On the emergence of non-Euclidean geometries
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On the emergence of non-Euclidean geometries
On the emergence of non-Euclidean geometries
For about 2000 years, Euclidean geometry was considered to be generally valid for describing real physical space - domyhomework.club/ , among other things. Then, however, increasing criticism of this view led to two important discoveries:
The criticism of the separation of geometry and arithmetic by Euclid , led to the creation of the concept of the real number, with the help of which not only commensurable but also incommensurable quantities could be characterised. The starting point for the emergence of a mathematics of continuously - algerbra homework help - variable quantities was laid. Rene Descartes And Carl Friedrich Gauss , among others, were active in this field.
The criticism of individual postulates, especially the fifth (parallel postulate), led to the development of further geometries that did not contradict reality - the non-Euclidean geometries (by Lobatschewski, Bolyai, Gauss and Riemann) , which meant the transition from a mathematics of constant relations to one of variable relations.
Thus the axiom of parallels was replaced by its opposite statement:
"In a plane, through any point outside a given straight line, one can put more than one straight line that does not intersect the given straight line". This geometry proved to be as free of contradictions as Euclid-Hilbert geometry.
If many geometries are possible, there can be no general definition of the basic terms. The effort (by Euclid and others) to define basic concepts is thus impossible in principle - do my homework math . Basic terms therefore only refer to the system under consideration.
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