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- M. Giaquinto (2007). Visual Thinking in Mathematics: An Epistemological Study. Oxford University Press.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.Reading list | Discuss | Edit | Categorize |My bibliography
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| | Scholar | At my library | More options ... Recent work in visual perception shows that its phenomenal and cognitive aspects cannot be distinguished sharply. A preferable treatment of visual perception is to describe perceptual strategies, which represent the various ways in which perceivers may focus interest, and which are treated in this paper by two examples. Perceptual strategies suggest that both Sense Data theories, on the one hand, and treatments of perception such as Hanson's, on the other, are over-simplified. This work further suggests that the traditional distinction between primary and secondary qualities should be replaced by an epistemological distinction between visual and physical qualities. Such a distinction clarifies epistemological investigations of perception and shows the relevance of experimental investigations of visual perception to epistemological problems.
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| | Scholar | At my library | More options ... Our visual experience seems to suggest that no continuous curve can cover every point of the unit square, yet in the late nineteenth century Giuseppe Peano proved that such a curve exists. Examples like this, particularly in analysis (in the sense of the infinitesimal calculus) received much attention in the nineteenth century. They helped 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... (p. 67) 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 to visual representations. This would be a radical reformation of past practice, necessary, according to its advocates, for avoiding “unclarities and mistakes” like the one exposed by Peano.
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| Discuss | Edit | Categorize | My bibliography | Export citation | Other links: analysis.oxfordjournals.org dx.doi.org | Scholar | At my library | More options ...
| | Other links: analysis.oxfordjournals.org dx.doi.org | Scholar | At my library | More options ... Two experiments are reported which investigate the factors that influence how persuaded mathematicians are by visual arguments. We demonstrate that if a visual argument is accompanied by a passage of text which describes the image, both research-active mathematicians and successful undergraduate mathematics students perceive it to be significantly more persuasive than if no text is given. We suggest that mathematicians’ epistemological concerns about supporting a claim using visual images are less prominent when the image is described in words. Finally we suggest that empirical studies can make a useful contribution to our understanding of mathematical practice.
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| Discuss | Edit | Categorize | My bibliography | Export citation | Other links: philmat.oxfordjournals.org dx.doi.org | Scholar | At my library | More options ...
| | Other links: philmat.oxfordjournals.org dx.doi.org | Scholar | At my library | More options ... 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.
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| | Other links: bjps.oxfordjournals.org jstor.org dx.doi.org | Scholar | At my library | More options ... Discussion of M. Giaquinto, Visual Thinking in Mathematics: An Epistemological Study
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