Three-Dimensional Manifolds and Orbifolds

A. D. Mednykh

Even the ancient Greeks transformed mathematics from an empirical science into a deductive one. 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. Pure mathematics investigates the forms and relations as distinct from material content. For instance, its direct subject is an ideal ball, but not one or another ball-shaped body. Another example: mathematics deals with integers in general but not with aggregates of numbers or individual numbers, and so on. 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. Mathematical Geometry was the theory of a real space as later mechanics became the theory of motion

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