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Foundations of geometry is the study of geometries as axiomatic systems. There are several sets of axioms which give rise to Euclidean geometry or to non-Euclidean geometries. These are fundamental to the study and of historical importance, but there are a great many modern geometries that are not Euclidean which can be studied from this viewpoint. The term axiomatic geometry can be applied to any geometry that is developed from an axiom system, but is often used to mean Euclidean geometry studied from this point of view. The completeness and independence of general axiomatic systems are important mathematical considerations, but there are also issues to do with the teaching of geometry which come into play.

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  • Foundations of geometry
  • 幾何学基礎論
  • Основания геометрии
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  • 幾何学基礎論(きかがくきそろん、英: foundation of geometry)とはユークリッド幾何学の公理主義的研究である。 平行線公準の問題より非ユークリッド幾何学が生まれたが、それは同時にユークリッド幾何学の厳密性にも疑問が投げかけられることでもあった。すなわち、 * 無矛盾な幾何学を作るにはどのような公理系が必要であるか * 更にそれらの公理系から構成される幾何学はどのような構造を持つか * それらの複数の異なる公理系の幾何学の体系間の関係はどうなっているのか という疑問を解決すべく幾何学基礎論の研究が進められてゆくこととなる。 同時期にはラッセルのパラドックスにみられるように集合論でも似たような問題が起こり、数学の基礎そのものに疑問が持たれる時代であったが、ヒルベルトは、形式主義思想によってこれらの問題を解決すべく著書「幾何学基礎論」を執筆した。この著書はユークリッド幾何学の公理系を最も厳密に吟味した著作としても有名である。
  • Foundations of geometry is the study of geometries as axiomatic systems. There are several sets of axioms which give rise to Euclidean geometry or to non-Euclidean geometries. These are fundamental to the study and of historical importance, but there are a great many modern geometries that are not Euclidean which can be studied from this viewpoint. The term axiomatic geometry can be applied to any geometry that is developed from an axiom system, but is often used to mean Euclidean geometry studied from this point of view. The completeness and independence of general axiomatic systems are important mathematical considerations, but there are also issues to do with the teaching of geometry which come into play.
  • Основания геометрии — область математики, изучающая аксиоматические системы евклидовой геометрии, а также различных неевклидовых геометрий. Основные вопросы состоят в полноте, независимости и непротиворечивости аксиоматических систем.Основания геометрии также связаны с вопросом преподавания геометрии.
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