|Abstraction: Subsumption of objects by non-consideration of certain properties. See also equivalence relation, concretion, concreta, indiscernibility._____________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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|Stuhlmann-Laeisz II 49
Definition abstraction/Frege/St: abstraction is a process to identify each other in the various elements of a region - e.g. same color, same size, same shape - new objects are gained by abstraction of the partial identity - ((s) such objects which are only about numerical equality or color identity are individuated) - if A is such an abstraction, then there is by definition a (base) object a, so that the following is true: A is the F of a.
Thiel I 131
Abstraction/Mathematics/Frege/Thiel: Abstraction is a purely logical process, an operation with statements, the logical character of which is revealed by the change from the structure of the complicated initial statement to the structure of the new statement.
Frege understood this first. Statements not only about numbers, but also about sets, functions, concepts, situations, meaning and truth value of a statement, about structures.
Numbers/digits/number names/names/mathematics/Thiel: the philosophical punch line of the transition from general statements via digits to arithmetic statements is that although we have introduced the speech about numbers in addition to the speech about digits, it is still a form of speech, a facon de parler, whose possibility does not depend on the fact that there are still abstract objects beyond the concrete digits, which we call "numbers".
We also had no reason to conceive the digits as "names" of numbers, so that 4, IV, and |||| would denote the same number four, as it was assumed in the traditional philosophy of 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.
Die Grundlagen der Arithmetik Stuttgart 1987
Funktion, Begriff, Bedeutung Göttingen 1994
Logische Untersuchungen Göttingen 1993
Freges Logische Untersuchungen Darmstadt 1995
Philosophie und Mathematik Darmstadt 1995