|Systems, philosophy of science: systems are compilations of rules for the formation of statements on a previously defined subject domain. Apart from the - usually recursive - rules for the combination of expressions or signs, the specification of the vocabulary or sign set of the system is also required. See also axioms, axiom systems, theories, strength of theories, expressiveness, rules, order, recursion, models, structure, system theory._____________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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|A. d'Abro Die Kontroversen über das Wesen der Mathematik 1939 in Kursbuch 8 Mathematik 1967
System/d'Abro: If we replace one postulate by another, we obtain a new postulate system, for which the Euclidean model is no longer valid. It may be that this model is verified by a new geometry or model.
Hilbert has e.g. shown that if the parallel postulate is replaced by a suitably chosen one, the new postulate system is verified by Lobachevski's geometry.
In this case nothing new was discovered, since this geometry was already known. A completely new geometry, on the other hand, was secured when Hilbert rejected the Archimedean axiom._____________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.