Khots B. Observability and Mathematics Modeling. Hilbert, Euclid,...2025

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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 , 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