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The Diagram Problem
Abstract
The goal of the paper is to argue against the claim that thoughts can be modelled as having a diagram-like structure. The argument has a form of the so-called Diagram Puzzle, according to which the same features make diagrams cognitively reliable (and desirable) and unreliable (and non-desirable). I argue that to solve the Puzzle we have to accept the instrumental interpretation of diagrams, according to which diagrams are instruments of reasoning comparable to calculators. Instrumental view on the nature of diagrams leads to the problem of content determination: the claim that instruments can determine thoughts’ content, entails that, for example, a calculation carried out with fingers has a different content that the same calculation carried out with abacus. If instruments do not determine content, they can be seen as instruments that reveal the content of thoughts, but they do not change the thoughts content. I argue that diagrams are epiphenomenal which means that they cannot influence the thought’s content. Therefore, we can think with the help of diagrams, but it does not follow that thoughts have a diagram-like nature.
In the current outlook of education, collaborative learning has been introduced as a strategy to build knowledge, in a multimodal culture where information is expressed, produced and distributed through new formats and representational languages. Regarding the latter, the diagram has been used as a way to represent and organize information in learning processes, making its comprehension necessary, since there are different interpretations from the theory. That is why, through this study of theoretical design, a systematic review of the literature (SRL) was carried out, which it allowed to get information about the concepts of diagram and its connotation in collaborative learning, taking into account the definitions stated about it. The results show that the diagram is a representational instrument and not only a graphic or textual organizer, but also, a semiotic-social mental artifact, a semiotic-social mental artifact that mediates meanings and regulates their construction in subjects' collaborative learning interactions. Finally, the role of the diagram as a potentiator in the development of metacognitive skills is highlighted.
This paper explores the question of what makes diagrammatic representations effective for human logical reasoning, focusing on how Euler diagrams support syllogistic reasoning. It is widely held that diagrammatic representations aid intuitive understanding of logical reasoning. In the psychological literature, however, it is still controversial whether and how Euler diagrams can aid untrained people to successfully conduct logical reasoning such as set-theoretic and syllogistic reasoning. To challenge the negative view, we build on the findings of modern diagrammatic logic and introduce an Euler-style diagrammatic representation system that is designed to avoid problems inherent to a traditional version of Euler diagrams. It is hypothesized that Euler diagrams are effective not only in interpreting sentential premises but also in reasoning about semantic structures implicit in given sentences. To test the hypothesis, we compared Euler diagrams with other types of diagrams having different syntactic or semantic properties. Experiment compared the difference in performance between syllogistic reasoning with Euler diagrams and Venn diagrams. Additional analysis examined the case of a linear variant of Euler diagrams, in which set-relationships are represented by one-dimensional lines. The experimental results provide evidence supporting our hypothesis. It is argued that the efficacy of diagrams in supporting syllogistic reasoning crucially depends on the way they represent the relational information contained in categorical sentences.
Most designers know that yellow text presented against a blue background reads clearly and easily, but how many can explain why, and what really are the best ways to help others and ourselves clearly see key patterns in a bunch of data? This book explores the art and science of why we see objects the way we do. Based on the science of perception and vision, the author presents the key principles at work for a wide range of applications--resulting in visualization of improved clarity, utility, and persuasiveness. The book offers practical guidelines that can be applied by anyone: interaction designers, graphic designers of all kinds (including web designers), data miners, and financial analysts.
When people talk, they gesture. We show that gesture introduces action information into speakers' mental representations, which, in turn, affect subsequent performance. In Experiment 1, participants solved the Tower of Hanoi task (TOH1), explained (with gesture) how they solved it, and solved it again (TOH2). For all participants, the smallest disk in TOH1 was the lightest and could be lifted with one hand. For some participants (no-switch group), the disks in TOH2 were identical to those in TOH1. For others (switch group), the disk weights in TOH2 were reversed (so that the smallest disk was the heaviest and could not be lifted with one hand). The more the switch group's gestures depicted moving the smallest disk one-handed, the worse they performed on TOH2. This was not true for the no-switch group, nor for the switch group in Experiment 2, who skipped the explanation step and did not gesture. Gesturing grounds people's mental representations in action. When gestures are no longer compatible with the action constraints of a task, problem solving suffers.
In these studies I examine the role of distributed cognition in problem solving. The major hypothesis explored is that intelligent behavior results from the interaction of internal cognition, external objects, and other people, where a cognitive task can be distributed among a set of representations, some internal and some external. The Tower of Hanoi problem is used as a concrete example for these studies. In Experiment 1 I examine the effects of the distribution of internal and external representations on problem solving behavior. Experiments 2 and 3 focus on the effects of the structural change of a problem on problem solving behavior and how these effects depend on the nature of the representations. The results of all studies show that distributed cognitive activities are produced by the interaction among the internal and external representations. External representations are not simply peripheral aids. They are an indispensable part of cognition. Two of the factors determining the...
In this article, I propose an operational framework for diagrams. According to this framework, diagrams do not work like sentences, because we do not apply a set of explicit and linguistic rules in order to use them. Rather, we become able to manipulate diagrams in meaningful ways once we are familiar with some specific practice, and therefore we engage ourselves in a form of reasoning that is stable because it is shared. This reasoning constitutes at the same time discovery and justification for this discovery. I will make three claims, based on the consideration of diagrams in the practice of logic and mathematics. First, I will claim that diagrams are tools, following some of Peirce’s suggestions. Secondly, I will give reasons to drop a sharp distinction between vision and language and consider by contrast how the two are integrated in a specific manipulation practice, by means of a kind of manipulative imagination. Thirdly, I will defend the idea that an inherent feature of diagrams, given by their nature as images, is their ambiguity: when diagrams are ‘tamed’ by the reference to some system of explicit rules that fix their meaning and make their message univocal, they end up in being less powerful.
This chapter gives a detailed study of diagram-based reasoning in Euclidean plane geometry (Books I, III), as well as an exploration how to characterise a geometric practice. First, an account is given of diagram attribution: basic geometrical claims are classified as exact (equalities, proportionalities) or co-exact (containments, contiguities); exact claims may only be inferred from prior entries in the demonstration text, but coexact claims may be asserted based on what is seen in the diagram. Diagram control by constructions is necessary for this to work. Case-branching occurs when a construction renders a diagram un-representative. The roles of diagrams in reductio arguments, and of objection in probing a demonstration, are discussed.
A variety of scientific disciplines have set as their task explaining mental activities, recognizing that in some way these activities depend upon our brain. But, until recently, the opportunities to conduct experiments directly on our brains were limited. As a result, research efforts were split between disciplines such as cognitive psychology, linguistics, and artificial intelligence that investigated behavior, while disciplines such as neuroanatomy, neurophysiology, and genetics experimented on the brains of non-human animals. In recent decades these disciplines integrated, and with the advent of techniques for imaging activity in human brains, the term cognitive neuroscience has been applied to the integrated investigations of mind and brain. This book is a philosophical examination of how these disciplines continue in the mission of explaining our mental capacities.
Clearly we can solve problems by thinking about them. Sometimes we have the impression that in doing so we use words, at other times diagrams or images. Often we use both. What is going on when we use mental diagrams or images? This question is addressed in relation to the more general multipronged question: what are representations, what are they for, how many different types are they, in how many different ways can they be used, and what difference does it make whether they are in the mind or on paper? The question is related to deep problems about how vision and spatial manipulation work. It is suggested that we are far from understanding what is going on. In particular we need to explain how people understand spatial structure and motion, and how we can think about objects in terms of a basic topological structure with more or less additional metrical information. I try to explain why this is a problem with hidden depths, since our grasp of spatial structure is inherently a grasp of a complex range of possibilities and their implications. Two classes of examples discussed at length illustrate requirements for human visualization capabilities. One is the problem of removing undergarments without removing outer garments. The other is thinking about infinite discrete mathematical structures, such as infinite ordinals. More questions are asked than answered.
This paper investigates the following question: when can one reliably infer the existence of an intersection point from a
diagram presenting crossing curves or lines? Two cases are considered, one from Euclid's geometry and the other from basic
real analysis. I argue for the acceptability of such an inference in the geometric case but against in the analytic case.
Though this question is somewhat specific, the investigation is intended to contribute to the more general question of the
extent and limits of reliable diagrammatic reasoning in mathematics.
We report an experimental study on the effects of diagrams on deductive reasoning with double disjunctions, for example: Raphael is in Tacoma or Julia is in Atlanta, or both. Julia is in Atlanta or Paul is in Philadelphia, or both. What follows?
We confirmed that subjects find it difficult to deduce a valid conclusion, such as Julia is in Atlanta, or both Raphael is in Tacoma and Paul is in Philadelphia.
In a preliminary study, the format of the premises was either verbal or diagrammatic, and the diagrams used icons to distinguish between inclusive and exclusive disjunctions. The diagrams had no effect on performance. In the main experiment, the diagrams made the alternative possibilities more explicit. The subjects responded faster (about 35 s) and drew many more valid conclusions (nearly 30%) from the diagrams than from the verbal premises. These results corroborate the theory of mental models and have implications for the role of diagrams in reasoning.
1. Introduction 2. Language, diagram and system 3. Hyperproof: industrial strength logic teaching 4. Euler and the syllogism: back to the age of reason 5. Why do we have to learn logic? A paradox resolved 6. Interpretation, reasoning, and communication 7. Cards and dice: assessing evidence and understanding experimenters 8. The proof of a power station is in the...diagrams and equations for practitioners 9. Involvement and detachment in learning to communicate 10. Bloor, Lakatos and the sociology of diagrams 11. Conclusions for practice 12. Conclusions for theory
The cliché that a picture is worth a thousand words vastly underestimates the expressive power of many pictures and diagrams. In this note we show that even a simple map such as the outline of Manhattan Island, accompanied by a pointer marking North, implies a vast infinity of true statements.
The paper attempts to analyze in some detail the main problems encountered in reasoning using diagrams, which may cause errors
in reasoning, produce doubts concerning the reliability of diagrams, and impressions that diagrammatic reasoning lacks the
rigour necessary for mathematical reasoning. The paper first argues that such impressions come from long neglect which led
to a lack of well-developed, properly tested and reliable reasoning methods, as contrasted with the amount of work generations
of mathematicians expended on refining the methods of reasoning with formulae and predicate calculus. Next, two main groups
of problems occurring in diagrammatic reasoning are introduced. The second group, called diagram imprecision, is then briefly
summarized, its detailed analysis being postponed to another paper. The first group, called collectively the generalization
problem, is analyzed in detail in the rest of the paper. The nature and causes of the problems from this group are explained,
methods of detecting the potentially harmful occurrences of these problems are discussed, and remedies for possible errors
they may cause are proposed. Some of the methods are adapted from similar methods used in reasoning with formulae, several
other problems constitute new, specifically diagrammatic ways of reliable reasoning.
Visual thinking - visual imagination or perception of diagrams and symbol arrays, and mental operations on them - is omnipresent in mathematics. Is this visual thinking merely a psychological aid, facilitating grasp of what is gathered by other means? Or does it also have epistemological functions, as a means of discovery, understanding, and even proof? By examining the many kinds of visual representation in mathematics and the diverse ways in which they are used, Marcus Giaquinto argues that visual thinking in mathematics is rarely just a superfluous aid; it usually has epistemological value, often as a means of discovery. Drawing from philosophical work on the nature of concepts and from empirical studies of visual perception, mental imagery, and numerical cognition, Giaquinto explores a major source of our grasp of mathematics, using examples from basic geometry, arithmetic, algebra, and real analysis. He shows how we can discern abstract general truths by means of specific images, how synthetic a priori knowledge is possible, and how visual means can help us grasp abstract structures.
Visual Thinking in Mathematics reopens the investigation of earlier thinkers from Plato to Kant into the nature and epistemology of an individual's basic mathematical beliefs and abilities, in the new light shed by the maturing cognitive sciences. Clear and concise throughout, it will appeal to scholars and students of philosophy, mathematics, and psychology, as well as anyone with an interest in mathematical thinking.
* Original work on a fascinating topic
* Multi-disciplinary approach; draws on the latest research in cognitive science, psychology, mathematics education, and philosophy
* Wide range of mathematical examples from geometry, algebra, arithmetic, and real analysis
* Offers solutions to long-standing philosophical problems
Thinking with Diagrams: The Semiotic Basis of Human Cognition
- A.-V Pietarinen
- A-V Pietarinen
The graphic-linguistic distinction: Exploring alternatives
- A Shimojima
Why a diagram is (sometimes) worth ten thousand words
- J H Larkin
- H A Simon
- JH Larkin