Economics Dictionary of ArgumentsHome
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| 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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Benson Mates on Numbers - Dictionary of Arguments
I 212 Number/statement/property/Mates: every statement in which a number occurs, represents a property of the corresponding number. E.g. no person under 18 receives permission- property of the number 18, that it is a number k with the property, that no person obtains a permit under k years - therefore complete induction can be applied to such statements. I 288 Numbers/Frege/Mates: Def cardinal number: of a set a: set of all sets which are numerically equivalent to a. Def One/1: the set of all sets a, satisfying the condition (x) (Ey) (y e a <> y = x). Def Two/2: set of all sets which satisfy the condition (x)(y)(x not equal y u (z)(z e a <> (z = x v z = y))). Def sum: the sum p + q of two integers p and q is the set of all sets g which satisfying the condition (a)(Eb)(a e p u b e q u a U b = g u a D b = L) Def set of all positive numbers: the average ((s) Common) of all sets a which satisfying the condition 1 e a u (n)(n e a > n + 1 e a). ((S) successor). Mates: Thus Frege shows that arithmetic can be lead back entirely on logic, and thus that it is part of the logic. >Numbers/Frege, >Successor, >Sets, >Set theory._____________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. Translations: Dictionary of Arguments The note [Concept/Author], [Author1]Vs[Author2] or [Author]Vs[term] resp. "problem:"/"solution:", "old:"/"new:" and "thesis:" is an addition from the Dictionary of Arguments. If a German edition is specified, the page numbers refer to this edition. |
Mate I B. Mates Elementare Logik Göttingen 1969 Mate II B. Mates Skeptical Essays Chicago 1981 |
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