Philosophy Dictionary of ArgumentsHome
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| Isomorphism: A. In mathematics, an isomorphism is a one-to-one correspondence between two mathematical structures of the same type that preserves all of the relationships between the elements of the structures. See also Structures.
B. Isomorphism in linguistics is the one-to-one correspondence between form and meaning. It can exist between different levels of a language, such as the phonological, morphological, syntactic, and semantic levels. See also Phonemes, Morphemes, Syntax, Semantics.
_____________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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F. Waismann on Isomorphism - Dictionary of Arguments
I 53 Isomorphism/Mathematics/general public/generalization/axioms/Hilbert/Waismann: New: in modern mathematics one came to the realization that geometrical sentences can be applied to a completely different field. For example, all the theorems that are about the straight lines of our space can be interpreted as being about all the points of a four-dimensional space. The two thought systems are completely isomorphic (built the same). The sensuous appearance thus plays no role for the validity of the sentences. One is now consciously dispensing with saying what a straight line is. >Geometry, >Space. I 54 Point, line, plane, are understood to mean any things for which the axioms set forth are true. Hilbert gives an example: the numerical distribution of deviations in the cultivation of Drosophila (flies) coincide with the linear Euclidean axioms of congruence and the geometric concept "between". So simple and so accurate as you would not have dreamed of. >Analogies, >Proofs, >Provability. I 55 The last step: also the signs of the logic calculus are content-wise undefined. (connection signs). >Logical constants, >Equal sign, Connectives, >Identity. Problem: consistency must first be defined e.g.: Def consistent: is a formula system, if, in it, 1 unequal 1 does never occur. >Consistency. Metamathematics is then content-related, with the main goal of consistency. Hilbert: The axioms and provable propositions are representations of the thoughts which constituted the usual method of the previous mathematics, but they are not themselves the truths in the absolute sense. >Truth/Waismann, >Truth/Hilbert, >Axioms/Hilbert._____________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. |
Waismann I F. Waismann Einführung in das mathematische Denken Darmstadt 1996 Waismann II F. Waismann Logik, Sprache, Philosophie Stuttgart 1976 |
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> Counter arguments against Waismann
> Counter arguments in relation to Isomorphism
Ed. Martin Schulz, access date 2024-04-20