I, 215ff
Numbers/Geach: numbers do not name anything. Not: E.g.: "There are two Daimon and Phobos".
How often a concept is realized is not a feature of the term. ((s) GeachVsMeixner).
Unity/Multiplicity/Geach: cannot be attributed to an object.
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Unity and Multiplicity.
Solution/Frege: Numbers are attributed to the terms under which the objects fall.
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Numbers/Frege, >
Concept/Frege, >
Object/Frege.
Numbers/Geach: in mathematics sometimes as objects with properties E.g. Divisibility.
Geach: then we need an identity criterion.
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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. Frege: One puts next to each plate a knife: no class but a relation.
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Equality, >
Classes, >
Sets, >
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. " - (+) -
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Sets/Frege, >
Classes/Frege.
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".
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. - A conceptual word instead of a class.
I 225
Numbers/Classes/Geach: Numbers are not classes of classes.
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.
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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Mention.
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.