Fundamental Level

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Mathematical Models of Reality

This is a technical overview, for a more illustrative overview go to the j-node section.


  • Definition State Space Ω:
Set of abstract and physically distinct states of of a subset of the observable world.


  • Definition Observables fi:
f_i : \Omega \to R,\quad i = 1, \dots, n;

are rules associating physically distinct states in the state space with real numbers.


  • Note:
    • Ω can be finite, infinite, uncountable.
    • The notion of physically distinct states depends on the observer and is not a property of the system


  • Definition Natural System N:
N = \left\{\Omega, f_1, \dots, f_n\right\}.


  • Definition Equations of State:
\Phi_i(f_1, \dots, f_n) = 0, \quad i = 0, \dots, n;
describes the dependency relations of the observables.


  • Example Ideal Gas Law:
\Phi \left(P,V,T \right) = 0.


  • Notion of a Formal System F:
    • Constructs of the human mind.
    • Provides basis for making predictions about N, using rules of inference in F.
    • F is a formal description of N, if N can be encoded into F:
\Epsilon :  N \to F.


  • Note: The inverse mapping
\Epsilon^{-1}: F \to N,
gives the decoding back into the natural system, i.e., the predictions



Sources: Casti, J.L. (1989). Alternate Realities - Mathematical Models of Nature and Man. New York: Wiley and Sons.

Illustrations

Can be found here: Slides: From Fundamental to Complex.