Philosophy Lexicon of Arguments

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Numerals: numerals are symbols used to represent numbers. There are different systems of building numerals according to the supply of elementary symbols for the construction of the numerals. Examples are the Roman numerals and the Arabic numerals, but also the binary system, decimal system, hexadecimal system. It is necessary that there are fewer elementary symbols than numbers, as it would otherwise be impossible to learn them and to calculate with them. The properties of the numerical systems influence our numeracy. These properties are not to be confused with the properties of the numbers, which are independent of their representation in numerical form. See also numbers, counting, symbols, names, numeric names, numeric words.

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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Waismann I 70
Principle/Induction/Calculus/Definition/Poincaré/Waismann: ... this is the right thing in Poincaré's assertion that the principle of induction cannot be proved logically. VsPoincaré: But he does not represent, as he claimed, a synthetic judgment a priori; it is not a truth at all, but a determination: If the formula f(x) applies for x = 1, and f(c + 1) follows from f(c), let us say that "the formula f(x) is proved for all natural numbers".
I 71
But is this really just a determination? It might seem paradoxical that the associative law of addition should emerge from a mere definition (of formula D) (II 62). But the formula D is not a definition in the sense of school logic, namely, a substitution rule, but an instruction for the formation of definitions.
In the formula, there are only letters, but in the proof there are numbers! Therefore, we can predict results without performing the calculation.
The commutative law could be compared with an arrow pointing the series of numbers along into infinity.
This is not the same as saying that the law comprehends infinitely many single sentences. Example: this is similar to the sentences.
The headlight shines to infinity (true) and the headlight illuminates the infinity (impossible).
By making that convention, that is, by constructing such formulas, we adjust the calculus with letters with the calculus with numbers.

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.

Wa I
F. Waismann
Einführung in das mathematische Denken Darmstadt 1996

F. Waismann
Logik, Sprache, Philosophie Stuttgart 1976

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Ed. Martin Schulz, access date 2017-10-23