Philosophy Dictionary of ArgumentsHome
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| Arithmetics: Arithmetics is a branch of mathematics that deals with numbers, their properties, and operations._____________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 Arithmetics - Dictionary of Arguments
Thiel I 225 Arithmetics/Lorenzen/Thiel: Arithmetics is the theory in which the infinite occurs in its simplest form, it is essentially nothing more than the theory of the infinite itself. Arithmetics as the theory of the set of signs (e.g. tally-list) is universal in the sense that the properties and relations of any other infinite set of signs can always be "mapped" in some way. The complexity of matter has led to the fact that a large part of the secondary literature on Gödel has put a lot of nonsense into the world on metaphors such as "reflection", "self-reference", etc. >Self-reference, cf. >Regis Debray. I 224 The logical arithmetic full formalism is denoted with F. It contains, among other things, inductive definitions of the counting signs, the variables for them, the rules of quantifier logic and the Dedekind-Peanosian axioms written as rules. >Formalization, >Formalism. I 226 The derivability or non-derivability of a formula means nothing other than the existence or non-existence of a proof figure or a family tree with A as the final formula. Therefore also the metamathematical statements "derivable", respectively "un-derivable" each reversibly correspond unambiguously to a basic number characterizing them. >Theorem of Incompleteness/Gödel. Terminology/Writing: S derivable, $ not derivable. "$ Ax(x)" is now undoubtedly a correctly defined form of statement, since the count for An(n) is uniquely determined. Either $An(n) is valid or not. >Derivation, >Derivability. I 304 The centuries-old dominance of geometry has aftereffects in the use of language. For example "square", "cubic" equations etc. Arithmetics/Thiel: has today become a number theory, its practical part degraded to "calculating", a probability calculus has been added. >Probability, >Probability law. I 305 In the vector and tensor calculus, geometry and algebra appear reunited. A new discipline called "invariant theory" emerges, flourishes and disappears completely, only to rise again later. I 306 Functional analysis: is certainly not a fundamental discipline because of the very high level of conceptual abstraction. Invariants. I 307 Bourbaki contrasts the classical "disciplines" with the "modern structures". The theory of prime numbers is closely related to the theory of algebraic curves. Euclidean geometry borders on the theory of integral equations. The ordering principle will be one of the hierarchies of structures, from simple to complicated and from general to particular. >Structures._____________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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> Counter arguments against Thiel
> Counter arguments in relation to Arithmetics
Ed. Martin Schulz, access date 2024-04-20