![]() ![]() More accurately, this can be defined as follows. According to this idea, any point of the Euclidean manifold is surrounded by the cube of the corresponding size. That is the main idea about the essence of the Euclidean manifold. And at the same time, one could say that two cubes derived at the different points of space are equal. In spite of a very large size, our unit cube is just a negligibly small part of the Universe.Īn imaginary unit cube of this kind can be derived at any other point of space. And now let us note the following obvious fact. For convenience, let us call the cube mentioned above a unit cube. If we mentally cut the cube whose edge length is equal to one light year the house where we live, the terrestrial globe, the Solar System and everything required for the human life, we can easily put inside. This amounts to unimaginable magnitude in the conventional linear measure - about 9,46 e., the distance covered by the light ray in one year. Let us choose a slightly unusual unit of length for our further discussion. According to the traditions of different countries, the dimensions of subjects are evaluated in meters, feet, leagues and other standard measures. ![]() e., the three primary dimensions of the surrounding space. We got used to think so since our birth - everybody knows what the length, height and width are, i. The quantitative and the qualitative forms Mathematical Geometry was the theory of a real space as later mechanics became the theory of motion However, no matter how abstract is mathematics, none of mathematicians was in doubt that all their conceptions, theorems and formulas are real quantitative and spatial relations. Another example: mathematics deals with integers in general but not with aggregates of numbers or individual numbers, and so on. For instance, its direct subject is an ideal ball, but not one or another ball-shaped body. Pure mathematics investigates the forms and relations as distinct from material content. They demanded that proofs of mathematical statements should be concluded from the basic conceptions, and excluding the reference to the experience gained as an argument. Even the ancient Greeks transformed mathematics from an empirical science into a deductive one. ![]()
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