|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 15
Numbers/John Stuart Mill: mathematical objects, especially the numbers, are abstractions taken from concrete experience, that is, the most general properties or characteristics of reality. By generalizing the observation, we arrive at definitions for mathematical objects. They express facts about the totality of physical objects. (Mill: "aggregates").
Each proposition based on this asserts that a definite totality could have been formed by combining certain other totals, or by withdrawing them.
Each numeral sign "2", "3", etc. denotes for Mill a physical phenomenon, a property which belongs to the totality of things that we describe with the numeral signs.
FregeVsMill: drastic counterexamples: Doubtfulness in the case of 0 and 1, but also for very large numbers. Who should have ever observed the fact for the definition of 777 865? Mill could have defended himself. That his position seems to be more suited to the justification of our number and form than for the justification of arithmetic._____________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.
J. St. Mill
Utilitarianism: 1st (First) Edition Oxford 1998
Philosophie und Mathematik Darmstadt 1995