend is still young, it is already well developed and has circuit solutions required for the manufacture of highly profitable industrial samples, which statistically, functionally and technically far ahead of all analogues known today.


Acceptance of the existence of Pre-Arithmetics and development of algebra, Dynamic Systems, Cryptography and Nonlinear Dynamics (Non-Linear Dynamics) following from them will allow one, without affecting the basic accumulated scientific knowledge, to enrich mathematics, and at the same time to penetrate into finer levels of physical description of the world, to deeper understand the existing natural effects and phenomena, yet unexplained or controversial from the standpoint of modern science.

Igor Kulakov, Игорь Кулаков