63 research outputs found

    The Informal Logic of Mathematical Proof

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    Informal logic is a method of argument analysis which is complementary to that of formal logic, providing for the pragmatic treatment of features of argumentation which cannot be reduced to logical form. The central claim of this paper is that a more nuanced understanding of mathematical proof and discovery may be achieved by paying attention to the aspects of mathematical argumentation which can be captured by informal, rather than formal, logic. Two accounts of argumentation are considered: the pioneering work of Stephen Toulmin [The uses of argument, Cambridge University Press, 1958] and the more recent studies of Douglas Walton, [e.g. The new dialectic: Conversational contexts of argument, University of Toronto Press, 1998]. The focus of both of these approaches has largely been restricted to natural language argumentation. However, Walton's method in particular provides a fruitful analysis of mathematical proof. He offers a contextual account of argumentational strategies, distinguishing a variety of different types of dialogue in which arguments may occur. This analysis represents many different fallacious or otherwise illicit arguments as the deployment of strategies which are sometimes admissible in contexts in which they are inadmissible. I argue that mathematical proofs are deployed in a greater variety of types of dialogue than has commonly been assumed. I proceed to show that many of the important philosophical and pedagogical problems of mathematical proof arise from a failure to make explicit the type of dialogue in which the proof is introduced.Comment: 14 pages, 1 figure, 3 tables. Forthcoming in Perspectives on Mathematical Practices: Proceedings of the Brussels PMP2002 Conference (Logic, Epistemology and the Unity of the Sciences Series), J. P. Van Bendegem & B. Van Kerkhove, edd. (Dordrecht: Kluwer, 2004

    Writing in Britain and Ireland, c. 400 to c. 800

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    ¿Nuevas direcciones en filosofía de la matemática?

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    Suele suponerse que la labor realizada en Filosofía de la Matemática debe ser accesible a los filósofos y a los matemáticos. Además, es conveniente centrarse en la práctica real de la matemática, ya que los rasgos de esta práctica influyen en las nuevas direcciones de la filosofía de la matemática. La práctica se distingue filosóficamente por dos importantes aspectos: tiene una historia y se solapa con otras prácticas, sobre todo con la ciencia natural. Se explica por que la historicidad de la matemática es filosóficamente significativa y se discuten algunos problemas que se siguen para la ontología y la epistemología, por una lado, y la metodología y la pedagogía, por otro, al tomar en consideración la relación de la matemática con la ciencia natura

    Affine Grassmannians and Hessenberg Schubert Cells

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    We give an overview of the linear algebra, geometry, and combinatorics of affine Grassmannians along the lines of Fulton’s Young Tableaux for classical Grassmannians. We discuss geometric and linear algebraic aspects of the decomposition of the affine Grassmannian into affine Schubert cells in terms of coset representatives and linear models. We describe (Grassmannian) Hessenberg Schubert cells and show that every affine Schubert cell can be realized as a Hessenberg Schubert cell in a complete flag variety and as a Grassmannian Hessenberg Schubert cell in a finite Grassmannian
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