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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II 32
Number/Wittgenstein: not a concept, but a logical form.
II 283
Numbers/cardinal/Wittgenstein: that there are infinitely many cardinals, is a rule that one sets up.
II 343
Number/Frege/WittgensteinVsFrege: a number is a property of a property. - Problem: E.g. for blue-eyed men in the room. - Then the five would be a property of a property - to be a blue-eyed man in the room - e.g. to express that Hans and Paul are two, they would then have a property in common, which not exactly belongs to the other. - ((s) each would have the property to be different from the other.) - Solution/Frege: the property of being Hans or Paul.
II 344
Number/Wittgenstein: are not merely signs. - One can have two items of the form three, but only one number. - ((s) WittgensteinVsFormalism).
II 360
Number/Definition/WittgensteinVsRussell: numerical equality is a prerequisite for a clear correspondence. - Therefore, Russell's definition of the number is useless. - ((s) because circular, if you want to define number via illustration).
II 361
Definition/Wittgenstein: instead of a definition of "number" we must figure out the rules of usage.
II 415
Number/Russell/Wittgenstein: has claimed, 3 is a property that is common to all triads. - ((s) Frege: classes of classes - does Frege not mean objects with classes (instead of properties)?).
II 416
Definition number/WittgensteinVsRussell: the number is an attribute of a function which defines a class, not a property of the extension. - E.g. Extension: it would be a tautology to say, ABC is three. - In contrast, meaingful: to say, in this room are three people.
IV 93
Definition number/numbers/Wittgenstein/Tractatus: 6,021 - the number is the exponent of an operation.
- - -
Waismann I 66
Def natürlichen Zahlen/Wittgenstein: diejenigen, auf die man die Induktion bei Beweisen anwenden kann.

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.

L. Wittgenstein
Vorlesungen 1930-35 Frankfurt 1989

L. Wittgenstein
Das Blaue Buch - Eine Philosophische Betrachtung Frankfurt 1984

L. Wittgenstein
Tractatus Logico Philosophicus Frankfurt/M 1960

Wa I
F. Waismann
Einführung in das mathematische Denken Darmstadt 1996

F. Waismann
Logik, Sprache, Philosophie Stuttgart 1976

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Ed. Martin Schulz, access date 2017-10-22