Toward computing oriented representation of sets

Abstract

Diagrams probably rank among the oldest forms of human communication. Traditional logic diagrams (e.g., Venn diagrams, Euler diagrams, Peirce existential diagrams) have been utilized as conceptual representations, and it is claimed that these diagrammatic representations, in general, have advantages over linguistic ones. Nevertheless, current representations are not satisfactory. Diagrams of logic problems incompletely depict their underlying semantics and fail to provide a clear, basic, static structure with elementary dynamic features, creating a conceptual gap that sometimes causes misinterpretation. This paper proposes a conceptual apparatus to represent mathematical structure, and, without loss of generality, it focuses on sets. Set theory is described as one of the greatest achievements of modern mathematics. Nevertheless, its metaphysical interpretations raise paradoxes, and the notion of a collection, in terms of which sets are defined, is inconsistent. Accordingly, exploring a new view, albeit tentative, attuned to basic notions such as the definition of set is justifiable. This paper aims at providing an alternative graphical representation of a set as a machine with five basic “operations”: releasing, transferring, receiving, processing, and creating of things. Here, a depiction of sets is presented, as in the case of Venn-like diagrams, and is not intended to be a set theory contribution. The paper employs schematization as an apparatus of descriptive specification, and the resultant high-level description seems a viable tool for enhancing the relationship between set theory and computer science.