Towards a category theory approach to analogy: Analyzing re-representation and acquisition of numerical knowledge
Category Theory, a branch of mathematics, has shown promise as a modeling framework for higher-level cognition. We introduce an algebraic model for analogy that uses the language of category theory to explore analogy-related cognitive phenomena. To illustrate the potential of this approach, we use t...
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| Published in: | PLoS computational biology Vol. 13; no. 8; p. e1005683 |
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| Main Authors: | , |
| Format: | Journal Article |
| Language: | English |
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Public Library of Science
25.08.2017
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| ISSN: | 1553-7358, 1553-734X, 1553-7358 |
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| Abstract | Category Theory, a branch of mathematics, has shown promise as a modeling framework for higher-level cognition. We introduce an algebraic model for analogy that uses the language of category theory to explore analogy-related cognitive phenomena. To illustrate the potential of this approach, we use this model to explore three objects of study in cognitive literature. First, (a) we use commutative diagrams to analyze an effect of playing particular educational board games on the learning of numbers. Second, (b) we employ a notion called coequalizer as a formal model of re-representation that explains a property of computational models of analogy called "flexibility" whereby non-similar representational elements are considered matches and placed in structural correspondence. Finally, (c) we build a formal learning model which shows that re-representation, language processing and analogy making can explain the acquisition of knowledge of rational numbers. These objects of study provide a picture of acquisition of numerical knowledge that is compatible with empirical evidence and offers insights on possible connections between notions such as relational knowledge, analogy, learning, conceptual knowledge, re-representation and procedural knowledge. This suggests that the approach presented here facilitates mathematical modeling of cognition and provides novel ways to think about analogy-related cognitive phenomena. |
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| AbstractList | Category Theory, a branch of mathematics, has shown promise as a modeling framework for higher-level cognition. We introduce an algebraic model for analogy that uses the language of category theory to explore analogy-related cognitive phenomena. To illustrate the potential of this approach, we use this model to explore three objects of study in cognitive literature. First, (a) we use commutative diagrams to analyze an effect of playing particular educational board games on the learning of numbers. Second, (b) we employ a notion called coequalizer as a formal model of re-representation that explains a property of computational models of analogy called "flexibility" whereby non-similar representational elements are considered matches and placed in structural correspondence. Finally, (c) we build a formal learning model which shows that re-representation, language processing and analogy making can explain the acquisition of knowledge of rational numbers. These objects of study provide a picture of acquisition of numerical knowledge that is compatible with empirical evidence and offers insights on possible connections between notions such as relational knowledge, analogy, learning, conceptual knowledge, re-representation and procedural knowledge. This suggests that the approach presented here facilitates mathematical modeling of cognition and provides novel ways to think about analogy-related cognitive phenomena. Category Theory, a branch of mathematics, has shown promise as a modeling framework for higher-level cognition. We introduce an algebraic model for analogy that uses the language of category theory to explore analogy-related cognitive phenomena. To illustrate the potential of this approach, we use this model to explore three objects of study in cognitive literature. First, (a) we use commutative diagrams to analyze an effect of playing particular educational board games on the learning of numbers. Second, (b) we employ a notion called coequalizer as a formal model of re-representation that explains a property of computational models of analogy called "flexibility" whereby non-similar representational elements are considered matches and placed in structural correspondence. Finally, (c) we build a formal learning model which shows that re-representation, language processing and analogy making can explain the acquisition of knowledge of rational numbers. These objects of study provide a picture of acquisition of numerical knowledge that is compatible with empirical evidence and offers insights on possible connections between notions such as relational knowledge, analogy, learning, conceptual knowledge, re-representation and procedural knowledge. This suggests that the approach presented here facilitates mathematical modeling of cognition and provides novel ways to think about analogy-related cognitive phenomena.Category Theory, a branch of mathematics, has shown promise as a modeling framework for higher-level cognition. We introduce an algebraic model for analogy that uses the language of category theory to explore analogy-related cognitive phenomena. To illustrate the potential of this approach, we use this model to explore three objects of study in cognitive literature. First, (a) we use commutative diagrams to analyze an effect of playing particular educational board games on the learning of numbers. Second, (b) we employ a notion called coequalizer as a formal model of re-representation that explains a property of computational models of analogy called "flexibility" whereby non-similar representational elements are considered matches and placed in structural correspondence. Finally, (c) we build a formal learning model which shows that re-representation, language processing and analogy making can explain the acquisition of knowledge of rational numbers. These objects of study provide a picture of acquisition of numerical knowledge that is compatible with empirical evidence and offers insights on possible connections between notions such as relational knowledge, analogy, learning, conceptual knowledge, re-representation and procedural knowledge. This suggests that the approach presented here facilitates mathematical modeling of cognition and provides novel ways to think about analogy-related cognitive phenomena. Category Theory, a branch of mathematics, has shown promise as a modeling framework for higher-level cognition. We introduce an algebraic model for analogy that uses the language of category theory to explore analogy-related cognitive phenomena. To illustrate the potential of this approach, we use this model to explore three objects of study in cognitive literature. First, (a) we use commutative diagrams to analyze an effect of playing particular educational board games on the learning of numbers. Second, (b) we employ a notion called coequalizer as a formal model of re-representation that explains a property of computational models of analogy called “flexibility” whereby non-similar representational elements are considered matches and placed in structural correspondence. Finally, (c) we build a formal learning model which shows that re-representation, language processing and analogy making can explain the acquisition of knowledge of rational numbers. These objects of study provide a picture of acquisition of numerical knowledge that is compatible with empirical evidence and offers insights on possible connections between notions such as relational knowledge, analogy, learning, conceptual knowledge, re-representation and procedural knowledge. This suggests that the approach presented here facilitates mathematical modeling of cognition and provides novel ways to think about analogy-related cognitive phenomena. Analogy is claimed to be the core of human cognition due to its pervasive involvement in phenomena such as language, reasoning and learning. However, this phenomenon has mainly been studied in isolation through computational methods, which has made it difficult to appreciate its different roles in diverse cognitive phenomena. Recent studies have signaled that abstract concepts from category theory are able to describe constructions carried out by the human mind, thus presenting this formal theory as a general framework to study cognition. Our contribution here is to provide a model of analogy that bridges a gap between the formal notions of category theory and the psychological notions of cognition. We illustrate the usefulness of this approach by using our model to represent and analyze three different cognitive phenomena. Besides showing that some abstract mathematical concepts can describe concrete cognitive notions such as re-representation, learning, conceptual knowledge, and procedural knowledge, the process of mathematical modeling is presented here as an alternative for outlining relations between cognitive notions and for developing novel conceptualizations of the involved cognitive phenomena. |
| Audience | Academic |
| Author | Dartnell, Pablo Navarrete, Jairo A. |
| AuthorAffiliation | 1 Departamento de Ciencias Básicas, Universidad del Bío-Bío, Av. Andrés Bello 702, Chillán, Chile 2 Centro de Modelamiento Matemático (CMM), Centro de Investigación Avanzada en Educación (CIAE) de la Universidad de Chile, Periodista José Carrasco Tapia Nº 75, Santiago, Chile University of Minnesota, UNITED STATES |
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| Author_xml | – sequence: 1 givenname: Jairo A. orcidid: 0000-0002-2785-2751 surname: Navarrete fullname: Navarrete, Jairo A. – sequence: 2 givenname: Pablo orcidid: 0000-0003-1025-7021 surname: Dartnell fullname: Dartnell, Pablo |
| BackLink | https://www.ncbi.nlm.nih.gov/pubmed/28841643$$D View this record in MEDLINE/PubMed |
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| CitedBy_id | crossref_primary_10_1016_j_tics_2021_03_002 crossref_primary_10_3390_logics3040012 crossref_primary_10_1007_s11218_020_09545_3 crossref_primary_10_5964_jnc_14249 crossref_primary_10_1016_j_cogsys_2020_10_003 crossref_primary_10_1016_j_cedpsych_2018_06_006 crossref_primary_10_3389_fpsyg_2018_01926 crossref_primary_10_1111_mbe_12335 |
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| Copyright | COPYRIGHT 2017 Public Library of Science 2017 Public Library of Science. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited: Navarrete JA, Dartnell P (2017) Towards a category theory approach to analogy: Analyzing re-representation and acquisition of numerical knowledge. PLoS Comput Biol 13(8): e1005683. https://doi.org/10.1371/journal.pcbi.1005683 2017 Navarrete, Dartnell 2017 Navarrete, Dartnell 2017 Public Library of Science. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited: Navarrete JA, Dartnell P (2017) Towards a category theory approach to analogy: Analyzing re-representation and acquisition of numerical knowledge. PLoS Comput Biol 13(8): e1005683. https://doi.org/10.1371/journal.pcbi.1005683 |
| Copyright_xml | – notice: COPYRIGHT 2017 Public Library of Science – notice: 2017 Public Library of Science. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited: Navarrete JA, Dartnell P (2017) Towards a category theory approach to analogy: Analyzing re-representation and acquisition of numerical knowledge. PLoS Comput Biol 13(8): e1005683. https://doi.org/10.1371/journal.pcbi.1005683 – notice: 2017 Navarrete, Dartnell 2017 Navarrete, Dartnell – notice: 2017 Public Library of Science. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited: Navarrete JA, Dartnell P (2017) Towards a category theory approach to analogy: Analyzing re-representation and acquisition of numerical knowledge. PLoS Comput Biol 13(8): e1005683. https://doi.org/10.1371/journal.pcbi.1005683 |
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| SubjectTerms | Algorithms Artificial intelligence Biology and Life Sciences Cognition Cognition & reasoning Cognition - physiology Cognitive ability Computational Biology Computer applications Computer Simulation Educational psychology Empirical analysis Funding Game Theory Humans Knowledge Language Learning Mathematical models Mathematics education Methods Models, Theoretical Natural language processing Neural circuitry Neurological research Social Sciences Theorems (Mathematics) Theory |
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| Title | Towards a category theory approach to analogy: Analyzing re-representation and acquisition of numerical knowledge |
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