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From Logic to PracticeAn Inquiry into the Practice of Proving in Low-Dimensional Topology

From Logic to Practice: An Inquiry into the Practice of Proving in Low-Dimensional Topology [The aim of this article is to investigate specific aspects connected with visualization in the practice of a mathematical subfield: low-dimensional topology. Through a case study, it will be established that visualization can play an epistemic role. The background assumption is that the consideration of the actual practice of mathematics is relevant to address epistemological issues. It will be shown that in low-dimensional topology, justifications can be based on sequences of pictures. Three theses will be defended. First, the representations used in the practice are an integral part of the mathematical reasoning. As a matter of fact, they convey in a material form the relevant transitions and thus allow experts to draw inferential connections. Second, in low-dimensional topology, experts exploit a particular type of manipulative imagination which is connected to intuition of two- and three-dimensional space and motor agency. This imagination allows recognizing the transformations which connect different pictures in an argument. Third, the epistemic—and inferential—actions performed are permissible only within a specific practice: this form of reasoning is subject-matter dependent. Local criteria of validity are established to assure the soundness of representationally heterogeneous arguments in low-dimensional topology.] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

From Logic to PracticeAn Inquiry into the Practice of Proving in Low-Dimensional Topology

Part of the Boston Studies in the Philosophy and History of Science Book Series (volume 308)
Editors: Lolli, Gabriele; Panza, Marco; Venturi, Giorgio
From Logic to Practice — Nov 6, 2014

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References (24)

Publisher
Springer International Publishing
Copyright
© Springer International Publishing Switzerland 2015
ISBN
978-3-319-10433-1
Pages
315 –336
DOI
10.1007/978-3-319-10434-8_15
Publisher site
See Chapter on Publisher Site

Abstract

[The aim of this article is to investigate specific aspects connected with visualization in the practice of a mathematical subfield: low-dimensional topology. Through a case study, it will be established that visualization can play an epistemic role. The background assumption is that the consideration of the actual practice of mathematics is relevant to address epistemological issues. It will be shown that in low-dimensional topology, justifications can be based on sequences of pictures. Three theses will be defended. First, the representations used in the practice are an integral part of the mathematical reasoning. As a matter of fact, they convey in a material form the relevant transitions and thus allow experts to draw inferential connections. Second, in low-dimensional topology, experts exploit a particular type of manipulative imagination which is connected to intuition of two- and three-dimensional space and motor agency. This imagination allows recognizing the transformations which connect different pictures in an argument. Third, the epistemic—and inferential—actions performed are permissible only within a specific practice: this form of reasoning is subject-matter dependent. Local criteria of validity are established to assure the soundness of representationally heterogeneous arguments in low-dimensional topology.]

Published: Nov 6, 2014

Keywords: Concrete Object; Klein Bottle; Solid Torus; Homology Sphere; Epistemic Action

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