An Inquiry into the Practice of Proving in Low-Dimensional Topology

In Gabriele Lolli, Giorgio Venturi & Marco Panza (eds.), From Logic to Practice. Zurich, Switzerland: Springer International Publishing. pp. 315-336 (2015)

Authors
Silvia De Toffoli
Princeton University
Valeria Giardino
Centre National de la Recherche Scientifique
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.
Keywords Visualization  Topological pictures  Manipulative imagination
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References found in this work BETA

The Philosophy of Mathematical Practice.Paolo Mancosu (ed.) - 2008 - Oxford University Press.
The Euclidean Diagram.Kenneth Manders - 2008 - In Paolo Mancosu (ed.), The Philosophy of Mathematical Practice. Oxford University Press. pp. 80--133.

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