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Vectors on Curved Space



In this paper I provide an ontology for the co-variant vectors, contra-variant vectors and tensors that are familiar from General Relativity. This ontology is developed in response to a problem that Timothy Maudlin uses to argue against universals in the interpretation of physics. The problem is that if vector quantities are universals then there should be a way of identifying the same vector quantity at two different places, but there is no absolute identification of vector quantities, merely a path-relative one.

My solution to the problem is to use the mathematical characterization of vectors as differential operators on scalar fields. On the proposed hypothesis a scalar field is a conjunctive state of affairs, and vector and tensor fields are relations instantiated by scalar fields.