Popkov Valerian, Baturin Andrey International Alexander Bogdanov Institute, Yekaterinburg, Russia,www.bogdinst.ru

The Duality of NaturePhilosophic Rethinking of Poincaré Topological Complex

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 Closed circuit of processes is the cycle The number of independent cycles is

fundamental characteristic of any integrated system

Processes The unity in duality

Flows- are balanced in the

node - (The law of

conservation mass)

Potentialsof adjacent nodes are

balanced on the branches

(i1 – i2)

i2i1

(e1 – e2)

e2e1

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 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

Let’s come back to our riverLet’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 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 method]

Thank you for attention!