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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