Cognitive Structures and Processes in the Interpretation of Mathematical Proofs
Adrian Robert, August 2001
We propose that mathematical concepts are cognitively represented as blended
(Fauconnier & Turner, 1998) combinations of schemas acquired through everyday
experience; the properties of these source schemas generate the relational
characteristics of the concepts that are explored in mathematical
proofs. Proofs are understood by constructing novel blends of the concepts
under consideration, and they are verified through paying attention to the
presence or absence of dissonance in the associated mappings. We analyze these
mappings through examples and compare the analysis with the results of an
empirical survey of students' mental processes in studying the proofs. The
results suggest explanations for the trends of increasing rigor and conceptual
"fine-grainedness" in the historical development of mathematics, and they
potentially shed light on processes of natural language understanding because
the mechanisms at work in the two cases appear qualitatively similar.
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