Almost all philosophy about the state system has concentrated on the authorities. Any function of the state can be represented as a superposition of the functions of violence / coercion. Ultimately, the state appears to be a kind of plurality of subjects with a definite crater power / coercion / violence operation on it. The algebra of trust on the multiplicity of owners of themselves, endowed with free future, is each of them is only a part of nature, еру carrier of the part of the general human culture, and for their completeness, they have and understand the need for the Other. This is the philosophy of solving political, environmental, and climate challenges not through violent / voluntaristic methods, but by the recognition of sovereign rights and the search for ways to achieve sustainable development. Any cracy / power / coercion / violence must be separated from the models of society, the state. Public agreement is not an agreement with the abstract notion of the state, but an agreement with definite elected people who have gained the trust of those to whom they temporarily render their services. Contract is temporary, limited by period, with obligatory full responsibility of the parties. Scientific novelty. For more than two thousand years, long before Aristotle and Plato, European philosophical thought, reflecting on the structure of society, wanders in the labyrinths of kratia. Modern achievements of mathematics provide an opportunity to build ideal political objects, and a direct product of material and ideal government building. (Example of a trust algebra [4].)

The purpose of the work is to consider possible directions of development of political thought on the example of application of methods of mathematics and philosophy of politics, to reveal widespread system errors. Methods. The axiomatic approach, the method of proving the opposite, the method of defining paradoxes are used. Based on a critical analysis of the philosophy of consciousness by means of a communicative approach, possible for political science algebraic constructs are formed, which determine not only elements (sets) and operations, but also relations, the problem of identifying and defining systemic errors in certain political ideas, theories. Scientific novelty lies in the fact that we now consider politics as algebra and divide political objects into ideal objects and real objects. The algebra of politics is a collection of specific political algebras. The difference between these algebras is in the choice of carrier, operations and relationships. The carriers of these algebras are idealized sets of the geosphere, biosphere, technosphere, sociosphere, their intersections and unions, as well as operations specific to politics, such as violence, trust, interest, greed, etc. We regard a real object, including a real state, as a direct product of the ideal and the material (at the atomic level). In this case, the State (S) is a direct product of the ideal political object (constitution, legislation, regulations, etc. — IPO), and the material object (territory, subsoil, population, etc. — MO) IPO × MO = S.