Volume 15, 2008
Popkov Valerian, Baturin Andrey
Philosophic Rethinking of Poincaré Topological Complex
The key philosophic concepts - wholeness and duality - are analyzed on the basis of general scientific and vision ideas of H. Poincaré. His cellular structure with full set of topological invariants (cycles) can be considered as a model of dual arrangement of the World. The World is seen as a multidimensional process, consisting not of parts, but of local processes, adjoining each other. It is demonstrated, that a set of cycles at each structural level not only resolve paradoxes of wholeness and development, but represents a recommencing process, which reproduces its own environment. The wholeness – world is considered as a duality of flows and potentials, which arrange and produce absolutely different structures, being closely conjugated, as cycles and co-cycles within the whole. The
streams are structured and coordinated towards decrease of structural level dimensions: from the general to the particular, from the concrete to the abstract, from the depth to the surface. This is the direction of differentiation of the whole. Potentials are coordinated in the opposite direction, with increase of dimension, through structural elements of higher dimensions. The world is gathered, integrated, joined, specified through stresses; differentiated parts tend to scatter, connections between them strain and turn them back to their whole. A penetration of the scientist into more complicated processes turns out to be a movement within a closed multidimensional surface (closed manifold), enriched with new dimensions with inclusion of new processes. Those concrete examples illustrate importance of Poincaré duality theorem for closed manifolds.