ABSTRACT
The axioms of geometry are independent of one another and of the fundamental laws of logic, by which G. Frege means formal logic, which is founded on true premises, that is axioms. The whole of geometry is excluded because its axioms can be contradicted giving all the systems of non-Euclidean geometry. It seems very strange that Michael Dummett considers Frege the greatest philosopher of mathematics when the whole of geometry is absent from Frege’s works, even though geometry is the science of space and the concept of extension which is applicable to the natural real world - the cosmos. Hubert’s work on the foundations of Euclidean geometry is based on the real number system. The non-commutative multiplication reverses the axis of spin and in consequence the direction of spin. In the earth’s orbit around the sun the two motions are combined, namely translatory motion, that is orbital angulum momentum, and the rotation or spin, that is intrinsic angular momentum.
