Sets with members as machines with things that flow
Sabah Al‐Fedaghi · 2017
Mathematics and computer science are strongly intertwined, especially in using the language of set theory as a suitable base for transitions between formal (human-computer) and informal (human-human) communications and in supporting modeling, e.g., object oriented concepts. Yet, it seems that difficulties occur when set notations and diagrams are applied, which has motivated a search for an alternative to traditional ways of visually depicting set theory. Some researchers propose distinguishing between "set" and "collection." Others claim that the fundamental error in metaphysical interpretations of set theory, the reification of a collection as a separate object, is a result of grammatical confusion. Another issue raised in this context with regard to set theory is the static state that has been imposed on its representation; e.g., sets are spaces of containers, and "movements" of members are conceptualized as "arches" called mapping and functions. This paper offers a visual representation of set theory that may benefit modeling in computer science based on the familiar notion of machines. A machine can be seen as an extended form of the input-process-output system with five basic "operations": releasing, transferring, receiving, processing, and creating of things that flow. The paper employs such a schematization apparatus to construct a high-level description of sets and functions. The resultant diagrammatic representation seems a viable tool for enhancing the relationship between mathematics and computer science.