Phenomenological study, epistemic and hermeneutics of the noncommutative geometry

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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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The essence of the method of physics is inseparably connected with the problem of interplay between local and global properties of the universe. In the present paper we discuss this interplay as it is present in three major departments of contemporary physics: general relativity, quantum mechanics and some attempts at quantizing gravity (especially geometrodynamics and its recent successors in the form of various pregeometry conceptions). It turns out that all big interpretative issues involved in this problem point towards the necessity of changing from the standard space-time geometry to some radically new, most probably non-local, generalization. We argue that the recent noncommutative geometry offers attractive possibilities, and gives us a conceptual insight into its algebraic foundations. Noncommutative spaces are, in general, non-local, and their applications to physics, known at present, seem very promising. One would expect that beneath the Planck threshold there reigns a ``noncommutative pregeometry'', and only when crossing this threshold the usual space-time geometry emerges.