Marcus Giaquinto presents an investigation into the different kinds of visual thinking involved in mathematical thought, drawing on work in cognitive psychology, philosophy, and mathematics. He argues that mental images and physical diagrams are rarely just superfluous aids: they are often a means of discovery, understanding, and even proof.

"Groundbreaking when first published in 1969, this book is now of even greater relevance to make the reader aware of the need to educate the visual sense, a ...

La Sociedad del Conocimiento se caracteriza por un continuo contacto con una información de marcado carácter visual. No resulta extraño, pues, implementar en el aula universitaria una metodología basada en la imagen, el Visual Thinking. La metodología utilizada es de naturaleza cuantitativa. Se presentan los resultados de la aplicación de esta herramienta a 401 estudiantes de tres titulaciones de la Universidad de Córdoba, que manifiestan un alto nivel de satisfacción con la experiencia y mejoras en el aprendizaje (...) de las asignaturas. El Visual Thinking se considera una herramienta que favorece la implicación del estudiantado a partir del uso de imágenes. (shrink)

Can visual thinking be a means of discovery in elementary analysis, as well as a means of illustration and a stimulus to discovery? The answer to the corresponding question for geometry and arithmetic seems to be ‘yes’ (Giaquinto [1992], [1993]), and so a positive answer might be expected for elementary analysis too. But I argue here that only in a severely restricted range of cases can visual thinking be a means of discovery in analysis. Examination of (...) persuasive visual routes to two simple theorems (Rolle, Bolzano) shows that they are not ways of discovering the theorems; the type of visual thinking involved can never be used to discover analytic theorems of a certain generality. The hypothesis that visual thinking is never a means of discovering the existence or nature of the limit of some infinite process is considered, and a likely counter-example is set out. It is still possible that restricted theorems can be discovered visually: an example from Littlewood is examined in detail and not found wanting. Even when visual thinking is not a means of discovery it can provide the idea for a proof in a direct way; an example is presented (Intermediate Value Theorem). In conclusion: it may be possible to discover theorems in elementary real analysis by visual means, but only theorems of a restricted kind; however, visual thinking in analysis can be very useful in other ways. (shrink)

Our visual experience seems to suggest that no continuous curve can cover every point of the unit square, yet in the late 19th century Giuseppe Peano proved that such a curve exists. Examples like this, particularly in analysis received much attention in the 19th century. They helped to instigate what Hans Hahn called a ‘crisis of intuition’, wherein visual reasoning in mathematics came to be thought to be epistemically problematic. Hahn described this ‘crisis’ as follows : " Mathematicians (...) had for a long time made use of supposedly geometric evidence as a means of proof in much too naive and much too uncritical a way, till the unclarities and mistakes that arose as a result forced a turnabout. Geometrical intuition was now declared to be inadmissible as a means of proof … "Avoiding geometrical evidence, Hahn continued, mathematicians aware of this crisis pursued what he called ‘logicization’, ‘when the discipline requires nothing but purely logical fundamental concepts and propositions for its development’. On this view, an epistemically ideal mathematics would minimize, or avoid altogether, appeals …. (shrink)

To assess the effectiveness of Visual Thinking Strategies in medical education curricula, a pretest–posttest experimental study design was used to evaluate the impact of participating in VTS workshops on first-year medical students. A total of forty-one intervention and sixty comparative students completed the study which included the analysis of clinical images followed by a measurement of word count, length of time analyzing images, and quality of written observations of clinical images. VTS training increased the total number of words (...) used to describe clinical images, the time spent analyzing the images, and the number of clinically relevant observations. (shrink)

The habit of separating the intuitive from the abstractive functions, as they were called in the Middle Ages, goes far back in our tradition. Descartes, in the sixth Meditation, defined man as "a thing that thinks," to which reasoning came naturally; whereas imagining, the activity of the senses, required a special effort and was in no way necessary to the human nature or essence. The passive ability to receive images of sensory things, said Descartes, would be useless if there did (...) not exist in the mind a further and higher active faculty capable of shaping these images and of correcting the errors that derive from sensory experience. A century later Leibnitz spoke of two levels of clear cognition.1 Reasoning was cognition of the higher degree: it was distinct, that is, it could analyze things into their components. Sensory experience, on the other hand, was cognition of the lower order: it also could be clear but it was confused, in the original Latin sense of the term; that is, all elements fused and mingled together in an indivisible whole. Thus artists, who rely on this inferior faculty, are good judges of works of art but when asked what is wrong with a particular piece that displeases them can only reply that it lacks nescio quid, a certain "I don't know what." · 1. Leibnitz, Nouveaux essais sur l'entendement humain , bk. 2, chap. 29. Rudolf Arnheim is the author of Art and Visual Perception: A Psychology of the Creative Eye, Toward a Psychology of Art, The Dynamics of Architectural Form, and Visual Thinking. His previous contributions to Critical Inquiry are "On the Nature of Photography" and "A Stricture on Space and Time". (shrink)

Apin's cabinet of printed curiosities -- Thinking through plural images of logic -- The visible order of student lecture notebooks -- Visual thinking in logic notebooks and Alba amicorum -- The generation of art as the generation of philosophy.

Plato has devised texts which call the readers to collaborate cognitively with them. An important epistemic stimulation is the schematization, the line segment, which summarizes Plato’s idea of intellectual development. In this research, visual thinking will help us to make the most of the Platonic invitation to investigate further cognitive growth. It will be analyzed how visual discoveries are rendered possible by mental number lines, realizing the epistemological importance of visualization. Thanks to visualization, structuralism will be grasped. (...) It will reveal a connection with Plato’s philosophy which suggests a novel elaboration of the Platonic concept of intellectual growth. (shrink)

© Mind Association 2018Some time ago I was at the Bibliothèque Nationale in Paris investigating the teaching of philosophy during Descartes’ time. Fine monographs had already been published on the various regimens and practices at Descartes’ college at La Flèche, and Jesuit institutions in general, as well as the collegiate curriculum in seventeenth-century France. But as interested as I was in the form of the teaching—how philosophy was taught, where, and when—I was more interested in its content—what was actually taught. (...) A fair number of early modern textbooks and some manuscripts of lectures and student notebooks were still available; beyond that, one could find reports of weekly or monthly disputations, condemnations of specific doctrines, various broadsheets, manifestos, and student theses and dissertations. After overcoming some administrative resistance, I found myself in a room at the Cabinet d’Estampes looking at a... (shrink)

Our visual experience seems to suggest that no continuous curve can cover every point of the unit square, yet in the late 19th century Giuseppe Peano proved that such a curve exists. Examples like this, particularly in analysis received much attention in the 19th century. They helped to instigate what Hans Hahn called a ‘crisis of intuition’, wherein visual reasoning in mathematics came to be thought to be epistemically problematic. Hahn described this ‘crisis’ as follows : " Mathematicians (...) had for a long time made use of supposedly geometric evidence as a means of proof in much too naive and much too uncritical a way, till the unclarities and mistakes that arose as a result forced a turnabout. Geometrical intuition was now declared to be inadmissible as a means of proof … "Avoiding geometrical evidence, Hahn continued, mathematicians aware of this crisis pursued what he called ‘logicization’, ‘when the discipline requires nothing but purely logical fundamental concepts and propositions for its development’. On this view, an epistemically ideal mathematics would minimize, or avoid altogether, appeals …. (shrink)

Published in 1891, Edmund Husserl's first book, Philosophie der Arithmetik, aimed to ‘prepare the scientific foundations for a future construction of that discipline’. His goals should seem reasonable to contemporary philosophers of mathematics: "…through patient investigation of details, to seek foundations, and to test noteworthy theories through painstaking criticism, separating the correct from the erroneous, in order, thus informed, to set in their place new ones which are, if possible, more adequately secured. 1"But the ensuing strategy for grounding mathematical knowledge (...) sounds strange to the modern ear. For Husserl cast his work as a sequence of ‘psychological and logical investigations’, providing a psychological analysis "…of the concepts multiplicity, unity, and number, insofar as they are given to use authentically and not through indirect symbolizations. "This emphasis on psychology is a reflection of Husserl's training. As a teenager studying in Leipzig, he attended the lectures of Wilhelm Wundt, a seminal figure in the field of experimental psychology. Wundt held that, via introspection, we can study and classify our inner experiences, in much the same way that scientists study the natural world. 2 People working in his laboratory were therefore trained in procedures for observing and reporting on their own thought processes, as a means of gathering scientific data regarding our cognitive faculties. Bridging the gap between psychology and epistemology, Wundt felt that the results of such inquiry could have normative consequences, since the principles of reasoning employed in the sciences not only have their origins in psychological processes, but, moreover, are justified by the fundamental role they play in thought. His two-volume work, Logik , thus combined empirical considerations with a Kantian emphasis on the way that knowledge depends on our cognitive faculties. 3In The Philosophy of …. (shrink)

This paper applies the Sonoma Model of Critical Thinking to visual arts in an educational setting. The analysis produces insights into the functioning of the model, insights into visual arts, and pragmaticconclusions regarding relationships among art historians, visual artists, and others. We summarize the Sonoma Model of critical thinking and apply it to thinking about art history and visual arts. We use these insights to apply the Sonoma Model to thinking critically about (...) visual arts in an educational environment. One application of the model for visual artists and art historians is in appreciating these disciplines, something that is often lacking among and between them. We find further insight in the application of the elements of the model to themselves, and close with practical affective conclusions for applications of the model by faculty and administrators. (shrink)

This article delineates the notion of conjectural artworks—that is, ways of thinking and explaining formal and relational phenomena by visual means—and presents an appraisal and review of the use of such visual ways in the work of Chilean biologists and philosophers Humberto Maturana and Francisco Varela. Particularly, the article focuses on their recurrent uses of Cellular Automaton, that is, discrete, locally interacting, rule-based mathematical models, as conjectural artworks for understanding the concepts of autopoiesis, structural coupling, cognition and (...) enaction: (i.e. Protobio and Bittorio). Additionally, the article proposes a new model of conjectural artwork based on an extension of cellular automaton: random Boolean networks namely, binary systems with variable local connections. Such model, as it is argued, is useful to connect the theoretical frameworks by Maturana and Varela, especially structural coupling and enaction, with other relevant fields such as biosemiotics’ Umwelt-research and cognitive landscapes in neurodynamics, and to advance and explore the concepts of structurally coupled categorization and generalization. (shrink)

Background: Visual arts education has been suggested to influence cognitive and behavioral outcomes, yet its specific effects on critical thinking and behavior regulation remain underexplored. The effect of visual arts education on these domains forms the topic of the present study, which employs a mixed approach research design. Aim: To assess the impact of visual arts education on the improvement of critical thinking and self-regulation in students. Methodology: A quasi-experimental approach was employed with 80 individuals, (...) with 40 in the experimental group Report Phrase receiving education in visual arts and 40 in the control group receiving non-visual arts education. CTST and BRI assessment tools were used to acquire quantitative data about the participants, utilizing the independent and paired sample T-tests. Data collected from participants include interviews, focus group discussions, self-administered, and thematic analysis. A repeated measures ANOVA was used to analyse the data. Result: Statistical findings depicted improvements in the quantitative aspect; critical thinking skills improved from 70.0 to 75.3 with p<0.001 and behavior regulation improved from 65.0 to 69.2 with p<0.001 in the experimental-group in contrast to the control-group. The thematic analysis found striking increases in attention, impulse control, task completion, creativity, collaboration, affect expression, and self-esteem of experimental-group participants. Based on the findings of the Repeated Measures ANOVA, significant enhancements were observed in both domains for critical thinking and behavior regulation with effect size estimates of 0.40 and 0.30, respectively. Conclusion: The development of critical thinking and behavior control in students is greatly impacted by art instruction. Quantitative and qualitative data both support the utility of using visual arts interventions for cognitive and behavioral development and inform practice in art education. (shrink)

For a large proportion of our daily lives, spontaneously occurring thoughts tend to disengage our minds from goal-directed thinking. Previous studies showed that EEG features such as the P3 and alpha oscillations can predict mind-wandering to some extent, but only with accuracies of around 60%. A potential candidate for improving prediction accuracy is the Steady-State Visual Evoked Potential, which is used frequently in single-trial contexts such as brain-computer interfaces as a marker of the direction of attention. In this (...) study, we modified the sustained attention to response task that is usually employed to measure spontaneous thought to incorporate the SSVEP elicited by a 12.5-Hz flicker. We then examined whether the SSVEP could track and allow for the prediction of the stickiness and task-relatedness dimensions of spontaneous thought. Our results show that the SSVEP evoked by flickering words was able to distinguish between more and less sticky thinking but not between whether a participant was on- or off-task. This suggests that the SSVEP is able to track spontaneous thinking when it is strongly disengaged from the task but not off-task thought in general. Future research should determine the exact dimensions of spontaneous thought to which the SSVEP is most sensitive. (shrink)

Data literacy – the ability to analyze, interpret, evaluate, and use data and data visualizations – has become increasingly important for understanding and communicating information in the discipline of history. In the United States, curricular standards and standardized assessments already reflect this importance, but educators lack a clear picture of how students use data visualizations when reasoning about the past. How do students use data visualizations when reasoning about a historical question? To what degree does using data visualizations enhance students’ (...) historical reasoning? What challenges do students face when attempting to use data visualizations? This study addresses these questions, investigating how fifth-grade, eighth-grade, and high school students use data visualizations from typical school textbooks. Twenty-seven students were interviewed, surveyed, and asked to think aloud about a historical question while using a textbook entry that included a data visualization. Results from qualitative and quantitative analyses highlight the potential for data visualizations to enhance historical reasoning. However, results also indicate that students are unlikely to read data visualizations on their own, underscoring the importance of providing targeted data literacy instruction. (shrink)

Certain forms of art privilege the ellipsis. For example, in poetry, one has only to think of Mallarmé's Un Coup de dés for ellipsis to mark its peculiar trace upon the processes of meaning. Nonetheless, because ellipsis situates itself so curiously between meaning and signification, a formal definition may seem too categorical in explaining what ellipsis can do in its between-state, flanked by visuality, semiotics and signification. This article examines how Merleau- Ponty, Derrida and Deleuze, employ textual-visual and ontological-perceptual (...) strategies for making sense of this slippery signifier. These strategies allow us to think through examples of ellipsis in the writing of Hélène Cixous and in the photographs of Robert Frank, in terms of semantic, affective, aesthetic and material qualities. (shrink)

As Hegel once said, in Byzantium, between homoousis and homoiousis, the difference of one letter could decide the life and death of thousands. As this article seeks to argue, Byzantine thinking was not only attentive to conceptual differences, but also to iconic ones. The iconoclastic controversy (726-842 AD) arose from two different interpretations of the nature of images: whereas iconoclastic philosophy is based on the assumption of a fundamental 'iconic identity', iconophile philosophy defends the idea of'iconic difference'. And while (...) the reception in the Latin West of the controversies over the image as a mere problem of referentiality of the letter explains why its originality has remained underestimated for centuries, reexamining Byzantine visual thinking in the light of today's 'pictorial turn' reveals its striking modernity. (shrink)

Visual programming language is a crucial part of learning programming. On this basis, it is essential to use visual programming to lower the learning threshold for students to learn about artificial intelligence to meet current demands in higher education. Therefore, a 3-h AI course with an RGB-to-HSL learning task was implemented; the results of which were used to analyze university students from two different disciplines. Valid data were collected for 65 students in the Science -student group and 39 (...) students in the Humanities -student group. Independent sample t-tests were conducted to analyze the difference between cognitive styles and computational thinking. No significant differences in either cognitive style or computational thinking ability were found after the AI course, indicating that taking visual AI courses lowers the learning threshold for students and makes it possible for them to take more difficult AI courses, which in turn effectively helping them acquire AI knowledge, which is crucial for cultivating talent in the field of AI. (shrink)

The question of whether cognition can influence perception has a long history in neuroscience and philosophy. Here, we outline a novel approach to this issue, arguing that it should be viewed within the framework of top-down information-processing. This approach leads to a reversal of the standard explanatory order of the cognitive penetration debate: we suggest studying top-down processing at various levels without preconceptions of perception or cognition. Once a clear picture has emerged about which processes have influences on those at (...) lower levels, we can re-address the extent to which they should be considered perceptual or cognitive. Using top-down processing within the visual system as a model for higher-level influences, we argue that the current evidence indicates clear constraints on top-down influences at all stages of information processing; it does, however, not support the notion of a boundary between specific types of information-processing as proposed by the cognitive impenetrability hypothesis. (shrink)

As some artists discovered early in the century, there is a particular pleasure and stimulation to be derived from works of art created by cultures untouched by our own traditions of form. In part this is probably a delight in exoticism, in being away from home, and in part it possibly is our sentiment for cultures we look on as traditional, in a Jungian sense, or primitive in their unquestioning allegiance to simple cultural necessity. But more significantly, without indulging in (...) philosophical or anthropological speculation, we are forced, in looking at such objects as these elegantly designed boxes and bowls, to revise our visual thinking, our assumptions about unity and grace.Joshua C. Taylor, director of the National Collection of Fine Arts of the Smithsonian Institution, has written Learning to Look, William Page: The American Titian, and catalogues of exhibits of futurism and the works of Umberto Boccioni. Part 1 of this paper has been published in somewhat different form in Boxes and Bowls. (shrink)

This paper presents a study of the generation, manipulation and use of visual representations in different episodes of scientific discovery. The study identifies a common set of transformations of visual representations underlying the distinctive methods and imagery of different scientific fields. The existence of common features behind the diversity of visual representations suggests a common dynamical structure for visual thinking, showing how visual representations facilitate cognitive processes such as pattern-matching and visual inference through (...) the use of tools, technologies and other cultural resources. This dynamical model suggests a way of theorizing the interaction of cognitive, socio-cultural and technological aspects of science without losing sight of the essential contribution each makes to the processes of discovery. Whereas scientific work is often construed epistemically, as having the aim of improving the fit between theories and phenomena or culturally given notions of what counts as reality, this study shows that scientists use transformations to modify visual representations in ways that achieve other kinds of match: between a representation and the cognitive demands of a task or between an emerging representation and the social need to communicate and negotiate new meanings in a context of culturally embedded conventions. By showing how images connect each of the overlapping contexts of scientific work the proposed model negotiates the sometimes contested boundary between cognitive and social aspects of science. (shrink)