Saving proof from paradox: Gödel’s paradox and the inconsistency of informal mathematics

In this paper I shall consider two related avenues of argument that have been used to make the case for the inconsistency of mathematics: firstly, Gödel’s paradox which leads to a contradiction within mathematics and, secondly, the incompatibility of completeness and consistency established by Gödel’s incompleteness theorems. By bringing in considerations from the philosophy of mathematical practice on informal proofs, I suggest that we should add to the two axes of completeness and consistency a third axis of formality and informality. I use this perspective to respond to the arguments for the inconsistency of mathematics made by Beall and Priest, presenting problems with the assumptions needed concerning formalisation, the unity of informal mathematics and the relation between the formal and informal

The Semiotics of Spider Diagrams

Spider diagrams are based on Euler and Venn/Peirce diagrams, forming a system which is as expressive as monadic first orderlogic with equality. Rather than being primarily intended for logicians,spider diagrams were developed at the end of the 1990s in the context of visual modelling and software specification. We examine the original goals of the designers, the ways in which the notation has evolved and itsconnection with the philosophical origins of the logical diagrams of Euler, Venn and Peirce on which spider diagrams are based. Using Peirce's concepts and classification of signs, we analyse the ways in which different sign types are exploited in the notation. Our hope is that this analysis may be of interest beyond those readers particularly interested in spider diagrams, and act as a case study in deconstructing a simple visual logic. Along the way, we discuss the need for a deeper semiotic engagement in visual modelling

Information systems framework for enterprise agility

Modern day enterprises operate and transact in an increasingly dynamic business environment. As a result, they are vulnerable to spontaneous changes and uncertainties. These usually reduce effectiveness and optimal performances in enterprises, and can have negative impacts such as loss of competitiveness, and bankruptcy. Enterprise agility, i.e., the ability of enterprises to respond to changes, is a core imperative for effective change management. Yet, it is challenging, difficult to achieve, and a major concern for corporate executives. Enterprises would thus require novel approaches to manage changes and enhance agility. In order to facility or achieve enterprise agility, it would be necessary and vital to develop frameworks or processes that can support effective change management. Such frameworks or processes should include techniques for modelling enterprises changes explicitly, so as to enhance the understanding of how changes relate to or affect enterprises. In addition, there should be techniques for deriving the elements of an enterprise, e.g., business process and data entities, that are required to adapt a given enterprise change. However, concepts, constructs, and techniques for representing changes are often neglected, if available at all, in the existing enterprise modelling approaches such as TOGAF and ZACHMAN. This contributes to the difficulty in applying the existing enterprise modelling approaches to enhance enterprise agility and effective change management. The work described in this thesis provides a novel approach for supporting enterprise agility and change management. Therefore, this thesis contributes a conceptual process or framework for representing enterprise changes, and deriving enterprise elements such as data entity, business goal, and business process required to adapt a given change. Other contributions made by this approach include a novel conceptual modelling language for representing enterprise changes, an enterprise modelling language, and a set of procedures and rules that can be used to derive the new domain elements required to adapt changes. An industry case study has been used to test the utility of this framework. The results obtained from this case study shows that this framework supports enterprise agility and change management in a number of ways

Enriching enterprise resource planning systems for strategic advantage: a semiotic motivated approach

Enterprise Resource Planning is often endorsed as a means to facilitate strategic advantage for businesses. The scarcity of resources is the method by which some businesses maintain their position. However, the ubiquitous trend towards the adoption of Enterprise Resourcing Planning systems coupled with market saturation makes the promise of advantage less compelling. Reported in this paper is a proposed solution based upon semiotic theory that takes a typical Enterprise Resource Planning deployment scenario and shapes it according to the needs of people in post-implementation contexts to leverage strategic advantage in different ways

String Diagrams for λ\lambda-calculi and Functional Computation

This tutorial gives an advanced introduction to string diagrams and graph languages for higher-order computation. The subject matter develops in a principled way, starting from the two dimensional syntax of key categorical concepts such as functors, adjunctions, and strictification, and leading up to Cartesian Closed Categories, the core mathematical model of the lambda calculus and of functional programming languages. This methodology inverts the usual approach of proceeding from syntax to a categorical interpretation, by rationally reconstructing a syntax from the categorical model. The result is a graph syntax -- more precisely, a hierarchical hypergraph syntax -- which in many ways is shown to be an improvement over the conventional linear term syntax. The rest of the tutorial focuses on applications of interest to programming languages: operational semantics, general frameworks for type inference, and complex whole-program transformations such as closure conversion and automatic differentiation