Semantic and syntactic proof productions

Keith Weber, Lara Alcock

Research output: Contribution to journalArticlepeer-review

97 Scopus citations


In this paper, we distinguish between two ways that an individual can construct a formal proof. We define a syntactic proof production to occur when the prover draws inferences by manipulating symbolic formulae in a logically permissible way. We define a semantic proof production to occur when the prover uses instantiations of mathematical concepts to guide the formal inferences that he or she draws. We present two independent exploratory case studies from group theory and real analysis that illustrate both types of proofs. We conclude by discussing what types of concept understanding are required for each type of proof production and by illustrating the weaknesses of syntactic proof productions.

Original languageEnglish (US)
Pages (from-to)209-234
Number of pages26
JournalEducational Studies in Mathematics
Issue number2-3
StatePublished - 2004

ASJC Scopus subject areas

  • Mathematics(all)
  • Education


  • abstract algebra
  • advanced mathematical concepts
  • advanced mathematical thinking
  • convergent sequence
  • formal reasoning
  • isomorphism
  • limits
  • proofs
  • real analysis


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