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Conservativity/Field: includes some features of necessary truth without actually ever involving truth - ((s) >preservation of truth, truth transfer.)
Def Conservativity/Mathematics/Field: means that every internally consistent combination of nominalist statements is also consistent with the mathematics. - If we can also show that mathematics is not indispensible, we have no reason to believe in mathematical entities anymore.
Def Conservative/Conservativity/Theory/Mathematics/Field: conservative is a mathematical theory that is consistent with every internally consistent physical theory. - This is equivalent to: a mathematical theory is conservative iff for each assertion A about the physical world and each corpus N of such assertions, A does not follow from N + M, if it does not follow from N alone. - ((s) A mathematical theory adds nothing to a physical theory.) - M: mathematical theory - N: nominalistic physical theory. - Def Anti-Realism/Field: (new): an interesting mathematical theory must be conservative, but it must not be true. - Conservative theory: 1) It facilitates inferences - 2) It can substantially occur in the premises of the physical theories.
Point: Conservativity: necessary truth without truth simpliciter. - (i.e. it is has the properties of a necessarily true theory without existing entities.) - Unlike mathematics: science is not conservative. - It must also have non-trivial nominalist consequences.
Truth does not imply conservativity, nor vice versa. - I 63 the fact that mathematics never leads to an error shows that it is conservative, not that it is true. - From conservativity follows that statement with physical objects are materially equivalent to statements of standard mathematics. - Point: they need not have the same truth value!
Conservativity: can explain what follows, but not what does not follow.
Mathematics/Truth/Field: Thesis: good mathematics is not only true, but necessarily true - Point: Conservativity: necessary truth without truth simpliciter.
Conservative expansion does not apply to ontology.
Def Conservative/Science/Field: every inference from nominalistic premises on a nominalistic conclusion that can be made with by means of mathematics can also be made without it - with theoretical entities, unlike mathematical entities, there is no conservativity principle - i.e. conclusions that are made with the assumption of theoretical entities cannot be made without them.
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