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

The Duality of Nature
Philosophic Rethinking of Poincaré
Topological Complex
Popkov Valerian, Baturin Andrey
International Alexander Bogdanov
Institute, Yekaterinburg, Russia,
www.bogdinst.ru
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
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 (1demensional)
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)
i2
i1
(i1 – i2)

Potentials
of adjacent nodes are
balanced on
the branches
e1
(e1 – e2)
e2
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 mdimension 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 upwarddownward 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!