From Euclidean Geometry to Knots and Nets

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Abstract

This paper assumes the success of arguments against the view that informal mathematical proofs secure rational conviction in virtue of their relations with corresponding formal derivations. This assumption entails a need for an alternative account of the logic of informal mathematical proofs. Following examination of case studies by Manders, De Toffoli and Giardino, Leitgeb, Feferman and others, this paper proposes a framework for analysing those informal proofs that appeal to the perception or modification of diagrams or to the inspection or imaginative manipulation of mental models of mathematical phenomena. Proofs relying on diagrams can be rigorous if (a) it is easy to draw a diagram that shares or otherwise indicates the structure of the mathematical object, (b) the information thus displayed is not metrical and (c) it is possible to put the inferences into systematic mathematical relation with other mathematical inferential practices. Proofs that appeal to mental models can be rigorous if the mental models can be externalised as diagrammatic practice that satisfies these three conditions.

Original languageEnglish
Article number10.1007/s11229-017-1558-x
Pages (from-to)2715-2736
Number of pages22
JournalSynthese
Volume196
Issue number7
Early online date19 Sept 2017
DOIs
Publication statusPublished - 15 Jul 2019

Keywords

  • Euclidean geometry
  • Knot theory
  • Mathematics
  • Proof
  • Topology

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