Number, Space, Motion & Time
Pure mathematics treats the basic notions of, and the truths relating to, number, space, motion and time that are known intuitively (i.e., by common sense). Over numerous centuries mathematicians have formulated logical systems of thought (viz., arithmetic, algebra, geometry and kinematics) based upon concepts and axioms that also serve as the very foundation of science. The non-Euclidean geometries are not only incompatible with each other, but also incompatible with the long-standing concepts and axioms of mathematics. Hence, they are invalid, and any physical theories based upon such pseudo-geometries result in science fiction.