Mathematics is the Queen of Science, and,
Arithmetic the Queen of Mathematics.

Carl. F. Gauss (1777-1855)


The research in the field of Stochastic Discrete-Time Systems made it possible to reveal the facts that need special attention and treatment, as reflected and justified in the articles «Pre-arithmetic» (Prearithmetic) and «Hypothesis about the nature of Arithmetic».

Let us single out the main facts:

1. In contrast to prevailing views and adopted rules, arithmetic operations (addition and subtraction) are nontrivial and give absolutely accurate results only in cases when these operations are not related to time and occur instantaneously.
2. Arithmetic is preceded by Pre-Arithmetic. Pre-Arithmetic is trivial, indecomposable into elementary components, and lays a claim to the leading and fundamental role in Higher Arithmetic.
3. Pre-Arithmetic gives rise to its other varieties, and with them to new Arithmetics.
4. Pre-Arithmetics and Arithmetics following from them are endowed with a universal structure and order, referred to as Dichotomic Sequences. And with this, as follows from the isomorphism, Nature is endowed with a similar structure and order.
5. In Arithmetic, the dichotomic order is ideal, but in Pre-Arithmetic, it has a chaotic character.
6. In Pre-Arithmetics, the basic variable set in compliance with the result of the operation is accompanied by the induction component that is inextricably entwined with and is dual with respect to the basic variable. All operations in Pre-Arithmetic are executed per act, followed by local and nonlocal discrete transitions.
7. The induction component has a strongly pronounced non-linear character, does not degenerate into a constant and does not disappear, as is the case in Ordinary Arithmetic.
8. The progress of processes in Pre-Arithmetic is accompanied by a short, depending on the initial conditions, transient time-dependent segment area (attractor).
9. As the transition segment is passed through, the basic variable and induction component set in compliance with the process become phenomenally self-synchronized, reach a maximum period of repetition, and then everywhere, within the boundaries of each of the subsequent periods, behave in a stationary and repetition-free manner.

As can be seen, Pre-Arithmetics (themselves, directly and not indirectly, through the Arithmetic, and regardless of anything else) are characterized by the unique properties inherent in nature and well-represented in physics and observed in experiments.

Pre-Arithmetics (Prearithmetic) formed the basis of an innovative breakthrough (Random Method, Stochastic Method) in the field of Stochastic Systems and Stochastic Technologies, Nonlinear Dynamics (Non-Linear Dynamics) and Cryptography they represent; these technologies are characterized by a significantly pronounced chaotic behavior inherent in truly random processes. Stochastic Technologies, based on the results and the potentials, significantly surpass the known analogs, including those that make use of the Recurrence Methods and Galois Fields.


With the introduction of Pre-Arithmetic, we will have to rethink many things and consider them in a different manner. Then, there will come the turn of Nonlinear Dynamics (Non-Linear Dynamics) and processes related to the harmony in Chaos.

Igor Kulakov, Игорь Кулаков