Philosophy Lexicon of Arguments

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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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Thiel I 18
Numbers/Mathematics/Ontology/Mathematical Entities/Poincaré/Thiel: Definition Conventionalism (Poincaré:) Poincaré considers this concept to be a mistake for his contemporaries, who think that geometric statements, such as that the angular sum of the triangle can be proved via two right angles.
The empirical finding about spatial reality cannot clearly prescribe a geometry which characterizes this finding. The N.B. is not the measurement accuracy, but Poincaré says that the geometry remains freely selectable even with results within this measuring accuracy. We could also introduce additional physical laws for a measured triangle with less than 180°, e.g. the effect of "fields". The name "conventionalism" refers to this "free selectability" of the system of geometry.

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.

Chr. Thiel
Philosophie und Mathematik Darmstadt 1995

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Ed. Martin Schulz, access date 2017-10-22