Understanding in mathematics: The case of mathematical proofs

Hamami, Yacin; Morris, Rebecca Lea

doi:10.1111/nous.12489

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Understanding in mathematics: The case of mathematical proofs

Hamami, Yacin; Morris, Rebecca Lea

2024 • In Nous, 58 (4), p. 1073 - 1106

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https://hdl.handle.net/2268/333872

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10.1111/nous.12489

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Keywords :

Philosophy

Abstract :

[en] Although understanding is the object of a growing literature in epistemology and the philosophy of science, only few studies have concerned understanding in mathematics. This essay offers an account of a fundamental form of mathematical understanding: proof understanding. The account builds on a simple idea, namely that understanding a proof amounts to rationally reconstructing its underlying plan. This characterization is fleshed out by specifying the relevant notion of plan and the associated process of rational reconstruction, building in part on Bratman's theory of planning agency. It is argued that the proposed account can explain a significant range of distinctive phenomena commonly associated with proof understanding by mathematicians and philosophers. It is further argued, on the basis of a case study, that the account can yield precise diagnostics of understanding failures and can suggest ways to overcome them. Reflecting on the approach developed here, the essay concludes with some remarks on how to shape a general methodology common to the study of mathematical and scientific understanding and focused on human agency.

Disciplines :

Philosophy & ethics
Mathematics

Author, co-author :

Hamami, Yacin ; Université de Liège - ULiège > Traverses ; Department of Humanities, Social and Political Sciences, ETH Zürich, Zürich, Switzerland ; Institut Jean Nicod, Department of Cognitive Studies, ENS, EHESS, PSL University, CNRS, Paris, France

Morris, Rebecca Lea ; Independent Scholar, Minneapolis, United States

Language :

English

Title :

Understanding in mathematics: The case of mathematical proofs

Publication date :

2024

Journal title :

Nous

ISSN :

0029-4624

eISSN :

1468-0068

Publisher :

John Wiley and Sons Inc

Volume :

Issue :

Pages :

1073 - 1106

Peer reviewed :

Peer Reviewed verified by ORBi

Additional URL :

https://onlinelibrary.wiley.com/doi/pdf/10.1111/nous.12489

Funders :

F.R.S.-FNRS - Fonds de la Recherche Scientifique
REA - European Commission. Research Executive Agency

Funding text :

For very helpful comments and suggestions related to this work, we are grateful to Jeremy Avigad, Vincent Coumans, Henk de Regt, Jean-Pierre Ferrier, Alain Genestier, Marcus Giaquinto, Gerhard Heinzmann, Bruno Leclercq, Jean Paul Van Bendegem, and Roy Wagner. We would also like to thank audiences at the University of Li\u00E8ge, the Archives Henri Poincar\u00E9 in Nancy, the Institute for Science in Society in Nijmegen, the University of Vienna, and the ETH Z\u00FCrich for fruitful discussions. Finally we would like to thank two anonymous reviewers for their very helpful feedback and\u00A0suggestions. This project has received funding from the European Union's Horizon Europe research and innovative programme under the Marie Sklodowska-Curie grant agreement No 101063894.For very helpful comments and suggestions related to this work, we are grateful to Jeremy Avigad, Vincent Coumans, Henk de Regt, Jean\u2010Pierre Ferrier, Alain Genestier, Marcus Giaquinto, Gerhard Heinzmann, Bruno Leclercq, Jean Paul Van Bendegem, and Roy Wagner. We would also like to thank audiences at the University of Li\u00E8ge, the Archives Henri Poincar\u00E9 in Nancy, the Institute for Science in Society in Nijmegen, the University of Vienna, and the ETH Z\u00FCrich for fruitful discussions. Finally we would like to thank two anonymous reviewers for their very helpful feedback and suggestions. This project has received funding from the European Union's Horizon Europe research and innovative programme under the Marie Sklodowska\u2010Curie grant agreement No 101063894.

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