The Duality of NaturePhilosophic Rethinking of Poincaré Topological Complex Popkov Valerian, Baturin Andrey International Alexander Bogdanov Institute, Yekaterinburg, Russia, www.bogdinst.ru
The wholeness is dualityDieser 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
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.
The sample of Poincaré Topological Complex - tetrahedron • The wholeness (tetrahedron) has two faces; there are dual operations: intersection 6 lines or connection 4 nodes
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
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
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
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 • Closedcircuit of processes is the cycle • The number of independent cycles is fundamental characteristic of any integrated system
i2 i1 e2 e1 (i1 – i2) (e1 – e2) ProcessesThe unity in duality • Flows • are balanced in the node • (The law of conservation mass) • Potentials of adjacent nodes are balanced on the branches
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
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”
Poincaré duality theorem • Flows and stresses(cycles and co-cycles) are the samecomplex 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
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.
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
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 loopsor 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 method]