What is Diagrammatic Reasoning in Mathematics?

In recent years, epistemological issues connected with the use of diagrams and visualization in mathematics have been a subject of increasing interest. In particular, it is open to dispute what role diagrams play in justifying mathematical statements. One of the issues that may appear in this context is: what is the character of reasoning that relies in some way on a diagram or visualization and in what way is it distinct from other types of reasoning in mathematics? In this paper it is proposed to distinguish between several ways of using visualization or diagrams in mathematics, each of which could be connected with a different concept of diagrammatic/visual reasoning. Main differences between those types of reasoning are also hinted at. A distinction between visual (diagrammatic) reasoning and visual (diagrammatic) thinking is also considered

Hacia la lógica plástica: emergencia de la lógica del razonamiento visual

Se presenta, define y aplica una red de conceptos tendente a elucidar la noción de consecuencia lógica sobre cuerpos de información no verbalmente codificada. Los problemas y nociones fundacionales de lógicas diagramáticas se discuten críticamente y se generalizan a patrones de razonamiento plástic

Raisonner avec des diagrammes : perspectives cognitives et computationnelles

31 pagesInternational audienceDiagrammatic, analogical or iconic representations are often contrasted with linguistic or logical representations, in which the shape of the symbols is arbitrary. The aim of this paper is to make a case for the usefulness of diagrams in inferential knowledge representation systems. Although commonly used, diagrams have for a long time suffered from the reputation of being only a heuristic tool or a mere support for intuition. The first part of this paper is an historical background paying tribute to the logicians, psychologists and computer scientists who put an end to this formal prejudice against diagrams. The second part is a discussion of their characteristics as opposed to those of linguistic forms. The last part is aimed at reviving the interest for heterogeneous representation systems including both linguistic and diagrammatic representations

Information enforcement in learning with graphics : improving syllogistic reasoning skills

This thesis is an investigation into the factors that contribute to good choices among graphical systems used in teaching, and the feasibility of implementing teaching software that uses this knowledge.The thesis describes a mathematical metric derived from a cognitive theory of human diagram processing. The theory characterises differences among representations by their ability to express information. The theory provides the factors and relationships needed to build the metric. It says that good representations are easily processed because they are more vivid, more tractable and less expressive, than poor representations.The metric is applied to abstract systems for teaching and learning syllogistic reasoning, TARSKI'S WORLD, EULER CIRCLES, VENN DIAGRAMS and CARROLL'S GAME OF LOGIC. A rank ordering reflects the value of each system predicted by the theory and the metric. The theory, the metric and the systems are then tested in empirical studies. Five studies involving sixty-eight learners, examined the benefit of software based on these abstract systems.Studies showed the theory correctly predicted learners' success with the circle systems and poorer performance with TARSKI'S WORLD. The metric showed small but clear differences in expressivity between the circle systems. Differences between results of the learners using the circle systems contradicted the predictions of the metric.Learners with mathematical training were better equipped and more successful at learning syllogistic reasoning with the systems. Performance of learners without mathematical training declined after using the software systems. Diagrams drawn by learners together with video footage collected during problem solving, led to a catalogue of errors, misconceptions and some helpful strategies for learning from graphical systems.A cognitive style test investigated the poor performance of non-mathematically trained learners. Learners with mathematics training showed serialist and versatile learning styles while learners without this training showed a holist learning style. This is consistent with the hypothesis that non-mathematically trained learners emphasise the use of semantic cues during learning and problem solving.A card-sorting task investigated learners' preferences for parts of the graphical lexicon used in the diagram systems. Preferences for the EULER lexicon increased difficulty in explaining the system's poor results in earlier studies. Video footage of learners using the systems in the final study illustrated useful learning strategies and improved performance with EULER while individual instruction was available.Further work describes a preliminary design for an adaptive syllogism tutor and other related work

Technological roadmap on AI planning and scheduling

At the beginning of the new century, Information Technologies had become basic and indispensable constituents of the production and preparation processes for all kinds of goods and services and with that are largely influencing both the working and private life of nearly every citizen. This development will continue and even further grow with the continually increasing use of the Internet in production, business, science, education, and everyday societal and private undertaking. Recent years have shown, however, that a dramatic enhancement of software capabilities is required, when aiming to continuously provide advanced and competitive products and services in all these fast developing sectors. It includes the development of intelligent systems – systems that are more autonomous, flexible, and robust than today’s conventional software. Intelligent Planning and Scheduling is a key enabling technology for intelligent systems. It has been developed and matured over the last three decades and has successfully been employed for a variety of applications in commerce, industry, education, medicine, public transport, defense, and government. This document reviews the state-of-the-art in key application and technical areas of Intelligent Planning and Scheduling. It identifies the most important research, development, and technology transfer efforts required in the coming 3 to 10 years and shows the way forward to meet these challenges in the short-, medium- and longer-term future. The roadmap has been developed under the regime of PLANET – the European Network of Excellence in AI Planning. This network, established by the European Commission in 1998, is the co-ordinating framework for research, development, and technology transfer in the field of Intelligent Planning and Scheduling in Europe. A large number of people have contributed to this document including the members of PLANET non- European international experts, and a number of independent expert peer reviewers. All of them are acknowledged in a separate section of this document. Intelligent Planning and Scheduling is a far-reaching technology. Accepting the challenges and progressing along the directions pointed out in this roadmap will enable a new generation of intelligent application systems in a wide variety of industrial, commercial, public, and private sectors

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