In this article, I will discuss the relationship between mathematical intuition and mathematical visualization. I will argue that in order to investigate this relationship, it is necessary to consider mathematical activity as a complex phenomenon, which involves many different cognitive resources. I will focus on two kinds of danger in recurring to visualization and I will show that they are not a good reason to conclude that visualization is not reliable, if we consider its use in mathematical (...) practice. Then, I will give an example of mathematical reasoning with a figure, and show that both visualization and intuition are involved. I claim that mathematical intuition depends on background knowledge and expertise, and that it allows to see the generality of the conclusions obtained by means of visualization. (shrink)
Visual imagination (or visualization) is peculiar in being both free, in that what we imagine is up to us, and useful to a wide variety of practical reasoning tasks. How can we rely upon our visualizations in practical reasoning if what we imagine is subject to our whims? The key to answering this puzzle, I argue, is to provide an account of what constrains the sequence in which the representations featured in visualization unfold—an account that is consistent with (...) its freedom. Three different proposals are outlined, building on theories that link visualization to sensorimotor predictive mechanisms (e.g., efference copies, forward models ). Each sees visualization as a kind of reasoning, where its freedom consists in our ability to choose the topic of the reasoning. Of the three options, I argue that the approach many will find most attractive—that visualization is a kind of off-line perception, and is therefore in some sense misrepresentational—should be rejected. The two remaining proposals both conceive of visualization as a form of sensorimotor reasoning that is constitutive of one’s commitments concerning the way certain kinds of visuomotor scenarios unfold. According to the first, these commitments impinge on one’s web of belief from without, in the manner of normal perceptual experience; according to the second, these commitments just are one’s (occurrent) beliefs about such generalizations. I conclude that, despite being initially counterintuitive, the view of visualization as a kind of occurrent belief is the most promising. (shrink)
In this paper, I give an account of how our hominin ancestors evolved a conscious ability I call scenario visualization that enabled them to manufacture novel tools so as to survive and flourish in the ever-changing and complex environments in which they lived. I first present the ideas and arguments put forward by evolutionary psychologists that the mind evolved certain mental capacities as adaptive responses to environmental pressures. Specifically, Steven Mithen thinks that the mind has evolved cognitive fluidity, viz., (...) an ability to exchange information flexibly between and among mental modules. Showing the deficiency in Mithen’s view, I then argue that the flexible exchange of information between and among modules together with scenario visualization is what explains the ability to construct the novel tools needed to survive and flourish in the environments in which our hominin ancestors resided. Finally, I trace the development of the multi-purposed javelin, from its meager beginnings as a stick, in order to illustrate scenario visualization in novel tool manufacturing. (shrink)
This paper explores the development of the chemical table as a tool designed for chemical information visualization. It uses a historical context to investigate the purpose of chemical tables and charts, analyzing them from the perspective of theory of tables, cartography, and design. It suggests reasons why the two-dimensional periodic table remains the de facto standard for chemical information display.
The goal of three-dimensional visualization is to present information in such a way that the viewer suspends disbelief and uses the screen imagery the same way as he or she would use an identical, real 3D scene. To do this effectively, programmers employ a variety of 3D depth cues. Our own anecdotal experience says that shadows and stereopsis are two of the best for visualization. The nice thing is that both of these are possible to do in interactive (...) programs. They sacrifice a certain amount of interactive speed, but they are possible. (shrink)
In a study of three indigenous and non-indigenous cultural groups in northwestern and northeastern Siberia, framed line tests and a landscape drawing task were used to examine the hypotheses that test-based assessments of context sensitivity and independence are correlated with the amount of contextual information contained in drawings, and with the order in which the focal and background objects are drawn. The results supported these hypotheses, and inspection of the regression relationships suggested that the intergroup variations in test performance were (...) likely to result from differences in the attention accorded to contextual information, as revealed by the drawings. Social and environmental explanations for the group differences in context sensitivity are also discussed. The conclusions support the argument that cultural differences in artistic styles and perceptual tests reflect the same underlying perceptual tendencies, and they are consistent with the argument that these tendencies reflect corresponding differences in patterns of social and environmental interaction. (shrink)
Philosophers have relied on visual metaphors to analyse ideas and explain their theories at least since Plato. Descartes is famous for his system of axes, and Wittgenstein for his first design of truth table diagrams. Today, visualisation is a form of ‘computer-aided seeing’ information in data. Hence, information is the fundamental ‘currency’ exchanged through a visualisation pipeline. In this article, we examine the types of information that may occur at different stages of a general visualization pipeline. We do so (...) from a quantitative and a qualitative perspective. The quantitative analysis is developed on the basis of Shannon’s information theory. The qualitative analysis is developed on the basis of Floridi’s taxonomy in the philosophy of information. We then discuss in detail how the condition of the ‘data processing inequality’ can be broken in a visualisation pipeline. This theoretic finding underlines the usefulness and importance of visualisation in dealing with the increasing problem of data deluge. We show that the subject of visualisation should be studied using both qualitative and quantitative approaches, preferably in an interdisciplinary synergy between information theory and the philosophy of information. (shrink)
In this paper, I first present the ideas and arguments put forward by evolutionary psychologists that humans evolved certain capacities to creatively problem solve. Specifically, Steven Mithen thinks that creative problem solving is possible because the mind has evolved a conscious capacity he calls cognitive fluidity, the flexible exchange of information between and among mental modules. While I agree with Mithen that cognitive fluidity acts as a necessary condition for creative problem solving, I disagree that cognitive fluidity alone will suffice (...) for such an activity. I argue further that the flexible exchange of information between and among modules, as well as what I call scenario visualization - a conscious ability to segregate and integrate visual images in future scenarios - evolved in our species and accounts for certain kinds of creative problem solving. (shrink)
Recent technical improvements, such as 3D microscopy imaging, have shown the necessity of studying 3D biological tissue architecture during carcinogenesis. In the present paper a computer simulation model is developed allowing the visualization of the microscopic biological tissue architecture during the development of metaplastic and dysplastic lesions.The static part of the model allows the simulation of the normal, metaplastic and dysplastic architecture of an external epithelium. This model is associated to a knowledge base which contains only data on the (...) nasal epithelium. The latter has been well studied by numerous authors and its lesional states are well known. An inference engine allows the initialization of the static model parameters. A statistical comparison between simulated epithelia and real epithelia is achieved by adjusting the parameter values during the simulation. (shrink)
As a committee of the National Academy of Engineering recognized, ethics education should foster the ability of students to analyze complex decision situations and ill-structured problems. Building on the NAE’s insights, we report about an innovative teaching approach that has two main features: first, it places the emphasis on deliberation and on self-directed, problem-based learning in small groups of students; and second, it focuses on understanding ill-structured problems. The first innovation is motivated by an abundance of scholarly research that supports (...) the value of deliberative learning practices. The second results from a critique of the traditional case-study approach in engineering ethics. A key problem with standard cases is that they are usually described in such a fashion that renders the ethical problem as being too obvious and simplistic. The practitioner, by contrast, may face problems that are ill-structured. In the collaborative learning environment described here, groups of students use interactive and web-based argument visualization software called “AGORA-net: Participate – Deliberate!”. The function of the software is to structure communication and problem solving in small groups. Students are confronted with the task of identifying possible stakeholder positions and reconstructing their legitimacy by constructing justifications for these positions in the form of graphically represented argument maps. The argument maps are then presented in class so that these stakeholder positions and their respective justifications become visible and can be brought into a reasoned dialogue. Argument mapping provides an opportunity for students to collaborate in teams and to develop critical thinking and argumentation skills. (shrink)
We study localized plasmons at the nanoscale (nano-plasmons) in graphene. The collective excitations of induced charge density modulations in graphene are drastically changed in the vicinity of a single impurity compared to graphene's bulk behavior. The dispersion of nano-plasmons depends on the number of electrons and the sign, strength and size of the impurity potential. Due to this rich parameter space the calculated dispersions are intrinsically multidimensional requiring an advanced visualization tool for their efficient analysis, which can be achieved (...) with parallel rendering. To overcome the problem of analyzing thousands of very complex spatial patterns of nano-plasmonic modes, we take a combined visual and quantitative approach to investigate the excitations on the two-dimensional graphene lattice. Our visual and quantitative analysis shows that impurities trigger the formation of localized plasmonic excitations of various symmetries. We visually identify dipolar, quadrupolar and radial modes, and quantify the spatial distributions of induced charges. (shrink)
Mathematical investigation, when done well, can confer understanding. This bare observation shouldn’t be controversial; where obstacles appear is rather in the effort to engage this observation with epistemology. The complexity of the issue of course precludes addressing it tout court in one paper, and I’ll just be laying some early foundations here. To this end I’ll narrow the field in two ways. First, I’ll address a specific account of explanation and understanding that applies naturally to mathematical reasoning: the view proposed (...) by Philip Kitcher and Michael Friedman of explanation or understanding as involving the unification of theories that had antecedently appeared heterogeneous. For the second narrowing, I’ll take up one specific feature (among many) of theories and their basic concepts that is sometimes taken to make the theories and concepts preferred: in some fields, for some problems, what is counted as understanding a problem may involve finding a way to represent the problem so that it (or some aspect of it) can be visualized. The final section develops a case study which exemplifies the way that this consideration – the potential for visualizability – can rationally inform decisions as to what the proper framework and axioms should be. The discussion of unification (in sections 3 and 4) leads to a mathematical analogue of Goodman’s problem of identifying a principled basis for distinguishing grue and green. Just as there is a philosophical issue about how we arrive at the predicates we should use when making empirical predictions, so too there is an issue about what properties best support many kinds of mathematical reasoning that are especially valuable to us. The issue becomes pressing via an examination of some physical and mathematical cases that make it seem unlikely that treatments of unification can be as straightforward as the philosophical literature has hoped. Though unification accounts have a grain of truth (since a phenomenon (or cluster of phenomena) called “unification” is in fact important in many cases) we are far from an analysis of what “unification” is.. (shrink)
Many scientific discoveries have depended on external diagrams or visualizations. Many scientists also report to use an internal mental representation or mental imagery to help them solve problems and reason. How do scientists connect these internal and external representations? We examined working scientists as they worked on external scientific visualizations. We coded the number and type of spatial transformations (mental operations that scientists used on internal or external representations or images) and found that there were a very large number of (...) comparisons, either between different visualizations or between a visualization and the scientists’ internal mental representation. We found that when scientists compared visualization to visualization, the comparisons were based primarily on features. However, when scientists compared a visualization to their mental representation, they were attempting to align the two representations. We suggest that this alignment process is how scientists connect internal and external representations. (shrink)
The cognitive function of mental images with respect to the referential aspect of language is examined and used in the listener model ANTLIMA of the natural language system SOCCER. An operational realization of the reference relation used to recognize instances of spatial concepts in the results of a vision system and also to visualize locative expressions is presented and compared to A. Herskovits' analysis of the semantics of spatial prepositions.
Abstract This paper addresses the role that argumentation schemes and argument visualization software tools can play in helping to find and counter objections to a given argument one is confronted with. Based on extensive analysis of features of the argumentation in these two examples, a practical four-step method of finding objections to an argument is set out. The study also applies the Carneades Argumentation System to the task of finding objections to an argument, and shows how this system has (...) some capabilities that are especially useful. Content Type Journal Article Pages 1-23 DOI 10.1007/s10503-012-9261-z Authors Douglas Walton, Centre for Research in Reasoning, Argumentation and Rhetoric (CRRAR), University of Windsor, 2500 University Ave. W., Windsor, ON N9B 3Y1, Canada Journal Argumentation Online ISSN 1572-8374 Print ISSN 0920-427X. (shrink)
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.
There is a particular irony that chemistry – the most visual, tactile, and pungent of sciences – is rarely associated with modern notions of aesthetics and science. Indeed, as any examination of aesthetics and modern science reveals, physics, rather than chemistry or biology, is considered the paradigm because of its extraordinary ability to comprehend and communicate through the symbolic language of mathematics. Echoing Heisenberg’s 1970 essay, "The Meaning of Beauty in the Exact Sciences", this perspective on physics takes the inherent (...) abstraction of quantum mechanics and relativity as the result of the physicists’ search for beauty in nature. (shrink)
Carneades is an open source argument mapping application and a programming library for building argumentation support tools. In this paper, Carneades’ support for argument reconstruction, evaluation and visualization is illustrated by modeling most of the factual and legal arguments in Popov v Hayashi.
Molecular models are typical topics of chemical research depending on the technical standards of observation, computation, and representation. Mathematically, molecular structures have been represented by means of graph theory, topology, differential equations, and numerical procedures. With the increasing capabilities of computer networks, computational models and computer-assisted visualization become an essential part of chemical research. Object-oriented programming languages create a virtual reality of chemical structures opening new avenues of exploration and collaboration in chemistry. From an epistemic point of view, virtual (...) reality is a new computer-assisted tool of human imagination and recognition. (shrink)
First ever philosophy treatise on photography, analytic in approach but sensitive to photo-history, not confined to aesthetics or art (illus.), Walker Evans photo on cover. Papercover printing, Dec. 2000.
First drafted in 2006 and currently in version 2.1, the London Charter calls for the adoption of international standards for intellectual integrity, transparency, sustainability, and access in 3D modeling for cultural heritage. While the London Charter has been in the process of revision and distribution to the heritage community, game engines have become less expensive and more approachable. Several engines offer the ability to publish easily across operating systems, mobile devices, and the web, causing a rapid expansion in their use (...) for archeological visualization. However, the very power of game engines to create and publish immersive content poses fundamental challenges to the emphasis on data-driven visualization and transparency expressed in the London Charter. These challenges should not be suppressed, since they can prove heuristically fruitful if they are explicitly recognized and explored. This potential is illustrated by a descriptive analysis of the recreation of the House of the Prince of Naples in Pompeii by an undergraduate humanities class, which concludes that the immersive effects of engine-based visualizations are as much to be found in their creation as in their “playing.” This suggests the value of democratizing the creation of game engine content for heritage visualization beyond research visualization laboratories, as a part of undergraduate curricula in the humanities. (shrink)
In this paper, we propose a novel medium for interactions based on an interpersonal psychological approach referred to as ‘naïve psychology’. We adopt the visual assessment of clustering tendency (VAT) to naïve psychology for the visual understanding of other people. The VAT algorithm produces a visual display that can be used to assess clustering tendencies in a set of persons (notions) by reconstructing a digital image representation of a square relational dissimilarity matrix for its set. This algorithm clearly represents two (...) types of imbalanced situations in naïve psychology: crisp and fuzzy. The visual image of a balanced or imbalance situation is useful for a deeper human understanding. (shrink)
The central aim of this paper is to present a Boolean algebraic approach to the classical Aristotelian Relations of Opposition, namely Contradiction and (Sub)contrariety, and to provide a 3D visualisation of those relations based on the geometrical properties of Platonic and Archimedean solids. In the first part we start from the standard Generalized Quantifier analysis of expressions for comparative quantification to build the Comparative Quantifier Algebra CQA. The underlying scalar structure allows us to define the Aristotelian relations in Boolean terms (...) and to propose a 3D visualisation by transforming a cube into an octahedron. In part two, the architecture of the CQA is shown to carry over, both to the classical quantifiers of Predicate Calculus and to the modal operators—which are given a Generalized Quantifier style re-interpretation. In this way we provide an algebraic foundation for Blanché’s Aristotelian hexagon as well as a 3D alternative to his 2D star-like visualisation. In a final part, a richer scalar structure is argued to underly the realm of Modality, thus generalizing the 3D algebra with eight (2 3 ) operators to a 4D algebra with sixteen (2 4 ) operators. The visual representation of the latter structure involves a transformation of the hypercube to a rhombic dodecahedron. The resulting 3D visualisation allows a straightforward embedding, not only of the classical Blanché star of Aristotelian relations or the paracomplete and paraconsistent stars of Béziau (Log Investig 10, 218–232, 2003) but also of three additional isomorphic Aristotelian constellations. (shrink)
This paper argues that spacetime visualisability is not a necessary condition for the intelligibility of theories in physics. Visualisation can be an important tool for rendering a theory intelligible, but it is by no means a sine qua non. The paper examines the historical transition from classical to quantum physics, and analyses the role of visualisability (Anschaulichkeit) and its relation to intelligibility. On the basis of this historical analysis, an alternative conception of the intelligibility of scientific theories is proposed, based (...) on Heisenberg's reinterpretation of the notion of Anschaulichkeit. (shrink)
This book presents an original theory of the nature of pictorial representation. The most influential recent theory of depiction, put forward by Nelson Goodman, holds that the relation between depictions and what they represent is entirely conventional. Flint Schier argues to the contrary that depiction involves resemblance to the things depicted, providing a sophisticated defence of our basic intuitions on the subject. Canvassing an attractive theory of 'generativity' rather than resemblance, Dr Schier provides a detailed account of depiction, showing how (...) it illuminates and resolves many of the enigmas of pictorial representation while remaining true to our basic intuitions on the subject. (shrink)
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. (shrink)