This thesis explores the various forms of reasoning that are associated with diagrams. It does this by a logical analysis of diagrammatic symbols. The thesis is divided into three sections dealing with different aspects of diagrammatic logic. They are (1) The relevance of diagrammatic symbols and their role in logic, (2) Methods of formalizing diagrammatic symbols, such as subway maps and Peirce's Existential Graphs through the means of Graph theory, (3) The conception of inference in diagrammatic logic systems.

This thesis explores the various forms of reasoning that are associated with diagrams. It does this by a logical analysis of diagrammatic symbols. The thesis is divided into three sections dealing with different aspects of diagrammatic logic. They are (1) The relevance of diagrammatic symbols and their role in logic, (2) Methods of formalizing diagrammatic symbols, such as subway maps and Peirce's Existential Graphs through the means of Graph theory, (3) The conception of inference in diagrammatic logic systems.

This thesis is on the automation of diagrammatic proofs, a novel approach to mechanised mathematical reasoning. Theorems in automated theorem proving are usually proved by formal logical proofs. However, there are some conjectures which humans can prove by the use of geometric operations on diagrams that somehow represent these conjectures, so called diagrammatic proofs. Insight is often more clearly perceived in these diagrammatic proofs than in the algebraic proofs. We are investigating and automating such diagrammatic reasoning about mathematical theorems. Concrete rather than general diagrams are used to prove ground instances of a universally quantified theorem. The diagrammatic proof in constructed by applying geometric operations to the diagram. These operations are in the inference steps of the proof. A general schematic proof is extracted from the ground instances of a proof. it is represented as a recursive program that consists of a general number of applications of geometric operations. When gien a particular diagram, a schematic proof generates a proof for that diagram. To verify that the schematic proof produces a correct proof of the conjecture for each ground instance we check its correctness in a theory of diagrams. We use the constructive omega-rule and schematic proofs to make a translation from concrete instances to a general argument about the diagrammatic proof. The realisation of our ideas is a diagrammatic reasoning system DIAMOND. DIAMOND allows a user to interactively construct instances of a diagrammatic proof. It then automatically abstracts these into a general schematic proof and checks the correctness of this proof using an inductive theorem prover.

This project looks at using diagrammatic reasoning to prove mathematical theorems. The work is motivated by a need for theorem provers whose reasoning is readily intelligible to human beings. It should also have practical applications in mathematics teaching. We focus on the continuous domain of analysis - a geometric subject, but one which is taught using a dry algebraic formalism which many students find hard. The geometric nature of the domain makes it suitable for a diagram-based approach. However it is a difficult domain, and there are several problems, including handling alternating quantifiers, sequences and generalisation. We developed representations and reasoning methods to solve these. Our diagram logic isn't complete, but does cover a reasonable range of theorems. It utilises computers to extend diagrammatic reasoning in new directions – including using animation. This work is tested for soundness, and evaluated empirically for ease of use. We demonstrate that computerised diagrammatic theorem proving is not only possible in the domain of real analysis, but that students perform better using it than with an equivalent algebraic computer system.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / This dissertation addresses the features of diagrammatic reasoning from the perspective of syllogistics. With the emergence of modern logic, diagrams where not considered as legitimate elements of decision methods, operating only as illustrative and heuristic tools. More recently there were questioning for what characteristics of diagrammatic methods are distinguished from purely sentential methods and how these distinctive features determine the possible misleading and inaccurate character of diagrams. Exploring this debate within syllogistics, I show that diagrammatic methods are more complex and dynamic systems than they appear for two reasons. On the one hand, diagrammatic systems are distinguished from sentential systems by their semiotic constitution. A diagram uses a spatial relationship to represent some aspect of the logical domain, while sentential systems uses a symbol for represent the same aspect. On the other hand, in order to generate an isomorphic representation with this spatial relation, diagrammatic reasoning involves substantial cognitive and perceptual capabilities that provide advantages for some utilities. / A presente dissertação versa sobre as características do raciocínio diagramático a partir da lógica silogística. No surgimento da lógica moderna diagramas foram descartados enquanto legítimos elementos de métodos de decisão, operando apenas como ferramentas ilustrativas e heurísticas. Mais recentemente houve questionamento por qual razão métodos diagramáticos seriam distintos de métodos puramente sentenciais e como essas características distintivas determinariam o caráter possivelmente enganoso e pouco preciso dos diagramas. Explorando esse debate a partir da silogística, mostramos que métodos diagramáticos são sistemas mais complexos e dinâmicos do que aparentam em dois sentidos. Por um lado, sistemas diagramáticos distinguem-se de sistemas sentencias pela sua constituição semiótica, na medida em que utilizam uma relação espacial para representar algum aspecto do domínio lógico, enquanto que sistemas sentenciais utilizam um símbolo para representar o mesmo aspecto. Por outra via, ao utilizar essa propriedade para gerar uma representação isomórfica, o raciocínio diagramático envolve substancialmente capacidades cognitivas e perceptuais que proporcionam vantagens para determinadas utilidades.

Professional writing students must analyze communications in multiple modalities, on page or screen. This project argues that student analysts benefit from using articulated heuristics, summaries of articles, books, or theories in chart form that remain in the visual field with the communication to be analyzed. Keeping the heuristic in view reduces students' cognitive load by narrowing the search for solution to the categories in the heuristic. These heuristics, often one page or one screen, contain key words, phrases, or questions that allow students to approach analysis from experts' points of view at more than one level of complexity. Students locate instantiations of the categories in the communication analyzed, incorporating the category/instantiation pairs into personal schemas for analysis. As students classify communications, relate parts together and to other communications, and perform operations on the content, they see how communication achieves its meaning and formulate appropriate responses. Rather than rely on one all-purpose heuristic, this dissertation presents a range of heuristics reflecting rhetorical, discourse, linguistic, usability, and visual strategies that enable students to critique both form and function in communication. The heuristics reflect a systematically ordered workplace context, articulate an appropriate and specific theory for the situation, interface with other heuristic systems for depth and efficacy, and instantiate the categories at some helpful secondary level of complexity. To theorize the visual nature of the heuristic chart displays, I employ the semiotic of Charles Sanders Peirce, working through the implications of chart construction as I diagram Peirce's theory of diagrammatic iconicity.

Science, Technology, Engineering, and Mathematics (STEM) experts employ many representations that novices find hard to use because they require a critical STEM skill, interpreting two-dimensional (2D) diagrams that represent three-dimensional (3D) information. The current research focuses on learning to interpret topographic maps. Understanding topographic maps requires knowledge of how to interpret the conventions of contour lines, and skill in visualizing that information in 3D (e.g. shape of the terrain). Novices find both tasks difficult. The present study compared two interventions designed to facilitate understanding for topographic maps to minimal text-only instruction. The 3D Visualization group received instruction using 3D gestures and models to help visualize three topographic forms. The Pattern Identification group received instruction using pointing and tracing gestures to help identify the contour patterns associated with the three topographic forms. The Text-based Instruction group received only written instruction explaining topographic maps. All participants then completed a measure of topographic map use. The Pattern Identification group performed better on the map use measure than participants in the Text-based Instruction group, but no significant difference was found between the 3D Visualization group and the other two groups. These results suggest that learning to identify meaningful contour patterns is an effective strategy for learning how to comprehend topographic maps. Future research should address if learning strategies for how to interpret the information represented on a diagram (e.g. identify patterns in the contour lines), before trying to visualize the information in 3D (e.g. visualize the 3D structure of the terrain), also facilitates students' comprehension of other similar types of diagrams. / Psychology

Model Driven Engineering (MDE) has gained a considerable attention in the software engineering domain in the past decade. MDE proposes shifting the focus of the engineers from concrete artifacts (e.g., code) to more abstract structures (i.e., models). Such a change allows using the human intelligence more efficiently in engineering software products. Model Transformation (MT) is one of the key operations in MDE and plays a critical role in its successful application. The current MT approaches, however, usually miss either one or both of the two essential features: 1) declarativity in the sense that the MT definitions should be expressed at a sufficiently high level of abstraction, and 2) formality in the sense that the approaches should be based on precise underlying semantics. These two features are both critical in effectively managing the complexity of a network of interrelated models in an MDE process. This thesis tackles these shortcomings by promoting a declarative MT approach that is built on mathematical foundations. The approach is called Query Structured Transformation (QueST) as it proposes a structured orchestration of diagrammatic queries in the MT definitions. The aim of the thesis is to make the QueST approach –that is based on formal foundations– accessible to the MDE community. This thesis first motivates the necessity of having declarative formal approaches by studying the variety of model synchronization scenarios in the networks of interrelated models. Then, it defines a diagrammatic query framework (DQF) that formulates the syntax and the semantics of the QueST collection-level diagrammatic operations. By a detailed comparison of the QueST approach and three rule-based MT approaches (ETL, ATL, and QVT-R), the thesis shows the way QueST contributes to the development of the following aspects of MT definitions: declarativity, modularity, incrementality, and logical analysis of MT definitions. / Thesis / Doctor of Philosophy (PhD)

Por um lado, visualiza-se a ciência química, sua bela história, conceitos e teorias que podem levar a inúmeros questionamentos, reflexões e análises, de outro lado, a realidade nos revela um ensino que, na maioria das escolas, prioriza fórmulas, teorias descontextualizadas, modelos de limitada compreensão, em meio a um currículo com necessidade de discussões e revisão, bem como um enorme desestímulo por parte dos discentes. Neste sentido, acreditamos que a Filosofia da Química pode fornecer subsídios teóricos para uma contextualização, sistematização, organização, entendimento e transposição didática dos conteúdos. O desenvolvimento de uma Filosofia da Química (FQ) incorporada ao ensino ainda não foi proposto e o currículo da disciplina também continua sem transmitir o pluralismo que representa, sendo que a práxis química está descontextualizada da história, dos valores, do pré-científico e da pré-química. Consideramos a necessidade de avaliar problemas no ensino e possibilidades de constituir uma ligação entre a Filosofia da Química e a educação em química, buscando material bibliográfico que fornecesse subsídios para essa inserção. O trabalho iniciou com pesquisa bibliográfica sobre a Filosofia da Química, seus autores, campos de discussão e perspectivas. Os problemas no ensino que podem ser beneficiados com a Filosofia da Química, destacados nos primeiros artigos, foram decisivos para que novas estratégias pudessem ser alcançadas, e, como uma influência no desenvolvimento desta pesquisa, considerou-se cada vez mais o valor de se investigar a linguagem utilizada na representação de átomo, conceito fundamental na educação em química. Reconhecer a representação de átomo e molécula como essenciais no ensino da química, transitando entre a linguagem diagramática e a discursiva, manteve-se como um postulado em nossas pesquisas. Como uma consequência dessa escolha verificou-se a necessidade de estudar a diagramaticidade, com bases teóricas na Filosofia da Química. / On the one hand, we see the chemical science, its beautiful history, concepts and theories that can lead to numerous questions, reflections and analysis, on the other hand, the reality reveals a teaching that, in most schools, prioritizes formulas, theories decontextualized, limited understanding models beside a curriculum requiring discussion and review, as well as a bigunstimulated he part of students. In this way, we believe that the chemistry of Philosophy can provide theoretical support for context, organization, systematization, understanding and didactic transposition of the contents. The development a Philosophy of Chemistry (PC) incorporated into the chemical teaching was not proposed and the curriculum of the course also continues without transmitting the pluralism that is, with chemical praxis is decontextualized history, the values, the pre-scientific and prechemistry. We consider the need of to evaluate problems in education and possibilities to establish a link between the philosophy of chemistry and education in chemistry searching for bibliography that provide subsidies for this insertion. The work began with bibliographic research on the philosophy of chemistry, its authors, discussion fields and perspectives. The problems in education that may benefit from the Philosophy of Chemistry, highlighted in the first articles were decisive for that new strategies could be achieved, and as an influence on the development of this research, is increasingly considered the value of investigating language used in the atom representation, fundamental concept in education in chemistry. Tore cognize the atom and molecule representation as essential in chemistry teaching, moving between the diagrammatic language and discourse, remained as a postulate in our research. As a consequence of this choice there is a need to study the diagramaticidade with theoretical basis foundations in the Chemical Philosophy.

Two applications of Category Theory are considered. The link between them is applications to Physics and more specifically to Entropy. The first research chapter is broader in scope and not explicitly about Physics, although connections to Statistical Mechanics are made towards the end of the chapter. Matroids are abstract structures that describe dependence, and strong maps are certain structure-preserving functions between them with desirable properties. We examine properties of various categories of matroids and strong maps: we compute limits and colimits; we find free and cofree constructions of various subcategories; we examine factorisation structures, including a translation principle from geometric lattices; we find functors with convenient properties to/from vector spaces, multisets of vectors, geometric lattices, and graphs; we determine which widely used operations on matroids are functorial (these include deletion, contraction, series and parallel connection, and a simplification monad); lastly, we find a categorical characterisation of the greedy algorithm. In conclusion, this project determines which aspects of Matroid Theory are most and least conducive to categorical treatment. The purpose of the second research chapter is to provide a categorical framework for generalising and unifying notions of Entropy in various settings, exploiting the fact that Entropy is a monotone subadditive function. A categorical characterisation of Entropy through a category of thermodynamical systems and adiabatic processes is found. A modelling perspective (adiabatic categories) that directly generalises an existing model is compared to an axiomatisation through topological and linear structures (topological weak semimodules), where the latter is based on a categorification of semimodules. Properties of each class of categories are examined; most notably a cancellation property of adiabatic categories generalising an existing result, and an adjunction between the categories of weak semimodules and symmetric monoidal categories. An adjunction between categories of adiabatic categories and topological weak semimodules is found. We examine in which cases each of these classes of categories constitutes a traced monoidal category. Lastly, examples of physical applications are provided. In conclusion, this project uncovers a way of, and makes progress towards, retrieving the statistical formulation of Entropy from simple axioms.