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Phenomenological study, epistemic and hermeneutics of the noncommutative geometry

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Abstract

By using three philosophical poles, namely phenomenology, epistemology and hermeneutics, this article analyses the character of mathematical concepts in non-commutative geometry, for which the concept of non-commutativity and the questioning of the concept of locality play a great role. The relationship between geometrical thought and algebraic/analytical operative experience in this geometry is examined. Firstly, we approach the phenomenological notions of epoche, of effective reality, and of empty substrate X in Husserl's Ideen I, shifting their meaning. Secondly, we envisage epistemologically the dialectic between «despatialisation» and «respatialisation», and the «duality of the operation and of the object» found in G.-G. Granger. Thirdly, we seek the hermeneutic for mathematical concepts, which brings out a fusion of quasi-visual experience of objects and of semantic thought in geometry. This fusion is mediatised by effective experience of operations and by syntaxic thought in algebra/analysis. A kind of dialectic between geometry and algebra/analysis makes possible creative production and the creative interpretation of mathematical concepts. Thus this article shows concretely a new philosophical method for analysing mathematical concepts.

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