|Numbers: whether numbers are objects or concepts, has been controversial in the philosophical discussion for millennia. The most widely accepted definition today is given by G. Frege (G. Frege, Grundlagen der Arithmetik 1987, p. 79ff). Frege-inspired notions represent numbers as classes of classes, or as second-level terms, or as that with one measure the size of sets. Up until today, there is an ambiguity between concept and object in the discussion of numbers. See also counting, sets, measurements, mathematics, abstract objects, mathematical entities, theoretical entities, number, platonism._____________Annotation: The above characterizations of concepts are neither definitions nor exhausting presentations of problems related to them. Instead, they are intended to give a short introduction to the contributions below. – Lexicon of Arguments. |
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).
Def 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.
Def 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._____________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.
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