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The author or concept searched is found in the following 6 entries.
| Disputed term/author/ism | Author |
Entry |
Reference |
|---|---|---|---|
| Arithmetics | Thiel | 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. |
T I Chr. Thiel Philosophie und Mathematik Darmstadt 1995 |
| Finiteness | Hilbert | Thiel I 245 Finite/Hilbert: in the sense of Hilbert, it is only a question of how statements about infinite objects can be justified by means of "finite" methods. >Infinity, >Circularity, cf. >Recursion, >Recursivity. Hilbert found the finiteness in the "operational" method, especially of the combinatorics, arithmetics, and elemental algebra already exemplarily realized. They were "genetically" (constructively) built up into the second third of the 19th century, while the construction of geometry was a prime example for the axiomatic structure of a discipline. >Constructivism, >Geometry, >Number theory, >Arithmetics, >Axioms, >Axiom systems. I 246 Each finite operation is an area that is manageable for the person who is acting. This area can change during the process. I 247 The fact that the arithmetic functions required for Goedel's proof are even primitively recursive is remarkable in that not all effectively computable functions are primitively recursive, and the primitive recursive functions are a true subclass of the computable functions. >K. Gödel, >Completeness/Gödel, >Incompleteness/Gödel. I 248 An effectively computable, but not primitive, recursive function is e.g. explained by the following scheme for the calculation of their values (not proved) (x 'is the successor of x): ψ(0,n) = n' ψ(m',0) = ψ(m,1) ψ(m',n')= ψ(m,ψ(m',n)). (I 247) If one wants to approach the general concept of comprehensibility, one has to accept the so-called μ operator as a new means of expression. Thiel I 249 Computability/Church/Thiel: how close is this to a concept of "general computability"? There is the concept of "Turing computability", the concept of the "l definability" in Church and the "canonical systems" in Post. >Calculability, >A. Turing, >E. Post. Each function, which is in one of these classes, is also demonstrable in the others. Church has then uttered the presumption that with this an adequate clarification of the general concept of computability is achieved. >Church Thesis. But it means that this is a "non-mathematical" presumption, and is not capable of any mathematical proof. It is an intuitive term: whether such a specification is "adequate" cannot be answered with mathematical means. >Proofs, >Provability, >Adequacy. I 250 Apart from finiteness and constructivity, there remain other questions: none of the definitions for the offered functional classes is finite: e.g. μ-recursive functions. The attempt to describe effective executability with classical means remains questionable, but if we interpret the existence quantifier constructively, we have already assumed the concept of constructivity. >Quantification, >Quantifiers, >Existential quantification. |
T I Chr. Thiel Philosophie und Mathematik Darmstadt 1995 |
| Goedel | Mates | I 289 Goedel/Mates: main result: Goedel showed by the incompleteness theorem that one can not identify mathematical truth with derivability from a particular system of axioms. >K. Gödel, >Incompleteness/Gödel, >Mathematical truth, >Validity, >Derivation, >Derivability, >Axioms, >Axiom systems. |
Mate I B. Mates Elementare Logik Göttingen 1969 Mate II B. Mates Skeptical Essays Chicago 1981 |
| Incompleteness | Debray | Sokal I 200 Incompleteness/Gödel/Debray/Bricmont/Sokal: (R. Debray 1981)(1): Debray makes an allusion to Gödel's incompleteness theorem and explains that "collective insanity finds its final reason in a logical axiom that itself is not justified: incompleteness". (1981(1), p. 10). (1981, p. 256): The "mystery of our collective misery, the a priori condition of every political history in the past, present and future, can be expressed in a few simple, even childlike words. When one realizes that more work and the unconscious are to be defined in a single sentence (...),... >Incompleteness, >Gödel. Sokal I 201 ...there is no danger of confusing simplicity with exaggerated simplification. The mystery takes the form of a logical law, an extension of Gödel's theorem: there can be no organized system without closure, and no system can be closed by elements that belong to only this system. SokalVsDebray: There is simply no logical relationship between this sentence and sociological questions. Note: in a more recent text(2), Debray admits that "Gödelitis... is a common disease" and that the "transfer of a scientific knowledge and its generalisation outside its specific valid domain can lead to great mistakes (p. 7). In addition, he explains that his use of the sentence is only meant "metaphorical or isomorphic". (1996, S. 7). >For the correct use of mathematical concepts see >Sokal/Bricmont. 1. R. Debray, Critique de la raison politique, Paris, 1981. 2. R. Debray, "L'incomplétude logique du religieux", Bulletin de la societé francaise de philosophie 90, 1996 pp. 1-25. |
Debr I Régis Debray Critique de la raison politique ou l’Inconscient religieux Paris 1987 Sokal I Alan Sokal Jean Bricmont Fashionabel Nonsense. Postmodern Intellectuals Abuse of Science, New York 1998 German Edition: Eleganter Unsinn. Wie die Denker der Postmoderne die Wissenschaften missbrauchen München 1999 Sokal II Alan Sokal Fashionable Nonsense: Postmodern Intellectuals’ Abuse of Science New York 1999 |
| Incompleteness | Serres | Sokal I 203 Incompleteness/Society/Debray/Serres/Bricmont/Sokal: (M. Serres,"Paris 1800"(1): according to Debray... societies organize themselves only under the explicit condition that they are based on something different from them, something that is beyond their definition or boundary. They cannot satisfy themselves. He describes the foundation as religious. With Gödel he completes Bergson.... SokalVsSerres: the so-called "Gödel-Debray principle" is just as irrelevant to the history of science as it is to politics. >Incompletenes, >Self-reference, >Circular reasoning, >Foundation, >Ultimate justification, >Incompleteness/Gödel, >Kurt Gödel. For the correct use of the concepts of physics and mathematics see >Sokal/Bricmont, >Feynman, or >Thorne, >Gribbin, >Hacking. 1. M. Serres, "Paris 1800" in: M. Authier (Ed.) Elemente einer Geschichte der Wissenschaften, Frankfurt/M. 1994, p, 636f |
Serres I M. Serres The Five Senses: A Philosophy of Mingled Bodies Sokal I Alan Sokal Jean Bricmont Fashionabel Nonsense. Postmodern Intellectuals Abuse of Science, New York 1998 German Edition: Eleganter Unsinn. Wie die Denker der Postmoderne die Wissenschaften missbrauchen München 1999 Sokal II Alan Sokal Fashionable Nonsense: Postmodern Intellectuals’ Abuse of Science New York 1999 |
| Truth | Tarski | Glüer II 22 Truth theory/Davidson: the defined T-predicate (truth predicate) in the metalanguage can be translated back into the object language and the state before the elimination of the true can be restored. >Truth predicate, >Object language, >Metalanguage. Object language and metalanguage should contain the predicate true. >Homophony. Davidson, however, can evade the dilemma by not giving a definition. He calls it a definition of truth in Tarski's style, hereafter referred to as T-theory. --- Rorty IV (a) 22 True/Tarski: the equivalences between the two sides of the T-sentences do not correspond to any causal relationship. >Tarski scheme, >Equivalence. Davidson: there is no way to subdivide the true sentences so that on the one hand they express "factual", while on the other side they do not express anything. Cf. >Correspondence, >Correspondence theory. --- Berka I 396 Truth/Tarski: we start from the classical correspondence theory. I 399 We interpret truth like this: we want to see all sentences as valid, which correspond to the Tarski scheme - these are partial definitions of the concept of truth. - Objectively applicable: is the truth definition, if we are able, to prove all the mentioned partial definitions on the basis of the meta language.(1) 1. A.Tarski, „Grundlegung der wissenschaftlichen Semantik“, in: Actes du Congrès International de Philosophie Scientifique, Paris 1935, Vol. III, ASI 390, Paris 1936, pp. 1-8 --- Berka I 475 Truth-Definition/truth/Tarski: wrong: to assume that a true statement is nothing more than a provable sentence. - This is purely structural. Problem: No truth-definition must contradict the sentence definition. N.B.: but this has no validity in the field of provable sentences. - E.g. There may be two contradictory statements that are not provable. - All provable statements are indeed content-wise true. Nevertheless the truth definition must also contain the non-provable sentences. >Provability, >Definitions. Berka I 482 Definition true statement/Tarski: x is a true statement, notation x ε Wr iff. x ε AS (meaningful statement) and if every infinite sequence of classes satisfies x. >Satisfaction/Tarski. That does not deliver a truth criterion. >Truth criterion. No problem: nevertheless the sense of x ε Wr (x belongs to the class of true statements) gets understandable and unambiguous. I 486 Relative Truth/accuracy in the range/Tarski: plays a much greater role than the (Hilbertian) concept of absolute truth, which was previously mentioned - then we modify Definition 22 (recursive fulfillment) and 23 (truth). As derived terms we will introduce the term of the statement that a) in a domain of individuals with k elements is correct and b) of the statement that is true in every domain of individuals.(2) 2. A.Tarski, Der Wahrheitsbegriff in den formalisierten Sprachen, Commentarii Societatis philosophicae Polonorum. Vol. 1, Lemberg 1935 --- Horwich I 111 Truth/Tarski: is a property of sentences - but in the explanation we refer to "facts". - ((s) Quotation marks by Tarski). >Facts. Horwich I 124 Truth/true/eliminability/Tarski: truth cannot be eliminated with generalizations if we want to say that all true sentences have a certain property. E.g. All consequences of true sentences are true. Also not eliminable: in particular statements of the form "x is true": E.g. the first sentence that Plato wrote, is true. Because we do not have enough historical knowledge.(3) ((s) The designation "the first sentence..." is here the name of the sentence. This cannot be converted into the sentence itself. Eliminability: from definition is quite different from that of redundancy.) >Elimination, >Eliminability, cf. >Redundancy theory. 3. A. Tarski, The semantic Conceptions of Truth, Philosophy and Phenomenological Research 4, pp. 341-75 --- Skirbekk I 156 Definition Truth/Tarski: a statement is true when it is satisfied by all objects, otherwise false. Skirbekk I 158 Truth/Tarski: with our definition, we can prove the (semantic, not the logical) sentence of contradiction and the sentence definition. - The propositional logic does not include the term true at all. Truth almost never coincides with provability. All provable statements are true, but there are true statements that cannot be proved. - Such disciplines are consistent but incomplete. >Incompleteness/Gödel). There is even a pair of contradictory statements, neither of which is provable.(4) 4. A.Tarski, „Die semantische Konzeption der Wahrheit und die Grundlagen der Semantik“ (1944) in: G. Skirbekk (ed.) Wahrheitstheorien, Frankfurt 1996 |
Tarski I A. Tarski Logic, Semantics, Metamathematics: Papers from 1923-38 Indianapolis 1983 D II K. Glüer D. Davidson Zur Einführung Hamburg 1993 Berka I Karel Berka Lothar Kreiser Logik Texte Berlin 1983 Horwich I P. Horwich (Ed.) Theories of Truth Aldershot 1994 Skirbekk I G. Skirbekk (Hg) Wahrheitstheorien In Wahrheitstheorien, Gunnar Skirbekk Frankfurt 1977 |
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| Disputed term/author/ism | Author Vs Author |
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| Nagel, Th. | Putnam Vs Nagel, Th. | IV 151/152 PutnamVsNagel: it is a mistake to assume that Goedel would have shown that the human mind is more complicated than the most complex machine so far. >Incompleteness/Gödel. |
Putnam I Hilary Putnam Von einem Realistischen Standpunkt In Von einem realistischen Standpunkt, Vincent C. Müller Frankfurt 1993 Putnam I (a) Hilary Putnam Explanation and Reference, In: Glenn Pearce & Patrick Maynard (eds.), Conceptual Change. D. Reidel. pp. 196--214 (1973) In Von einem realistischen Standpunkt, Vincent C. Müller Reinbek 1993 Putnam I (b) Hilary Putnam Language and Reality, in: Mind, Language and Reality: Philosophical Papers, Volume 2. Cambridge University Press. pp. 272-90 (1995 In Von einem realistischen Standpunkt, Vincent C. Müller Reinbek 1993 Putnam I (c) Hilary Putnam What is Realism? in: Proceedings of the Aristotelian Society 76 (1975):pp. 177 - 194. In Von einem realistischen Standpunkt, Vincent C. Müller Reinbek 1993 Putnam I (d) Hilary Putnam Models and Reality, Journal of Symbolic Logic 45 (3), 1980:pp. 464-482. In Von einem realistischen Standpunkt, Vincent C. Müller Reinbek 1993 Putnam I (e) Hilary Putnam Reference and Truth In Von einem realistischen Standpunkt, Vincent C. Müller Reinbek 1993 Putnam I (f) Hilary Putnam How to Be an Internal Realist and a Transcendental Idealist (at the Same Time) in: R. Haller/W. Grassl (eds): Sprache, Logik und Philosophie, Akten des 4. Internationalen Wittgenstein-Symposiums, 1979 In Von einem realistischen Standpunkt, Vincent C. Müller Reinbek 1993 Putnam I (g) Hilary Putnam Why there isn’t a ready-made world, Synthese 51 (2):205--228 (1982) In Von einem realistischen Standpunkt, Vincent C. Müller Reinbek 1993 Putnam I (h) Hilary Putnam Pourqui les Philosophes? in: A: Jacob (ed.) L’Encyclopédie PHilosophieque Universelle, Paris 1986 In Von einem realistischen Standpunkt, Vincent C. Müller Reinbek 1993 Putnam I (i) Hilary Putnam Realism with a Human Face, Cambridge/MA 1990 In Von einem realistischen Standpunkt, Vincent C. Müller Reinbek 1993 Putnam I (k) Hilary Putnam "Irrealism and Deconstruction", 6. Giford Lecture, St. Andrews 1990, in: H. Putnam, Renewing Philosophy (The Gifford Lectures), Cambridge/MA 1992, pp. 108-133 In Von einem realistischen Standpunkt, Vincent C. Müller Reinbek 1993 Putnam II Hilary Putnam Representation and Reality, Cambridge/MA 1988 German Edition: Repräsentation und Realität Frankfurt 1999 Putnam III Hilary Putnam Renewing Philosophy (The Gifford Lectures), Cambridge/MA 1992 German Edition: Für eine Erneuerung der Philosophie Stuttgart 1997 Putnam IV Hilary Putnam "Minds and Machines", in: Sidney Hook (ed.) Dimensions of Mind, New York 1960, pp. 138-164 In Künstliche Intelligenz, Walther Ch. Zimmerli/Stefan Wolf Stuttgart 1994 Putnam V Hilary Putnam Reason, Truth and History, Cambridge/MA 1981 German Edition: Vernunft, Wahrheit und Geschichte Frankfurt 1990 Putnam VI Hilary Putnam "Realism and Reason", Proceedings of the American Philosophical Association (1976) pp. 483-98 In Truth and Meaning, Paul Horwich Aldershot 1994 Putnam VII Hilary Putnam "A Defense of Internal Realism" in: James Conant (ed.)Realism with a Human Face, Cambridge/MA 1990 pp. 30-43 In Theories of Truth, Paul Horwich Aldershot 1994 SocPut I Robert D. Putnam Bowling Alone: The Collapse and Revival of American Community New York 2000 |
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