Lakatos-style collaborative mathematics through dialectical, structured and abstract argumentation
The simulation of mathematical reasoning has been a driving force throughout the history of Artificial Intelligence research. However, despite significant successes in computer mathematics, computers are not widely used by mathematicians apart from their quotidian applications. An oft-cited reason f...
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| Published in: | Artificial intelligence Vol. 246; pp. 181 - 219 |
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| Main Authors: | , , , , |
| Format: | Journal Article |
| Language: | English |
| Published: |
Amsterdam
Elsevier B.V
01.05.2017
Elsevier Science Ltd |
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| ISSN: | 0004-3702, 1872-7921 |
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| Abstract | The simulation of mathematical reasoning has been a driving force throughout the history of Artificial Intelligence research. However, despite significant successes in computer mathematics, computers are not widely used by mathematicians apart from their quotidian applications. An oft-cited reason for this is that current computational systems cannot do mathematics in the way that humans do. We draw on two areas in which Automated Theorem Proving (ATP) is currently unlike human mathematics: firstly in a focus on soundness, rather than understandability of proof, and secondly in social aspects. Employing techniques and tools from argumentation to build a framework for mixed-initiative collaboration, we develop three complementary arcs. In the first arc – our theoretical model – we interpret the informal logic of mathematical discovery proposed by Lakatos, a philosopher of mathematics, through the lens of dialogue game theory and in particular as a dialogue game ranging over structures of argumentation. In our second arc – our abstraction level – we develop structured arguments, from which we induce abstract argumentation systems and compute the argumentation semantics to provide labelings of the acceptability status of each argument. The output from this stage corresponds to a final, or currently accepted proof artefact, which can be viewed alongside its historical development. Finally, in the third arc – our computational model – we show how each of these formal steps is available in implementation. In an appendix, we demonstrate our approach with a formal, implemented example of real-world mathematical collaboration. We conclude the paper with reflections on our mixed-initiative collaborative approach. |
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| AbstractList | The simulation of mathematical reasoning has been a driving force throughout the history of Artificial Intelligence research. However, despite significant successes in computer mathematics, computers are not widely used by mathematicians apart from their quotidian applications. An oft-cited reason for this is that current computational systems cannot do mathematics in the way that humans do. We draw on two areas in which Automated Theorem Proving (ATP) is currently unlike human mathematics: firstly in a focus on soundness, rather than understandability of proof, and secondly in social aspects. Employing techniques and tools from argumentation to build a framework for mixed-initiative collaboration, we develop three complementary arcs. In the first arc – our theoretical model – we interpret the informal logic of mathematical discovery proposed by Lakatos, a philosopher of mathematics, through the lens of dialogue game theory and in particular as a dialogue game ranging over structures of argumentation. In our second arc – our abstraction level – we develop structured arguments, from which we induce abstract argumentation systems and compute the argumentation semantics to provide labelings of the acceptability status of each argument. The output from this stage corresponds to a final, or currently accepted proof artefact, which can be viewed alongside its historical development. Finally, in the third arc – our computational model – we show how each of these formal steps is available in implementation. In an appendix, we demonstrate our approach with a formal, implemented example of real-world mathematical collaboration. We conclude the paper with reflections on our mixed-initiative collaborative approach. |
| Author | Budzynska, Katarzyna Pease, Alison Lawrence, John Corneli, Joseph Reed, Chris |
| Author_xml | – sequence: 1 givenname: Alison orcidid: 0000-0003-1856-9599 surname: Pease fullname: Pease, Alison email: a.pease@dundee.ac.uk organization: Centre for Argument Technology, University of Dundee, UK – sequence: 2 givenname: John surname: Lawrence fullname: Lawrence, John organization: Centre for Argument Technology, University of Dundee, UK – sequence: 3 givenname: Katarzyna surname: Budzynska fullname: Budzynska, Katarzyna organization: Centre for Argument Technology, University of Dundee, UK – sequence: 4 givenname: Joseph surname: Corneli fullname: Corneli, Joseph organization: Department of Computing, Goldsmiths, University of London, UK – sequence: 5 givenname: Chris surname: Reed fullname: Reed, Chris organization: Centre for Argument Technology, University of Dundee, UK |
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| Keywords | Automated reasoning Abstract argumentation Argumentation Mathematical argument Structured argumentation Philosophy of mathematical practice Automated theorem proving Lakatos Collaborative intelligence Social creativity Dialogue games |
| Language | English |
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| SubjectTerms | Abstract argumentation Argumentation Automated reasoning Automated theorem proving Collaboration Collaborative intelligence Computer simulation Dialogue games Game theory Lakatos Logic Mathematical analysis Mathematical argument Mathematical models Mathematics Philosophy of mathematical practice Reasoning Semantics Social creativity Social factors Structured argumentation Theorem proving |
| Title | Lakatos-style collaborative mathematics through dialectical, structured and abstract argumentation |
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