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Existence/Field: should not be part of the logic - therefore, mathematics cannot be reduced to logic - otherwise too many properties would have to be assumed.
Mathematics/Knowledge/Field: nevertheless, mathematical knowledge is simply logical knowledge because of deflationism - know a lot/little about math: two kinds of knowledge: mathematical knowledge: non-logical knowledge: e.g. what other mathematicians accept -
- this knowledge is empirical.
Pure mathematics/application/Field: e.g. number theory: is not applicable at all to the world - e.g. set theory: must allow the use of elementary elements. - Solution: "impure math": functions that map physical objects to numbers - then the comprehension axioms must also contain non-mathematical vocabulary - E.g. instances of the separation axiom.
Mathematics/Field: can prove to be inconsistent - even if it is extremely improbable - then it would also be non-conservative - so mathematics is not a priori true.
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