A Systems Theoretical Formal Logic for Category Theory

by

Carlos Pedro Gonçalves

Mathematics researcher at UNIDE-ISCTE
cpdsg.research@gmail.com (primary); cpdsg@iscte.pt

Maria Odete Madeira
Interdisciplinary researcher in philosophy of science and systems science
mosmg.research@gmail.com (primary), mariaodete.sm@gmail.com

Abstract

A systems theoretical thinking on the categorial object and morphism is developed, leading to areflection on the philosophical and mathematical foundations of category theory, which allowsfor the introduction of a formal language for category theory and of a categorial calculus as amorphic web-based logical calculus. A formal system, built from such calculus, is proposed and the logical semantics is addressed. Both syntax and semantics are independent from set theory.

Keywords: System, object, morphism, morphic web, individuation, entity, identity, categories,n-categories

1. Introduction

"In the present work, we propose a systems theoretical approach to category theory, introducing a formal language (LCat) and a formal system (FCat′ ), that incorporate the main system theoretic foundations of the categorial object and morphism. The formal system is based upon a morphic web calculus that we call categorial calculus. Both the logical syntax and semantics of such calculus are addressed and shown to be independent from set theory, which makes the theory itself independent from set theory.

In section 2., we address the categorial object as a system, providing for the philosophical ground of the main work. In section 3., we introduce the formal language LCat and a formal system FCat1−6 that is able to address the simpler structures of category theory.

In section 4., we address the morphic wholes as systems, through the so-called border marker. This leads to a development of the identity laws into a more ontologically and systemically complete logic, that addresses both systemic individuation and identity. The formal system, developed in section 4., is called FCat′ and it is capable of dealing with category theory, n-category theory and a different class of structures that cross systemic levels, which are more complex hierarchical structures than the ones worked upon in n-category theory.

In section 5., we address the logical semantics. In section 6. we conclude with a few final remarks.

http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1396841