Nikolai Khots

Observability in Mathematics was developed based on the denial of the concept of infinity. The book introduces Observers into arithmetic, and arithmetic becomes dependent on Observers. And after that, the basic mathematical parts also become dependent on Observers. One of such parts is arithmetic itself and algebra. Arithmetic and Algebra play important roles not only in pure Mathematics but in contemporary Physics, for example, in Relativity theory and Quantum Mechanics. They will be called New Arithmetic and Algebra, both observers at the logical level and in arithmetic and algebra. The book reconsiders the foundations of classic arithmetic and algebra from this mathematical perspective. The relationships between numbers, polynomials, quaternions, groups, and algebras are discovered and exhibit new properties. It is shown that almost all classic arithmetic and algebra theorems are satisfied in Mathematics with Observers' arithmetic and algebra, where probabilities are less than 1.

Observability in Mathematics were developed by authors based on denial of infinity idea. We introduce Observers into arithmetic, and arithmetic becomes dependent on Observers. And after that the basic mathematical parts also become dependent on Observers. One of such parts is geometry. Geometry plays important role not only in pure Mathematics but in contemporary Physics, for example, in Relativity theory, Quantum Yang-Mills theory. We call New Geometry both Observers in arithmetics and in geometry. We reconsider the basis of classic geometry (points, straight lines, planes and space) from this Mathematics point of view. The relations of connection, order, parallels (Euclid, Gauss-Bolyai-Lobachevsky, Riemann), congruence, continuity are discovered and have new properties. We show that almost all classic geometry theorems are satisfied in Mathematics with Observers geometry with probabilities less than 1. That means classic geometries are not a limiting cases of the Observer’s geometry, but are only particular cases. And new geometry opens the road to reconsider differential geometry, algebraic geometry, geometric algebra, topology, and also to reconsider geometrical applications to various parts of contemporary physics. We proved that Mathematics with Observers gives a birth a new geometry.