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Variable/Mathematics/Logic/Frege: have nothing to do with change - arithmetic: has nothing to do with quantities like e.g. lengths - only with numbers.
II 83 f
Variable/Frege: not name of an "indeterminate" or "variable" number - "x" has no properties (only in context) - "indeterminately": not an adjective but an adverb to the process of the calculation - "universality": not meaning but suggestion. - Letters are rarely names of numbers: π, i, e are not variables but constants. - Solution: E.g. "n" is used in the antecedent of a conditional sentence.
Number/Frege: E.g. "A variable takes a value." Here, the number must play both roles: as object it becomes a variable, as property it is called value.
Variable/Designation/Description/Frege: "x" designates nothing! - x only hints at numbers - therefore, e.g. "x² + 3x" designates nothing - the whole function only indicates. - On the other hand, "sin" (sinus) is a sign that designates - but not a law yet. - law: e.g. "y = sin x".
Die Grundlagen der Arithmetik Stuttgart 1987
Funktion, Begriff, Bedeutung Göttingen 1994
Logische Untersuchungen Göttingen 1993