|Metalanguage: metalanguage is the language in which linguistic forms, the meaning of expressions and sentences, the use of language, as well as the admissibility of formations, and the truth of statements are discussed. The language you refer to is called object language. A statement about the form, correctness, or truth of another statement thus includes both, i.e. object language and meta language. See also richness, truth-predicate, expressiveness, paradoxes, mention, use, quasi-reference, quotation, hierarchy, fixed points._____________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. |
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.
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.
Hofstadter/Genz: uses the successor relation SS0 instead of 2, so no meanings crept in.
Evidence/Hofstadter: is something informal. The result of reflection.
Formalisation/Hofstadter: serves to logically defend intuitions.
Derivation/Hofstadter: artificially produced equivalent of the evidence...
...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.
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.
Object language: not all true sentences can be derived in it.
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: forces the occurrence of "unnatural" numbers, which cannot be represented as successors of 0 (see Hofstadter, Gödel, Escher, Bach, German Edition, p. 240)._____________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. The note [Author1]Vs[Author2] or [Author]Vs[term] is an addition from the Dictionary of Arguments. If a German edition is specified, the page numbers refer to this edition.
Gedankenexperimente Weinheim 1999
Wie die Naturgesetze Wirklichkeit schaffen. Über Physik und Realität München 2002