Semantic and Syntactic Levels of Mathematical Thinking and Learning

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Mathematical thinking appears to comprise both, processes of formal deduction and processes of semantic interpretation. Distinct formal shapes of mathematical statements provide a substantial basis for syntactic conceptions of logical entailment. On the other hand, mathematical cognition successfully relies on models of quasi-empirical concept formation and knowledge acquisition. In this work, the role of semantic and syntactic aspects in mathematical reasoning is examined specifically within domains of predicate logic and elementary algebra. An experimental framework is developed that allows one to systematically distinguish between semantic and syntactic forms of mathematical thinking. Whereas, from a psychological perspective, both semantic and syntactic levels of processing seem to play an important role in various kinds of mathematical reasoning, it remains unclear how these different modalities of conception are interrelated. It is shown that important links are given by everyday's language which allows one to meaningfully shift from one level of description to another, multiple forms of processing are thus integrated into a comprehensive net of conceptual strategies. The results presented are discussed in terms of educational concepts as well as epistemological issues. Semantic perspectives suggest that, besides traditional tools of formal conception, also empiristic forms of mathematical methodology should be considered. The tools developed are intended to contribute to a deeper understanding of the basic modalities of mathematical reasoning, their interrelations give an impression of the outstanding richness and complexity of the processes involved.