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Logical models of mathematical texts: the case of conventions for division by zero
Journal of Logic, Language and Information
Swansea University Author:
John Tucker
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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...
Published in: | Journal of Logic, Language and Information |
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URI: | https://cronfa.swan.ac.uk/Record/cronfa66861 |
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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 Springer division by zero; arithmetic; traditional conventions for writing mathematics; legal texts; illegal texts; undecidability; common meadows 0 0 0 0001-01-01 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 2024-07-06T18:50:34.9415566 2024-06-23T18:12:30.7582055 Faculty of Science and Engineering School of Mathematics and Computer Science - Computer Science John Tucker 0000-0003-4689-8760 1 Jan Bergstra 2 |
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 |
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431b3060563ed44cc68c7056ece2f85e |
author_id_fullname_str_mv |
431b3060563ed44cc68c7056ece2f85e_***_John Tucker |
author |
John Tucker |
author2 |
John Tucker Jan Bergstra |
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Journal article |
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Journal of Logic, Language and Information |
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Swansea University |
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10.1007/s10849-024-09420-w |
publisher |
Springer |
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Faculty of Science and Engineering |
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Faculty of Science and Engineering |
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Faculty of Science and Engineering |
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School of Mathematics and Computer Science - Computer Science{{{_:::_}}}Faculty of Science and Engineering{{{_:::_}}}School of Mathematics and Computer Science - Computer Science |
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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 |
0001-01-01T18:50:33Z |
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1803852954500857856 |
score |
11.017731 |