Dialectics of Counting and the Mathematics of Vagueness

TL;DR

This paper introduces new rough natural number systems and dialectical counting methods to enhance rough set theory and fuzzy set representations, offering a novel mathematical framework for vagueness and granularity.

Contribution

It develops a formal axiomatic approach to granularity in rough set theory and derives algebraic semantics from dialectical counting procedures, expanding foundational understanding.

Findings

Enhanced rough set-theoretical measures

Representation of fuzzy set theory via granule-theoretic terms

Algebraic structures for rough naturals and their variants

Abstract

New concepts of rough natural number systems are introduced in this research paper from both formal and less formal perspectives. These are used to improve most rough set-theoretical measures in general Rough Set theory (\textsf{RST}) and to represent rough semantics. The foundations of the theory also rely upon the axiomatic approach to granularity for all types of general \textsf{RST} recently developed by the present author. The latter theory is expanded upon in this paper. It is also shown that algebraic semantics of classical \textsf{RST} can be obtained from the developed dialectical counting procedures. Fuzzy set theory is also shown to be representable in purely granule-theoretic terms in the general perspective of solving the contamination problem that pervades this research paper. All this constitutes a radically different approach to the mathematics of vague phenomena and suggests new directions for a more realistic extension of the foundations of mathematics of vagueness from both foundational and application points of view. Algebras corresponding to a concept of \emph{rough naturals} are also studied and variants are characterised in the penultimate section.