@misc{Lexicon of Arguments, title = {Quotation from: Lexicon of Arguments – Concepts - Ed. Martin Schulz, 28 Mar 2024}, author = {Genz,Hennig}, subject = {Metalanguage}, note = {II 210 Meta language/addition/algorithm/sum/Gauss/Genz: the sum of the numbers from 1 to 100 is 5050 = 101 x 50: Example 1 to 10: 1+2+3+4+5+6+7+8+9+10 = (1+10)+(2+9)+(3+8)+(4+7)+(5+6) = 11+11+11+11+11 = 5 x 11 = 55 The sum can be rearranged in such a way that the result of the addition is independent of the sequence of the numbers due to the algorithm. N.B.: this is a statement about the results of additions, in meta-language. >Object language. II 211 Meta language/blackening/characters/formalisms/Hofstadter/Genz: Example for a purely typographical derivation: if 0+0=0, 1+0= 1 etc. as well as 1 = 1 is specified, you can add 1 + x = 1 + x for any x. Derivation/Formalism/Genz: that negative numbers must be excluded here has no significance for formalism and cannot be used to justify derivations within it. >Derivation, >Derivability. >Formalization. Hofstadter/Genz: Hofstadter uses the successor relation SS0 instead of 2, so no meanings crept in. Evidence/Hofstadter: evidence is something informal. The result of reflection. Formalisation/Hofstadter: formalisation serves to logically defend intuitions. Derivation/Hofstadter: derivation artificially produced an equivalent of the evidence... II 212 ...that makes the logical structure explicit. Simplicity/derivation/Hofstadter: it may be that myriads of steps are necessary, but the logical structure turns out to be quite simple. >Simplicity. Meaning/Genz: the infinite sequence of the above statements is summed up in the sentence that all numbers, if multiplied by 0, remain unchanged. N.B.: however, this is not based on the meaning of the symbols, but only on the typographic derivation rules of the object language. Meta language/Genz: it is an insight into formalism that guarantees that all tokens are true. Object language: be so that the above generalization ("all numbers, multiplied by 0, remain unchanged") can be formulated in it, but cannot be derived. 1st meta-language: here it can be derived. It contains complete induction. 2nd meta-language: here it cannot be derived, but its negation! (see below) Both meta languages contain the object language. Therefore, the consequences can be derived from them. II 213 Object language: not all true sentences can be derived in the object language. Solution: we add the sentence to the language ourselves, then it is true as well as (trivially) derivable. N.B.: in the second meta-language, which is incompatible with the first, its negation can be added instead of the sentence without creating a contradiction. 2nd meta-language: the 2nd meta-language forces the occurrence of "unnatural" numbers, which cannot be represented as successors of 0.(1) 1. Douglas Hofstadter (2008). Gödel, Escher, Bach. Stuttgart: Klett-Cotta. p. 240.}, note = { Gz I H. Genz Gedankenexperimente Weinheim 1999 Gz II Henning Genz Wie die Naturgesetze Wirklichkeit schaffen. Über Physik und Realität München 2002 }, file = {http://philosophy-science-humanities-controversies.com/listview-details.php?id=393601} url = {http://philosophy-science-humanities-controversies.com/listview-details.php?id=393601} }