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Theory/Object level/Field: we assume a theory here instead of the truth of the theory. Problem: the theory requires mathematical entities.
Physics/Theory/Language/Ontology/Field: Thesis: in the typical physical language, sentences are essential for the description of observations that contain mathematical entities. Then a theory without mathematical entities does not allow any inference about distances and masses. - Solution: new (comparative) predicates: For example, the distance between x and y is r-times the distance between z and w, etc. - For example, the velocity of y relative to y multiplied by the time difference between z and w is r-times spatial distance between u and v (Definition acceleration without numbers). - r: is a rational number. - This distinguishes the predicates in the family - NominalismVs: these are too many predicates.
Theory/Truth/Field: it is the assertion that the axioms of the theory are true of their objects at certain points of time (or at all times) - not the theory itself. - Variables: We leave it out here very often, but they must be understood as implicitly existing. - Instead of "pain has that and that causal role" we must say: "For every t and every c (organism) of type S to t, pain has that and that causal role in c to t".
Ideal Theory/Quine/Field: (Quine 1960, 23-4): Suppose there is an ideal theory (in the future) that could be considered as completely true: - Problem: this ideal theory could not correct the truth values of our actual (present) individual sentences. - reason: there is no general sense in which one can equate a single sentence of a theory with a single sentence of another theory. - Quine/(s): there is no inter-theoretical translatability. - Thus there is no Truth-predicate for single sentences of a theory - Falsehood is distributed to the whole theory. - There is no fact that distributes falsehood to single sentences. - FieldVsQuine: therefore the sentences are not "intertheoretically meaningless". - Solution/Field: "partial denotation": Newton's mass partially denoted. - FieldVsKuhn/FieldVsIncommensurability: denotational refinement: (later only partial quantity) means no incommensurability.
Realism, Mathematics and Modality Oxford New York 1989
Truth and the Absence of Fact Oxford New York 2001
Science without numbers Princeton New Jersey 1980