|Geach, Peter T.
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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.
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".
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.
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.
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.
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.
Logic Matters Oxford 1972