Conceptual Blending in Mathematical Creation/Invention

Danny A. J. Gómez Ramírez
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Danny A. J. Gómez Ramírez: Instituto Tecnológico Metropolitano (ITM), Research’s Labs Center Parque i

Chapter Chapter 7 in Artificial Mathematical Intelligence, 2020, pp 109-131 from Springer

Abstract: Abstract Additional domain-specific evidence is presented for the fact that conceptual blending represents a fundamental cognitive mechanism for mathematical creation/invention. Explicitly, a specific version of a well-known formalization of conceptual blending is given in terms of colimits of many-sorted first-order theories. With these formalisms, new nine meta-theorems are proved, describing how to generate recursively fundamental concepts of Fields and Galois theory as blends starting from five basic structural concepts coming from several sub-areas of mathematics such as Group Theory and Topology. In addition, most of the corresponding implementations (using the language CASL) are presented and computed with the Heterogeneous Tool Set (HETS). Additional concepts are generated by weakening inconsistent blends and transforming them into consistent ones. Finally, clear advantages of conceptual blending as a strong tool for enhancing the creative potential of students are described through a case study (Some of the results of this chapter can also be found in Gomez-Ramirez and Smaill (Formal conceptual blending in the (co-)invention of (pure) mathematics. However, all the meta-theorems and cognitive meta-generations presented here are an original product of the author, and in any particular case where secondary cooperation was valuable, this is explicitly acknowledged.).

Date: 2020
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DOI: 10.1007/978-3-030-50273-7_7

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