New Mathematics

Based on the generalization of associative models of classical and hypercomplex numbers, governed by distributive properties, models of new, object numbers have been proposed. They are named sady ("gardens").

These are finite sets whose elements take the form of matrices of various dimensions and complex structures. These sets are closed under operations that are, in particular, partially non-associative. They do not possess distributivity, which fundamentally distinguishes them from conventional sets.

They share analogical connections with models of Galois fields and their extensions, surpassing them by the criterion of directly generating functional relationships operating within the set.

From the study of such models, not only nontrivial algebraic laws and new algebras have been found, but also solutions that are unattainable by classical methods. For example, the Pythagorean theorem is generalized within them, and Fermat’s problem is resolved in a new way. With a modular multiplication operation and co-modular addition, object sets comply with the Diophantus-Brahmagupta condition.

Analysis shows that they contain the "seeds" of known algebras such as those of Leibniz, Malcev, Seigal, and Akivis… prompting further theoretical development and practical applications.

The absence of distributivity suggests, in the future, a generalization of Gaussian vector space models, as well as Hamiltonian, Clifford, and Grassmann algebras.

Object sets allow the interpretation of numerical magic squares as designs for technological devices capable of producing identical outputs under different conditions. Object-based magic squares are presented not for the sum of elements, but for their product.

Object mathematics “expanded” the 8 trigrams of China to 27 trigrams, enabling the actual formulation of the laws of life by linking mythology with Western analytics.

In these object sets, the theme of "cell" division manifests in new ways: their self-organization algorithm includes the construction of the "shell" of a cell based on its core, given environmental conditions.

Projective geometry is significantly generalized, in which points are replaced by elements of object sets, and lines are defined by various types of functional conditions. Object analogs of Pappus’s and Desargues’s theorems have been identified.

It has been justified that the finite object geometry of Fano does not comply with Desargues’s conditions.

Various argument-invariant functions have been found, including cyclic object exponentials, extending Euler’s model.

Models are proposed for generating spectra of associative and non-associative operations, which allow the mathematical modeling of diverse physiological and informational interactions between living systems—systems modeled by these object sets.

Numerous information encoding algorithms are illustrated, along with insights into information control.

The material is presented in a form accessible to many audiences. It could objectively become a catalyst for creative activity among young scientists interested in studying and modeling living systems and the laws of their life.

The proposed viewpoint is that associative operations are more suited to accounting for and describing Bodies and their physiology, while non-associative operations are more relevant to information interaction. From this perspective, object mathematics with complex structural matrices may be seen as a sketch of future mathematics for describing and managing living systems, their Consciousness and Feelings.
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