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Emergence and Evolution of Meaning The GDI Revisiting Programme Part II: The Regressive Perspective: Bottom-Up

“ Όδός “The way άνω up and the καί way down κάτο is one and μία καί the same” ώυτή ” Heraclitus of Ephesus Hermeneutical circle 2

The regressive perspective: bottom-up 0. The case of perception a) limits b) Direct and inverse EM problems 1. Direct problem (dimensionality question) a) Forward formulation b) Sampling theorems c) Forward problem 2. Inverse problem (interpretation question) a) Forward formulation b) Sampling theorems c) The limits of observation 3. Interpreting reality a) Reactive b) Reflexive c) Perspectivistic 3 Emergence and Evolution of Meaning: GDI revisiting programme - Part 2: regressive perspective – Bottom-Up

The limits of our perception Huygens 1 ° We only perceive surfaces Theorem Uniqueness 2 ° Every angle around the object is necessary Theorem 3 ° The discernible details of a thing Sampling are not smaller than  Theorem 4 ° In case of no sensibility to phase, Phaseless spatial perception is through the Uniqueness observation at 2 surfaces feasible Theorem 4 Emergence and Evolution of Meaning: GDI revisiting programme - Part 2: regressive perspective – Bottom-Up

The forward- and the inverse problems The forward Problem The inverse Problem Maxwell laws Huygens-Schelkunoff Theorems   z z    1  1 q e  q m ? S S ? E T J x x M y D D y linear,      2 - dimesional { J , M } { E , H } ? 5 Emergence and Evolution of Meaning: GDI revisiting programme - Part 2: regressive perspective – Bottom-Up

Unicity and Equivalence theorems Unicity [ E 1 , H 1 ] [  E ,  H ] no energy crosses through D theorem [ E 2 , H 2 ] Equivalence Theorem 6 Emergence and Evolution of Meaning: GDI revisiting programme - Part 2: regressive perspective – Bottom-Up

Forward formulation Vector potentials A and F ( D’Alambert- Eq.):     S J ( r ) inside     2 2  A A S  0 outside          2 2 ( ) G ( r , r ) ( r r )     S M ( r ) inside     2 2  F F S  0 outside    j R ( r r ' ) e   4 R ( r r ' ) Convolutional solution:  ~ ~ ~           F A ( r ) J ( r ) G ( r r ) d v J ( r ) G ( r )    A ( r ) J ( r ) G ( r )  V ~ ~ ~    F ( r ) M ( r ) G ( r )            F ( r ) M ( r ) G ( r r ) d v M ( r ) G ( r )  V    1        E j  1  A F   2   Fields E , H :    1        H j  1  F A   2   7 Emergence and Evolution of Meaning: GDI revisiting programme - Part 2: regressive perspective – Bottom-Up

Forward Formulation (real sources) z Only electrical currents: F =0 S J F ~ ~ ~     E J G ( r ) J ( r ) E J G ( r ) J ( r ) J x J D y                   2 E G ( R ) G ( R )( x x ) G ( R )( x x )( y y ) G ( R )( x x )( z z ) J ( r )   J 1 2 2 2 x x                      2  E  G ( R )( y y )( x x ) G ( R ) G ( R )( y y ) G ( R )( y y )( z z ) J ( r ) d v     J 2 1 2 2 y y                    2 V E G ( R )( z z )( x x ) G ( R )( z z )( y y ) G ( R ) G ( R )( z z ) J ( r )         J 2 2 1 2 z z           2 2 2 2 j 3 j 3 β R R j 1 j R R             j R j R G ( r r ) e G ( r r ) e 2  2 5 1  4 R 2 3 4 R 8 Emergence and Evolution of Meaning: GDI revisiting programme - Part 2: regressive perspective – Bottom-Up

Forward Formulation (equivalent magnetic sources) z Only magnetic currents: A =0 M F S ~ ~ ~     E G ( r ) M ( r ) E G ( r ) M ( r ) M M M M x D y             E   0 ( z z ) ( y y ) M ( r )     M x x                            E G ( R ) ( r r ) M ( r ' ) d s G ( R ) ( z z ) 0 ( x x ) M ( r ) d s     M 3 3 y y                 S S ( y y ) ( x x ) 0 M ( r )   E         z M z   1 j R 1      j R G ( r r ) e 3  3 4 R 9 Emergence and Evolution of Meaning: GDI revisiting programme - Part 2: regressive perspective – Bottom-Up

Sampling theorems for radiating fields Theorem 1 : The minimal distance between independent intensity values of a field generated by an arbitrary object is λ /2 . Theorem 2 : The maximum number of details of an object, inscribed in an sphere of radius a, which is causing an observed field distribution is 16 π (a  ) 2 . This is the essential dimension of the observation problem . Theorem 3: The minimal distance between independent values of the field corresponding to the manifestation of an object inscribed in a sphere of radius a, whose centre is at a distance d, is:  d/2a  . 10 Emergence and Evolution of Meaning: GDI revisiting programme - Part 2: regressive perspective – Bottom-Up

Forward (discretized) problem • Phenomena observed at D ( u,v ) Green equation corresponding to a set of sources x’, y’, z’  u, v Source Observation (object) domain                 u v G u v x y z G u v x y z x y z  f ( , ) ( , , , , ) ( , , , , ) ( , , )       N N N 1 1 1 1 1 1 1 1 1 1 1 1                              u v G u v x y z G u v x y z x y z  f ( , ) ( , , , , ) ( , , , , ) ( , , )       M M M M M M N N N N N N 1 1 1 Phenomena Source Wave function           ψ G u v x y z ( , , , , )       n n n 1 1 1 1 N N           Ψ     ψ ψ     f f T f  where n n n n          n n    1 ψ 1 G u v x y z ( , , , , )       M M M M n n n   N   T  Ψ ψ f f Direct problem (manifestation of reality) n n  n 1 11 Emergence and Evolution of Meaning: GDI revisiting programme - Part 2: regressive perspective – Bottom-Up

Inverse problem • A good way to suit our problem (to be invertible) is locating N punctual sources over S regularly spaced at a distance λ / 2 χ   Ψ  OBSERVED   Ψ     T f   projection             1 Ψ d Ψ f T T T T f [ ] min , OBS OBS projection . projection . f • Which can have a unique solution. 12 Emergence and Evolution of Meaning: GDI revisiting programme - Part 2: regressive perspective – Bottom-Up

The limits of observation 1. A Finite Number of Details related to the object can be found. 2. Such number depends on the surface bounding the object. 3. The volumetric distribution of an object cannot be known only based on its manifestations on the environment. 4. The description of the object that can be achieved corresponds to a projection of the inner inhomogeneities over S . • Fundamental limits to the observation problem, not related to sense structure, but to the differences that can be found • Related to the maximal a posteriori knowledge • Complexity Object can be > Complexity its Manifestation Given Unknown 13 Emergence and Evolution of Meaning: GDI revisiting programme - Part 2: regressive perspective – Bottom-Up

The meaning-offer of observation vs perception 14 Emergence and Evolution of Meaning: GDI revisiting programme - Part 2: regressive perspective – Bottom-Up

“Faraday, in his mind's eye, saw lines of force traversing all space where the mathematicians saw centres of force attracting at a distance...” J.C. Maxwell On the search of the electromagnetic reality 15