From Euclidean Geometry to Knots and Nets

This document is the Accepted Manuscript of an article accepted for publication in Synthese. Under embargo until 19 September 2018. The final publication is available at Springer via https://doi.org/10.1007/s11229-017-1558-x.This paper assumes the success of arguments against the view that informal mathematical proofs secure rational conviction in virtue of their relations with corresponding formal derivations. This assumption entails a need for an alternative account of the logic of informal mathematical proofs. Following examination of case studies by Manders, De Toffoli and Giardino, Leitgeb, Feferman and others, this paper proposes a framework for analysing those informal proofs that appeal to the perception or modification of diagrams or to the inspection or imaginative manipulation of mental models of mathematical phenomena. Proofs relying on diagrams can be rigorous if (a) it is easy to draw a diagram that shares or otherwise indicates the structure of the mathematical object, (b) the information thus displayed is not metrical and (c) it is possible to put the inferences into systematic mathematical relation with other mathematical inferential practices. Proofs that appeal to mental models can be rigorous if the mental models can be externalised as diagrammatic practice that satisfies these three conditions.Peer reviewe

Learning interaction patterns using diagrams varying in level and type of interactivity

An experiment was conducted to investigate the differences between learners when using computer based learning environments (CBLEs) that incorporated different levels of interactivity in diagrams. Four CBLEs were created with combinations of the following two interactivity properties: (a) the possibility to rotate the whole diagram (b) the possibility to move individual elements of the diagram in order to apprehend the relationships between them. We present and discuss the qualitative findings from the study in terms of the learners’ interaction patterns and their relevance for the understanding of performance scores. This supports our previous quantitative analysis showing an interaction between cognitive abilities and interactivity. Based on our findings we reflect on the possibilities to inform CBLEs with relevant information regarding learners’ cognitive abilities and representational preferences

A Diagram Is Worth A Dozen Images

Diagrams are common tools for representing complex concepts, relationships and events, often when it would be difficult to portray the same information with natural images. Understanding natural images has been extensively studied in computer vision, while diagram understanding has received little attention. In this paper, we study the problem of diagram interpretation and reasoning, the challenging task of identifying the structure of a diagram and the semantics of its constituents and their relationships. We introduce Diagram Parse Graphs (DPG) as our representation to model the structure of diagrams. We define syntactic parsing of diagrams as learning to infer DPGs for diagrams and study semantic interpretation and reasoning of diagrams in the context of diagram question answering. We devise an LSTM-based method for syntactic parsing of diagrams and introduce a DPG-based attention model for diagram question answering. We compile a new dataset of diagrams with exhaustive annotations of constituents and relationships for over 5,000 diagrams and 15,000 questions and answers. Our results show the significance of our models for syntactic parsing and question answering in diagrams using DPGs

Diagrammatic Reasoning and Modelling in the Imagination: The Secret Weapons of the Scientific Revolution

Just before the Scientific Revolution, there was a "Mathematical Revolution", heavily based on geometrical and machine diagrams. The "faculty of imagination" (now called scientific visualization) was developed to allow 3D understanding of planetary motion, human anatomy and the workings of machines. 1543 saw the publication of the heavily geometrical work of Copernicus and Vesalius, as well as the first Italian translation of Euclid

What is a logical diagram?

Robert Brandom’s expressivism argues that not all semantic content may be made fully explicit. This view connects in interesting ways with recent movements in philosophy of mathematics and logic (e.g. Brown, Shin, Giaquinto) to take diagrams seriously - as more than a mere “heuristic aid” to proof, but either proofs themselves, or irreducible components of such. However what exactly is a diagram in logic? Does this constitute a semiotic natural kind? The paper will argue that such a natural kind does exist in Charles Peirce’s conception of iconic signs, but that fully understood, logical diagrams involve a structured array of normative reasoning practices, as well as just a “picture on a page”

A cognitive exploration of the “non-visual” nature of geometric proofs

Why are Geometric Proofs (Usually) “Non-Visual”? We asked this question as a way to explore the similarities and differences between diagrams and text (visual thinking versus language thinking). Traditional text-based proofs are considered (by many to be) more rigorous than diagrams alone. In this paper we focus on human perceptual-cognitive characteristics that may encourage textual modes for proofs because of the ergonomic affordances of text relative to diagrams. We suggest that visual-spatial perception of physical objects, where an object is perceived with greater acuity through foveal vision rather than peripheral vision, is similar to attention navigating a conceptual visual-spatial structure. We suggest that attention has foveal-like and peripheral-like characteristics and that textual modes appeal to what we refer to here as foveal-focal attention, an extension of prior work in focused attention

Using patterns in the automatic marking of ER-Diagrams

This paper illustrates how the notion of pattern can be used in the automatic analysis and synthesis of diagrams, applied particularly to the automatic marking of ER-diagrams. The paper describes how diagram patterns fit into a general framework for diagram interpretation and provides examples of how patterns can be exploited in other fields. Diagram patterns are defined and specified within the area of ER-diagrams. The paper also shows how patterns are being exploited in a revision tool for understanding ER-diagrams

Form, science, and narrative in the anthropocene

A significant strand of contemporary fiction engages with scientific models that highlight a constitutive interdependency between humanity and material realities such as the climate or the geological history of our planet. This article looks at the ways in which narrative may capture this human-nonhuman interrelation, which occupies the foreground of debates on the so-called Anthropocene. I argue that the formal dimension of scientific knowledge-as manifested by diagrams or metaphors used by scientists-is central to this narrative remediation. I explore two analogical strategies through which narrative may pursue a formal dialogue with science: clusters of metaphorical language and the global structuring of the plot. Rivka Galchen's novel Atmospheric Disturbances (2008), for instance, builds on a visual representation of meteorological patterns in a storm (lifted from an actual scientific paper) to stage the narrator's mental illness. Two other contemporary works (Orfeo by Richard Powers and A Tale for the Time Being by Ruth Ozeki) integrate scientific models through the overall design of the plot. By offering close readings of these novels, I seek to expand work in the area of New Formalism and show how formal choices are crucial to bringing together the human-scale world and more-than-human phenomena

Experiments in the automatic marking of ER-Diagrams

In this paper we present an approach to the computer understanding of diagrams and show how it can be successfully applied to the automatic marking (grading) of student attempts at drawing entity-relationship (ER) diagrams. The automatic marker has been incorporated into a revision tool to enable students to practice diagramming and obtain feedback on their attempts