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Popkov Valerian, Baturin Andrey International Alexander Bogdanov Institute, Yekaterinburg, Russia, www.bogdinst.ru The Duality of Nature Philosophic Rethinking of Poincaré Topological Complex

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The wholeness is duality Dieser Dualität ist keine Dualismus! (R.Awenarius) corpuscle observability institutions resources goods flows position structure √1 = 1 wave accessibility markets problems services potentials relation function √-1 = i

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Poincaré Jules Henri (1854-1912) - the founder of mathematical topology He invented his “cellular system” with full set of topological invariants and suggested a simple regular procedure of its dual inversion The cellular system represents an aggregate of “cells” of different dimensions: 0-dimension nodes 1-dimension lines (branches) 2-dimension pieces of surfaces 3-dimension volumes …and so on Cells of lower dimensions adjoin the higher ones, shaping their facets, borders: branches are bounded with nodes, surfaces – with branches, volumes – with surfaces, and so on. Cells of the same dimension side with each other at common sides, constructing chains.

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The sample of Poincaré Topological Complex - tetrahedron The wholeness (tetrahedron) has two faces; there are dual operations: intersection 6 lines or connection 4 nodes

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The world is a multi-dimensional process It’s consisting of local processes, adjoining one another for example – a river has 3-dimensions a pilot of a plane see it as 2-dimensional water ribbon a hydrograph examines one as 2-dimensional bottom topography The top and the bottom meet, making up a costal line (1- demensional) there are also fish resources, birds and animals populations in the high-water bed There are also the goods and financial flows, associated with the river

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The cycle and independent cycle The cycle is closed circuit The independent cycle: it’s not the border of low dimension cycle and does not cover one the higher dimension cycle In some sense the independent cycle is “a defect” in ideal mathematic construction, but it’s very important for the thinking of wholeness

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The sample of independent cycle: “doughnut” (torus ) The torus has only two independent cycles; “blue” cycle and “red” one All another cycles on the torus surface may be transform in this two ones

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The main statements concerning to the structure of wholeness The key role belongs to a set of networks (circuits) and cycles Processes, adjoining one another, create networks Closed circuit of processes is the cycle The number of independent cycles is fundamental characteristic of any integrated system

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Processes The unity in duality Flows - are balanced in the node - (The law of conservation mass) Potentials of adjacent nodes are balanced on the branches (i1 – i2) i2 i1 (e1 – e2) e2 e1

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The wholeness – the world from two points of view Kinetic world (a flow) 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 wholeness Stressed world (a potential) Potentials are coordinated in the opposite direction: with increase of dimension, through structural elements of higher dimensions. The world is gathered, integrated, joined through stresses This is the direction of integrity of the wholeness

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Dual cyclic structure of the wholeness Cycles of the first kind are a closed “equiflow” circuit balanced in nodes a vortex flow into interior of the wholeness each closed flow closes the circle of potentials Cycles of the second kind (“co-cycles”) are equipotential “hoops”, which balance internal stresses of the wholeness within itself The “hoop” tightens the scattering flows, closing them to the “vortex”

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Poincaré duality theorem Flows and stresses (cycles and co-cycles) are the same complex of processes But these forms are quite independent, they produce absolutely different structures, being closely conjugated within the wholeness cycle and the co-cycle in each pair occur at different structural levels of the wholeness, namely, at the levels of “complementary dimensions Poincaré duality theorem is devoted that if total dimensions of the closed manifolds is n, each m- dimension cycle corresponds to a co-cycle of n - m dimension

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Let’s come back to our river Let’s single out 1-dimension linear flow, which penetrate 2-dimension equipotential surfaces, cutting the landscape horizontally, just like coils of compressed gravitational spring, pushing the flow to the bottomland. And if the flow turned out to be closed (1-dimensional cycle took place), it means, that somewhere there arose an upward flow, which, overcoming gravitation, push water upwards to potential field with an opposite intensity. Here we have a 2-dimension co-cycle.

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This global human world Countries and local unions, their borders, - instability arcs and voltage nodes, occurring within them Military and political, economic, climatic, ecological potentials of countries and regions National markets and transboundary trade flows The world system of labour division, global cycles of trade flows and co-cycles of regional potentials and tension of political and economic alliances That is how we see the problem field for the Poincaré program, started more than a hundred years ago

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Resourses [conceptions] Heraclitus the Ephesian (Dark) (535-475 BC) [the upward- downward path] Georg Wilhelm Friedrich Hegel (1770-1832) Science of Logic, tr. W. H. Johnston and L. G. Struthers, 2 vols., 1929; tr. A. V. Miller, 1969 [logic loops or dialectic] Friedrich Wilhelm Joseph Schelling (1775-1854). Ideas for a Philosophy of Nature: as Introduction to the Study of this Science (1988) translated by E.E. Harris and P. Heath, Cambridge: Cambridge University Press [duality – the soul of nature] Henri Poincaré(1854-1912), Analysis Situs, Journal de l'École Polytechnique ser 2, 1 (1895) pages 1-123[dual inversion of cellular system] Alexander Bogdanov (1873-1928) The General Science of Organization, trans. George Gorelik, Seaside, CA, Intersystems Publications, 1980 [activity-resistance] Gabrial Kron (1901-1968), Tensor Analysis of Networks, John Wiley and Sons, New York, 1939[dual networks and tearing met hod]

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Thank you for attention!

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Notes 5 Poynting Theorem

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