Philosophy Lexicon of Arguments

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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.

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I 219
Not all abstract objects are properties: numbers, classes, functions, geometric shapes, ideas, possibilities - give up or re-trace abstract objects - can be distinguished by the faithful use of "-ness" from concrete objects.
II 26
Numbers: quantification is objectification, numeric name - diagonals: are irrational, scope: is transcendental.
Measure: measuring scale: multidigit general term, puts physical objects in relation to pure numbers - counting: measuring a class.
II 28
Numbers/Ontology: Numbers are merely "facon de parler" - higher classes needed to replace numbers - otherwise only physical objects.
IX 54
Numbers/Frege/Quine: like predecessors (ancestor): Definition predecessor/Frege: the common elements of all classes for which the initial condition was fulfilled: "y e z" and the seclusion condition: which resulted in "a" "z 0 e z]} - Problem: the successor relation could also lead to things that are not >numbers - Numbers/Quine: we will mainly use them as a measure of multiplicities (that is how Frege had defined them) - a hasx elements"- the scheme goes back to Frege: a has 0 elements a = L. - s has SA°x elements Ey(y e a n _{y} has x elements.
IX 59
Numbers/Zermelo: (1908) takes L as 0, then {x} as S°x for each x. (i.e. "{x}" always one more than x! - {x} successor of x! - as numbers we then receive L, {L}, {{L}} .. etc.
IX 59~
Numbers/Von Neumann (1923) regards every natural number as the class of the previous numbers: 0 becomes L again, - but successor S°x does not become {x}, but x U {x}. (Combined with) - 1: as in Zermelo: equal {L} - but 2: {0,1} or {L,{L}}. - 3: {0,1,2} or {L,{L},{L,{L,{L}}} - for von Neuman this says that a has x elements, that a ~ x. (Number, equipotent) - that’s just the "a ~ {y: y < x}" from chapter 11, because for von Neumann is x = {y: y.

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.

W.V.O. Quine
Wort und Gegenstand Stuttgart 1980

W.V.O. Quine
Theorien und Dinge Frankfurt 1985

W.V.O. Quine
Grundzüge der Logik Frankfurt 1978

W.V.O. Quine
Mengenlehre und ihre Logik Wiesbaden 1967

W.V.O. Quine
Die Wurzeln der Referenz Frankfurt 1989

W.V.O. Quine
Unterwegs zur Wahrheit Paderborn 1995

W.V.O. Quine
From a logical point of view Cambridge, Mass. 1953

W.V.O. Quine
Bezeichnung und Referenz
Zur Philosophie der idealen Sprache, J. Sinnreich (Hg), München 1982

W.V.O. Quine
Philosophie der Logik Bamberg 2005

W.V.O. Quine
Ontologische Relativität Frankfurt 2003

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Ed. Martin Schulz, access date 2017-11-21