Economics Dictionary of ArgumentsHome
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| Generalization: a generalization is the extension of a statement (an attribution of properties) that applies to a domain D of objects to an object domain E that is larger than D and contains D. Time points may also belong to the subject domain. A property which fully applies to the objects of an object domain may be partially applicable to the objects of a larger domain. See also validity, general invalidity, general, predication, methods._____________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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Christian Thiel on Generalization - Dictionary of Arguments
I 180 Generality/Generalization/Infinity/Mathematics/Statements/Thiel: Who would have ever doubted that the Pythagorean theorem can be applied to an infinite number of cases? I 181 Problem: that in the formulation of the theorem irrational numbers are allowed as measures for the cathets of the right-angled triangles, for which we do not know any counting as for the rational numbers. If there were one, we could count the totality of the real numbers by combining them with a count of the rational numbers. Cantor's merit was to show the impossibility of this by his diagonal method. I 181 Table with columns and columns cut by diagonals. I 182 Def Dual Sequence/(s): Sequence of (binary) decisions as to whether a point is on the left or right half of a halved course. This leads to any rational number. I 184 But it leads to a contradiction. Then 1 bii = bii . the assumption that the dual sequence constructed as "negative" of its diagonals already occurs in the (arbitrary) list considered leads to an absurdity. After that, however, even the totality of all real numbers in the interval 0.1 cannot be recorded in a list (as Cantor also shows, not in an infinite list), it cannot be counted. I 185 So also not outside the interval. I 186 Continuum/Russell: (e.g.) sees an arithmetic term in the continuum, others a geometric one. I 189 New: Modern Mathematics I 189 (Topology) has defined the concept of the "border" of a set of points corresponding to the Aristotelian "border" in such a way that a point can be its own border. _____________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. |
T I Chr. Thiel Philosophie und Mathematik Darmstadt 1995 |
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