Hartry Field on Mathematics - Dictionary of Arguments
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. - E.g. Knowing a lot/little about maths: 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 mathematics": 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._____________Explanation of symbols: Roman numerals indicate the source, arabic numerals indicate the page number. The corresponding books are indicated on the right hand side. ((s)…): Comment by the sender of the contribution. Translations: Dictionary of Arguments The note [Concept/Author], [Author1]Vs[Author2] or [Author]Vs[term] resp. "problem:"/"solution:", "old:"/"new:" and "thesis:" is an addition from the Dictionary of Arguments. If a German edition is specified, the page numbers refer to this edition.
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
"Realism and Relativism", The Journal of Philosophy, 76 (1982), pp. 553-67
Theories of Truth, Paul Horwich, Aldershot 1994