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Logical Models of Mathematical Texts: The Case of Conventions for Division by Zero

Jan A. Bergstra, John Tucker Orcid Logo

Journal of Logic, Language and Information, Volume: 33, Issue: 4-5, Pages: 277 - 298

Swansea University Author: John Tucker Orcid Logo

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DOI (Published version): 10.1007/s10849-024-09420-w

Abstract

Arithmetical texts involving division are governed by conventions that avoid the risk of problems to do with division by zero (DbZ). A model for elementary arithmetic texts is given, and with the help of many examples and counter examples a partial description of what may be called traditional conve...

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Published in: Journal of Logic, Language and Information
ISSN: 0925-8531 1572-9583
Published: Springer Science and Business Media LLC 2024
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URI: https://cronfa.swan.ac.uk/Record/cronfa66861
first_indexed 2024-07-03T09:08:03Z
last_indexed 2025-02-22T05:54:56Z
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spelling 2025-02-21T14:55:19.9421950 v2 66861 2024-06-23 Logical Models of Mathematical Texts: The Case of Conventions for Division by Zero 431b3060563ed44cc68c7056ece2f85e 0000-0003-4689-8760 John Tucker John Tucker true false 2024-06-23 MACS Arithmetical texts involving division are governed by conventions that avoid the risk of problems to do with division by zero (DbZ). A model for elementary arithmetic texts is given, and with the help of many examples and counter examples a partial description of what may be called traditional conventions on DbZ is explored. We introduce the informal notions of legal and illegal texts to analyse these conventions. First, we show that the legality of a text is algorithmically undecidable. As a consequence, we know that there is no simple sound and complete set of guidelines to determine unambiguously how DbZ is to be avoided. We argue that these observations call for further explorations of mathematical conventions. We propose a method using logics to progress the analysis of legality versus illegality: arithmetical texts in a model can be transformed into logical formulae over special total algebras that are able to approximate partiality but in a total world. The algebras we use are called common meadows. This deep dive into informal mathematical practice using formal methods opens up questions about DbZ which we address in conclusion. Journal Article Journal of Logic, Language and Information 33 4-5 277 298 Springer Science and Business Media LLC 0925-8531 1572-9583 Division by zero; Arithmetic; Traditional conventions for writing mathematics; Legal texts; Illegal texts; Undecidability; Common meadows 1 12 2024 2024-12-01 10.1007/s10849-024-09420-w COLLEGE NANME Mathematics and Computer Science School COLLEGE CODE MACS Swansea University SU Library paid the OA fee (TA Institutional Deal) Swansea University 2025-02-21T14:55:19.9421950 2024-06-23T18:12:30.7582055 Faculty of Science and Engineering School of Mathematics and Computer Science - Computer Science Jan A. Bergstra 1 John Tucker 0000-0003-4689-8760 2 66861__30941__a7870aa89abe40cc84134409e76669ba.pdf 66861.VoR.pdf 2024-07-23T11:17:36.0877547 Output 367487 application/pdf Version of Record true © The Author(s) 2024. This article is licensed under a Creative Commons Attribution 4.0 International License. true eng http://creativecommons.org/licenses/by/4.0/
title Logical Models of Mathematical Texts: The Case of Conventions for Division by Zero
spellingShingle Logical Models of Mathematical Texts: The Case of Conventions for Division by Zero
John Tucker
title_short Logical Models of Mathematical Texts: The Case of Conventions for Division by Zero
title_full Logical Models of Mathematical Texts: The Case of Conventions for Division by Zero
title_fullStr Logical Models of Mathematical Texts: The Case of Conventions for Division by Zero
title_full_unstemmed Logical Models of Mathematical Texts: The Case of Conventions for Division by Zero
title_sort Logical Models of Mathematical Texts: The Case of Conventions for Division by Zero
author_id_str_mv 431b3060563ed44cc68c7056ece2f85e
author_id_fullname_str_mv 431b3060563ed44cc68c7056ece2f85e_***_John Tucker
author John Tucker
author2 Jan A. Bergstra
John Tucker
format Journal article
container_title Journal of Logic, Language and Information
container_volume 33
container_issue 4-5
container_start_page 277
publishDate 2024
institution Swansea University
issn 0925-8531
1572-9583
doi_str_mv 10.1007/s10849-024-09420-w
publisher Springer Science and Business Media LLC
college_str Faculty of Science and Engineering
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hierarchy_top_id facultyofscienceandengineering
hierarchy_top_title Faculty of Science and Engineering
hierarchy_parent_id facultyofscienceandengineering
hierarchy_parent_title Faculty of Science and Engineering
department_str School of Mathematics and Computer Science - Computer Science{{{_:::_}}}Faculty of Science and Engineering{{{_:::_}}}School of Mathematics and Computer Science - Computer Science
document_store_str 1
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description Arithmetical texts involving division are governed by conventions that avoid the risk of problems to do with division by zero (DbZ). A model for elementary arithmetic texts is given, and with the help of many examples and counter examples a partial description of what may be called traditional conventions on DbZ is explored. We introduce the informal notions of legal and illegal texts to analyse these conventions. First, we show that the legality of a text is algorithmically undecidable. As a consequence, we know that there is no simple sound and complete set of guidelines to determine unambiguously how DbZ is to be avoided. We argue that these observations call for further explorations of mathematical conventions. We propose a method using logics to progress the analysis of legality versus illegality: arithmetical texts in a model can be transformed into logical formulae over special total algebras that are able to approximate partiality but in a total world. The algebras we use are called common meadows. This deep dive into informal mathematical practice using formal methods opens up questions about DbZ which we address in conclusion.
published_date 2024-12-01T05:32:24Z
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