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Numbers/Frege/Wright: Frege suggests that the fact that our arithmetical language has these qualities is sufficient to establish natural numbers as a sortal concept whose instances, if they have some, are the objects. - WrightVsFrege: but the objects do not have to exist. - Problem: Frege thus demands that empirical concerns are irrelevant. - Then there is also no possibility of an error.
Numbers/BenacerrafVsReduction/Benacerraf/Field: there may be several correlations so that one cannot speak of "the" referent of number words. - Solution/Field: we have to extend "partially denoted" also to sequences of terms. - Then "straight", "prim", etc. become base-dependent predicates whose basis is the sequence of the numbers. - Then one can get mathematical truth (> truth preservation, truth transfer) - E.g. "The number two is Caesar" is neither true nor false. (without truth value).
Definition natural numbers/Zermelo/Benacerraf/Field: 0 is the empty set and every natural number > 0 is the set that is the only element which includes the set which is n-1 - Definition natural numbers/von Neumann/Benacerraf/Field: Every natural number n is the set that has the sets as elements which are the predecessors of n as elements. Fact/Nonfactualism/Field: it is clear that there is no fact about whether Zermelos or von Neumann's approach "presents" the things "correctly" - there is no fact which decides whether numbers are sets. - That is what I call the Definition Structural Insight: it makes no difference what the objects of a mathematical theory are, if they are only in a right relationship with each other.
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