Mathematicians’ conceptualisations of isomorphism and homomorphism: a story of contexts, contrasts, and utility

Abstract

ABSTRACT Isomorphism and homomorphism are fundamental topics in abstract algebra, but how mathematicians characterise the utility of these concepts remains understudied. Based on interviews with nine research-active mathematicians, we confirmed use of previously noted conceptual metaphors and noted new metaphors and metaphor clusters. We expand on prior work by highlighting five thematic contrasts between metaphors for these concepts: context dependent sameness, classification of objects versus computing maps, indistinguishable versus distinct objects, structure-preservation versus structure loss, and information gain versus structure loss. Implications include the centrality of sameness to thinking about isomorphism and homomorphism but the need for care in how that sameness is framed; the value of being able to attend to the object and map; the importance of being able to determine which structures are preserved and which are lost in context; and the existence of a conceptual purpose for homomorphism as a tool for changing perspectives.