# Philosophy Lexicon of Arguments

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.

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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.

Author Item Summary Meta data
I, 215ff
Numbers/Geach: naming nothing- not: E.g.: "There are two Daimon and Phobos" - how often a concept is realized is not a feature of the term. ((s) GeachVsMeixner) - Unit/Multiplicity/Geach: cannot be attributed to an object - Solution/Frege: they are attributed to the terms under which the objects fall. - Numbers: in mathematics sometimes as objects with properties E.g. Divisibility - Geach: then we need an identity criterion - Frege: Equality in numbers: "There is a one-to-one correspondence of Fs and Gs". - N.B.: this does not mean that the Fs or the Gs refer to a single object - a class. - Solution: Relation instead of class - E.g. One puts next to each plate a knife: no class but relation.
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I 220
Numbers/Frege: Self-critique: Classes must not be used to explain what numbers are, otherwise contradiction: "one and the same object is both, the class of the M's and the class of the G's, although an object (this object, e.g. number(!)) can be an M without being a G. " - (+) - This shows that the original concept of a class contained contradictions. - Numbers can be objects (with properties such as divisibility), classes cannot. - Not contradictory: "one and the same object: the number (not class!) of the F's and the number of K's".
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I 221f
Numbers/GeachVsFrege: Number is not "number of objects". - With this he rejects his own concerns to say that "the object of a number belongs to a class" (wrong). - "The number of the A's" is to mean: "the number of the class of all A's" (wrong) Solution/Geach: (as Frege elsewhere): the empty place in "the number of .." and "how many ... are there?" Can only be filled with a keyword in the plural - not with the name of an object or a list of objects. - conceptual word instead of class.
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I 225
Numbers/Classes/Geach: not classes of classes. (Frege dito) - if we connect a number (falsely) to a class a, we actually combine it with the property expressed by "___ is an element of a". This is not trivial - because when we associate a number with a property, the property is usually not expressed in that form.
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I 225
Numbers/Classes/Geach: false: "The number of F's is 0" - correct: "The class of F's is 0" - Class as number are equally specified by the mention of a property.
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I 235
Numbers/Frege/Geach: not classes of classes (Frege does not say this either). - The error stems from the idea that one could start with concrete objects and then group them into groups and supergroups.

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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.

Gea I
P.T. Geach
Logic Matters Oxford 1972

> Counter arguments against Geach
> Counter arguments in relation to Numbers ...

Ed. Martin Schulz, access date 2018-09-25