|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 18 f
Numbers/Frege: E.g. 16 = 4², 4 x 4 = 4² Here we see that equality of meaning does not lead to equality of thought.
I 66 ff
The figure contains the expression of a concept.
II 66 ff
Properties will be expressed by a concept. - A concept may fall under a higher one. E.g. There is at least one square root of 4. This is not a statement about a certain number 2, nor about -2, but about a concept, namely the square root of 4.
II 81 f
There are no variable numbers. - Variable: do we not denote variable numbers by x, y, z? This way of speaking is used, but these letters are not proper names of variable numbers, like "2" and "3" are proper names of constant numbers. - We cannot specify which properties "x" has in contrast to y.
Variable: not a proper name of an indefinite or variable number. - X has no properties (only in the context). - "Indefinitely" is not an adjective, but an adverb for the process the calculating. - Generality/Frege: not meaning but hint. - Proper Names: π, i, e are not variables! - Generality: here, the number has to play two roles: as an object it is called a variable, as a property, it is called a value. - Function: generality, law. - Tp any number of the x-range a number from the y-range is assigned. - A function is not a variable! (An elliptic function is not an elliptic variable). - The function is unsaturated.
Number/Object/Calculating/Addition/Frege: only from the meaning of the words "the number 4" (Frege: = object) we can say that it is the result of combining 3 and 1. - Not of the concept. - Calculation result: is an object, the result of the calculation: is not a concept.
Number/Frege: E.g. "a variable assumes a value". - Here, the number has to play two roles: as an object it is called a variable, as a property, it is called a value.
Numbers/Frege: from physical observations no conclusions can be drawn about numbers.
Quantity/Frege: Concept - Number: Object.
Numbers/Newton: the ratio of each size to another. - FregeVsNewton: here, the notions of size and ratio are assumed.
Numbers/Frege: Problem: numbers as sets: here, the concept of quantity is assumed.
Number/Frege: is no multiplicity. - That would exclude 0 and 1.
Number/One/Unit/Property/Frege: "One" cannot be a property. - Otherwise, there would be no thing that does not have this property.
Not the objects but the concepts are the bearers of the number. - Otherwise, different numbers could be assigned to the same example. - Thus the abstraction is accompanied by a judgment.
A number is not the property of a concept. - Number: abstract object - not property -> see below Number Equality/Equality: concept (not object).
Def Quantity/Frege: the quantity which belongs to the concept F is the scope of the concept equal numbered to the concept F.
Scope/Concept Scope/Frege: If straight a is parallel to straight b, then the scope of the concept of straight parallel to straight a is equal to the scope of the concept straight parallel to the straight b and vice versa. - Scope equality.
Number/Frege/(s): from the distinction Concept Scope (Quantity)/Object (Number). - If the object is zero, the quantity that belongs to this concept is one. - ((s) This is how you get from 0 to 1.).
I 121 Numbers/Frege: are not concepts. - They are (abstract) objects. - (see above) quantities are concepts.
Term: E.g. square root of -1. - This cannot be used with the definite article, however.
Number/Frege: neither heaps of things, nor property of such.
Number System/Expansion/Frege: in the expansion, the meaning not be established arbitrarily. - E.g. the meaning of the square root is not already invariably established before the definitions, but it is determined by them. - ((s) Frege: wants to get at meaning as use). - The new numbers are given to us as scopes of concepts.
Each figure is an equation.
Berka I 83
Number/Frege: must be defined in order to be able to present completeness of evidence at all - (sequence)._____________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.
Die Grundlagen der Arithmetik Stuttgart 1987
Funktion, Begriff, Bedeutung Göttingen 1994
Logische Untersuchungen Göttingen 1993
K. Berka/L. Kreiser
Logik Texte Berlin 1983