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Platonism/Field: his only argument is the applicability of mathematics.
FieldVsPlatonism: has to answer the fictionalist in his language - cannot rely on his "initial plausibility".
Definition Priority thesis/PT/Wright: Thesis: the priority of the syntactic over the ontological categories. - Platonism/Wright: that allows Frege to be a Platonist. - Definition Gödelian Platonism/Wright: in addition: the thesis that mathematical knowledge must be explained by a quasi-perceptual relation - FregeVsGödel - WrightVsGödel: we do not need that.
Definition weak Priority thesis/PT: that each syntactic singular term also works automatically in a semantical way as a singular term.
Equivalence/Platonism/Nominalism/Field: Question: In which sense is a Platonist statement (e.g. "direction 1 = direction 2") and a nominalistic statement equivalent (c1 is parallel to c2)? Problem: if there are no directions, the second cannot be a sequence of the first.
Definition Moderate Platonism/mP/Field: the thesis that there are abstract objects like numbers. - Then there are probably also relations between numbers and objects. - Moderate Platonism: these relations are conventions, derived from physical relations. - Definition Heavy Duty Platonism/HDP/Field: takes relations between objects and numbers as a bare fact.
Strong moderation condition/(Field (pro): it is possible to formulate physical laws without relation between objects and numbers.
Heavy Duty Platonism/Field: assumes size relationships between objects and numbers. - FieldVs: instead only between objects.
Platonism/Mathematics/VsStructuralism/Field: isomorphic mathematical fields do not need to be indistinguishable.
Quinish Platonism/Field: as a basic concept a certain concept of quantity, from which all other mathematical objects are constructed. So natural numbers and real numbers would actually be sets.
Number/Points/Field: no Platonist will identify real numbers with points on a physical line. - That would be too arbitrary ( "What line?") - What should be zero point - What should be 1?
Platonistic/Field: are terms such as e.g. gradient, Laplace Equation, etc.
1st order Platonism/Field: accepts abstract entities, but no 2nd order logic - Problem: but he needs these (because of power quantifiers).
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