Value, Representation, Information and Communication

TL;DR

The paper addresses the limitations of Shannon's transmission-centric view by introducing a monadology-based framework that separates value, representation, and information using set-theoretic foundations $ZFC$ and $NBG$ and employs surreal numbers. It presents a quantitative framework in which metric spaces on subsets, characterized by $d_H$, enable precision control and existence tests via $Cauchy$ sequences, and argues that the Von Neumann Universe represents optimal cognition for a monad. Key contributions include a consistently recursive formalization linking value, representation and information, a method for optimal representations via the $Cauchy$ Inequality, and the concept of functionality agreements among monads to enable efficient communication. This framework offers a new lens for information processing and intelligent systems with potential impact on computation on compressed data and cross-monad coordination.

Abstract

A new analytic framework is first formalized via the usage of the Monadology (Leibniz 1898), to expand the understanding of Zermelo-Fraenkel-choice set theory (ZFC) and Von Neumann-Bernays-Godel set theory (NBG). Implicitly, the framework levels value, representation and information separately. Given the fact that there exists a coincidental equivalence between Von Neumann universe and originally-formalized motivation in ZFC, this work hypothesizes the essential of ordered values for one monand, to carry out efficient communication with the rest. This work then focuses on the relationship among values, representation and information (and suggests potential methods for quantitative analysis). First, this framework generalizes the definition of values and representations from "Indexes approximate Values" principle by (Peng 2023) via surreal numbers (Knuth 1974). Second, credited to surreal numbers, this work recursively connects representations and information via subsets of sets. Therefore, the definition to metric space(s) is naturally formed by representations, and quantitative methods (e.g., Hausdorff Distance) can be applied for quantitative analysis among (sub)sets. Third, this framework conjectures that: as long as the metric space is (or can be formed as) complete, the existence tests can be performed via Cauchy Sequence (or its generalized methods). This work finally revisits the communication theory, and suggests new perspectives from the new analytic framework. Particularly, this work hypothesizes a (quantitative) relationship between values and representation, and conjectures that: the optimal construction of representations exists, and it can be derived as the core value of one monad via Cauchy Inequality (or its generalized methods).

Figures (2)

• Figure 1: A conceptual formalization of Shannon's mathematical framework for communication from (BellTechn48/Shannon).
• Figure 2: A pictorial example of Von Neumann Universe (left); and a pictorial example of surreal number from Gretchen Grimm on "Day 0 to infinity" (right).