# 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.