Observability and Mathematics Modeling : Hilbert, Euclid, Gauss-Bolyai-Lobachevsky, and Riemann Geometries

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.