Читать книгу Riemannian space. Recognition of formulas (structures) of riemannian manifolds by a neural network - Ludmila Naumova - Страница 4

In general, geometry presupposes both the concept of space and the first basic concepts that are necessary for performing spatial constructions. It gives nominal definitions of concepts, while the essential properties of defined objects are included in the form of axioms. But the relationship between concepts and axioms can be different. Riemann drew attention to the General concept of repeatedly extended quantities, which include spatial quantities. Based on the General concept of magnitude, Riemann constructed the concept of a repeatedly extended quantity. Different world definitions are possible for a repeatedly extended quantity, and space is nothing more than a SPECIAL CASE of a thrice extended quantity. Riemann proposed to construct space (according to the laws of mathematics and logic) on the basis of the General concept of magnitude (the concept of magnitude is much broader than the concept of spatial quantities). There is a quantity of mass, magnitude of force, of velocity, of temperature value, the value of time, etc. What is a multiply extended magnitude? Classes of repeatedly extended quantities (spaces) of various types, where the unit of measurement belongs to this type of quantities, are not necessarily a unit of length, as in ordinary space. Therefore, in such “constructed spaces”, different measure definitions are possible, that is, different laws of construction and measurement of figures, i.e. different geometries are possible. This is exactly the Riemann space in the broad sense of the word. The properties that distinguish such a Riemann space from other conceivable extended quantities can only be derived from experience. We have to get out of the world, the framework of flat space. Mathematicians are discovering new spaces in mathematical symbols so far. But the real geometry of the physical world is not deduced from the General properties of extended quantities; on the contrary, the properties that space distinguishes from other conceivable forms can only be derived from experience. Pure mathematics will never be able to make a choice and say what is the true structure of real space. Only calculations based on actual observations can produce results. We will tell you how a trained neural network is a calculation that calculates Riemannian manifolds in the broad sense of the word. a neural network uses big data to determine the structure of an object (diversity).