Syntactic theory of mathematical expressions

Abstract

Mathematical expressions consist of recursive combinations of numbers, variables, and operators. Theoretical linguists argue that the syntactic mechanisms for natural language also provide a basis for mathematics; however, to date, no attempt is made to support such arguments in a theoretically rigorous manner. The current study uses a methodology adopted in theoretical linguistics to analyze the syntactic properties of mathematical expressions. By reviewing recent behavioral and neuroimaging studies on mathematical syntax, we report that their analyses contain several inconsistencies with theoretical linguistics such as the use of ternary structures. We propose that a syntactic category called Applicative plays a central role in analyzing mathematical expressions with seemingly ternary structures by combining binary structures. In addition to basic arithmetic expressions, we demonstrate analyses on algebraic equations and complex expressions such as integral and differential calculi. The current study is the first attempt to provide a comprehensive framework for analyzing the syntactic structures of mathematical expressions.