David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Jack Alan Reynolds
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British Journal for the Philosophy of Science 45 (3):789-813 (1994)
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 , ), 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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Kajsa Bråting & Johanna Pejlare (2008). Visualizations in Mathematics. Erkenntnis 68 (3):345 - 358.
David Sherry (2009). The Role of Diagrams in Mathematical Arguments. Foundations of Science 14 (1-2):59-74.
Dominique Tournès (2012). Diagrams in the Theory of Differential Equations (Eighteenth to Nineteenth Centuries). Synthese 186 (1):257-288.
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