Does the way one writes something in mathematics make any theoretical difference? This is a question that Dr. Manders is addressing in this research. Representational differences have been regarded as of little theoretical significance in the philosophy or foundations of mathematics. Differences of notation or language have been seen as of merely practical interest, as long as they can be bridged by inference-reserving translations. Dr. Manders aims to identify features of representation central to conceptual function in mathematics, and assign them a philosophical role. Notably, he is studying contrasts in "representational granularity" between content-linked representations; roughly, these are differences among representations such as photographs, diagrams, formulae, and text, as to how they break up contents they share into individually representable pieces. He hopes to show how granularity differences underlie the function of concepts, generality, and proof; and thus should share center stage in the philosophical picture of intellectual understanding. It would follow that philosophy of mathematics, in order to get its sights on mathematical understanding, must engage mathematical practices in a way that is sensitive to these features of mathematical representation. Doing so promises to re-involve philosophy of mathematics with current mathematical practice and with the history of mathematics. Dr. Manders argument uses case studies: an analysis of Descartes' geometry, and subsequent geometrical practices in which algebraic representation predominates; and a contrasting analysis of diagram-based geometrical practices. He wants to show how strengths and weaknesses of several such practices as vehicles for the understanding may be understood from grain size differences between their respective representations.
