Probabilistic Arguments in Mathematics

This thesis addresses a question that emerges naturally from some observations about contemporary mathematical practice. Firstly, mathematicians always demand proof for the acceptance of new results. Secondly, the ability of mathematicians to tell if a discourse gives expression to a proof is less than perfect, and the computers they use are subject to a variety of hardware and software failures. So false results are sometimes accepted, despite insistence on proof. Thirdly, over the past few decades, researchers have also developed a variety of methods that are probabilistic in nature. Even if carried out perfectly, these procedures only yield a conclusion that is very likely to be true. In some cases, these chances of error are precisely specifiable and can be made as small as desired. The likelihood of an error arising from the inherently uncertain nature of these probabilistic algorithms can therefore be made vanishingly small in comparison to the chances of an error arising when implementing an equivalent deductive algorithm. Moreover, the structure of probabilistic algorithms tends to minimise these Implementation Errors too. So overall, probabilistic methods are sometimes more reliable than deductive ones. This invites the question: ‘Are mathematicians rational in continuing to reject these probabilistic methods as a means of establishing mathematical claims?

Proceedings of the 12th International Conference on Technology in Mathematics Teaching ICTMT 12

Innovation, inclusion, sharing and diversity are some of the words that briefly and suitably characterize the ICTMT series of biennial international conferences – the International Conference on Technology in Mathematics Teaching. Being the twelfth of a series which began in Birmingham, UK, in 1993, under the influential enterprise of Professor Bert Waits from Ohio State University, this conference was held in Portugal for the first time. The 12th International Conference on Technology in Mathematics Teaching was hosted by the Faculty of Sciences and Technology of the University of Algarve, in the city of Faro, from 24 to 27 June 2015, and was guided by the original spirit of its foundation. The integration of digital technologies in mathematics education across school levels and countries, from primary to tertiary education, together with the understanding of the phenomena involved in the teaching and learning of mathematics in technological environments have always been driving forces in the transformation of pedagogical practices. The possibility of joining at an international conference a wide diversity of participants, including school mathematics teachers, lecturers, mathematicians, mathematics educators and researchers, software designers, and curriculum developers, is one facet that makes this conference rather unique. At the same time, it seeks to foster the sharing of ideas, experiences, projects and studies while providing opportunities to try-out and assess tools or didactical proposals during times of hands-on work. The ICTMT 12 had this same ambition, when embracing and welcoming just over 120 delegates who actively and enthusiastically contributed to a very packed program of scientific proposals and sessions on various topics

Formalizing graphical notations

The thesis describes research into graphical notations for software engineering, with a principal interest in ways of formalizing them. The research seeks to provide a theoretical basis that will help in designing both notations and the software tools that process them. The work starts from a survey of literature on notation, followed by a review of techniques for formal description and for computational handling of notations. The survey concentrates on collecting views of the benefits and the problems attending notation use in software development; the review covers picture description languages, grammars and tools such as generic editors and visual programming environments. The main problem of notation is found to be a lack of any coherent, rigorous description methods. The current approaches to this problem are analysed as lacking in consensus on syntax specification and also lacking a clear focus on a defined concept of notated expression. To address these deficiencies, the thesis embarks upon an exploration of serniotic, linguistic and logical theory; this culminates in a proposed formalization of serniosis in notations, using categorial model theory as a mathematical foundation. An argument about the structure of sign systems leads to an analysis of notation into a layered system of tractable theories, spanning the gap between expressive pictorial medium and subject domain. This notion of 'tectonic' theory aims to treat both diagrams and formulae together. The research gives details of how syntactic structure can be sketched in a mathematical sense, with examples applying to software development diagrams, offering a new solution to the problem of notation specification. Based on these methods, the thesis discusses directions for resolving the harder problems of supporting notation design, processing and computer-aided generic editing. A number of future research areas are thereby opened up. For practical trial of the ideas, the work proceeds to the development and partial implementation of a system to aid the design of notations and editors. Finally the thesis is evaluated as a contribution to theory in an area which has not attracted a standard approach

Proof and Proving in Mathematics Education

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Fourth NASA Langley Formal Methods Workshop

This publication consists of papers presented at NASA Langley Research Center's fourth workshop on the application of formal methods to the design and verification of life-critical systems. Topic considered include: Proving properties of accident; modeling and validating SAFER in VDM-SL; requirement analysis of real-time control systems using PVS; a tabular language for system design; automated deductive verification of parallel systems. Also included is a fundamental hardware design in PVS