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Logical Models of Mathematical Texts: The Case of Conventions for Division by Zero
Jan A. Bergstra,
John Tucker 
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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| ISSN: | 0925-8531 1572-9583 |
| Published: |
Springer Science and Business Media LLC
2024
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| Online Access: |
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| URI: | https://cronfa.swan.ac.uk/Record/cronfa66861 |
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| 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 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. |
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| Keywords: |
division by zero; arithmetic; traditional conventions for writing mathematics; legal texts; illegal texts; undecidability; common meadows |
| College: |
Faculty of Science and Engineering |
| Funders: |
Swansea University |