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systems-theory--different-scales

dc4cd086cd7ce5b1832adf4fdd1211289880d2c7e295bcb0e684c01acee77c06
liminal@gitcitadel.com
Published 11/22/2024
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Core Framework

System Definition

S = (I, R, C, C', T)

Where:

  • I: interaction space

  • R: response repertoire

  • C: internal context

  • C': external context (complement)

  • T: transformation operators

System Categories

Simple Systems (š’œ)

  • Interaction growth: Iterated application of rules

  • Internal context: [source]

C = {axioms, rules, definitions}

  • External context: [source]

C' = āˆ…

  • Example: formal logical systems

Key Characteristics of š’œ

Godelian Limitations

  • All propositions decidable

  • Consistency provable

Emergence of Brittleness

  • Immediate failure at boundary

  • No graceful degradation

  • Binary success/failure states

Complicated Systems (š’ž)

  • Interaction growth: [source]

O(2^n)

  • Internal context: [source]

C = {axioms, theorems, parameters, architectures}

  • External context: [source]

C' = {training distribution, formal universe}

  • Properties:

    • Context enumerable but vast

    • Self-interacting axioms/theorems

    • Potential for inconsistency

    • Behavior bounded by axioms/training

    • Limited interaction with complement

Key Characteristics of š’ž

  • Typically computable

  • Godelian Limitations:

    • Incompleteness through self-reference

    • Undecidable propositions

    • Consistency unprovable within system

  • Scaling Properties:

    • Exponential growth in complexity

    • Increasing difficulty proving consistency

    • Accumulation of edge cases: Wear, rust, software bugs, etc.

Complex Systems (š•)

  • Interaction growth: [source]

O(nā†‘ā†‘k)

  • Internal context: [source]

C = {evolved structures, patterns}

  • External context: [source]

C' = Universe\C

  • Properties:

    • Born from and entangled with C'

    • Bidirectional C ā†” C' interaction

    • Context-dependent behavior

Universal System (š•Œ)

Note

The following system categories represent higher-order conceptual frameworks. Their formal properties are less rigorously defined, but have been considered for the context of sub-systems

  • Contains all possible interactions across all subsystems [source]

š’«(š•)

  • Higher order hyperoperator than š• as time evolves

Natural System (ā„•)

  • Contains all possible interactions from the set of possible Universal Systems, [source]

š’«(š•Œ)

  • Could this be interpreted as an uncountable number of interactions?

Context Relations

Simple Systems

  • C fully defines behavior

  • C' ignored/undefined

  • System fails if encounters C'

Complicated Systems

  • C approximates subset of C'

  • Limited C' interaction through training

  • Degrades outside training distribution

Complex Systems

  • C evolved within C'

  • Continuous C ā†” C' exchange

  • Adapts to C' changes

Key Principles

  1. Simple systems fight against C' through axiom boundaries

  2. Complicated systems approximate C' within training bounds

  3. Complex systems leverage C' through evolved structures

Transformation Theorems

  1. š’ž systems are tools created by š• for a purpose.

  2. š’ž systems bridge gap of adaptation through statistical approximation, a formalized representation of the behavior of objects in š•Œ.

  3. š• systems have evolved to account for all possible interactions within its neighborhood (ecosystem).

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