From Euclidean geometry to knots and nets

Synthese:1-22 (2017)
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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 it is easy to draw a diagram that shares or otherwise indicates the structure of the mathematical object, the information thus displayed is not metrical and 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.

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Brendan Larvor
University of Hertfordshire

Citations of this work

What are mathematical diagrams?Silvia De Toffoli - 2022 - Synthese 200 (2):1-29.
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On the Contemporary Practice of Philosophy of Mathematics.Colin Jakob Rittberg - 2019 - Acta Baltica Historiae Et Philosophiae Scientiarum 7 (1):5-26.
Rigour and Intuition.Oliver Tatton-Brown - 2019 - Erkenntnis 86 (6):1757-1781.

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References found in this work

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Remarks on the Foundations of Mathematics.Ludwig Wittgenstein - 1956 - Oxford: Macmillan. Edited by G. E. M. Anscombe, Rush Rhees & G. H. von Wright.
Mathematical Knowledge and the Interplay of Practices.José Ferreirós - 2015 - Princeton, USA: Princeton University Press.
Proofs and refutations (IV).I. Lakatos - 1963 - British Journal for the Philosophy of Science 14 (56):296-342.
Proofs and Refutations.Imre Lakatos - 1980 - Noûs 14 (3):474-478.

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