@misc{Lexicon of Arguments, title = {Quotation from: Lexicon of Arguments – Concepts - Ed. Martin Schulz, 29 Mar 2024}, author = {Geach,Peter}, subject = {Numbers}, note = {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. >Unity and Multiplicity. Solution/Frege: Numbers are attributed to the terms under which the objects fall. >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. >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. >Equality, >Classes, >Sets, >Relation. --- 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. " - (+) - >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. >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.}, note = { Gea I P.T. Geach Logic Matters Oxford 1972 }, file = {http://philosophy-science-humanities-controversies.com/listview-details.php?id=286631} url = {http://philosophy-science-humanities-controversies.com/listview-details.php?id=286631} }