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The author or concept searched is found in the following 8 entries.
| Disputed term/author/ism | Author |
Entry |
Reference |
|---|---|---|---|
| Axioms | Hilbert | Berka I 294 Definition/Axiom/Hilbert: the established axioms are at the same time the definitions of the elementary concepts whose relations they regulate. ((s) Hilbert speaks of relationships, not of the use of concepts). >Definitions, >Definability, >Basic concepts. Independence/Axiom/Hilbert: the question is whether certain statements of individual axioms are mutually dependent, and whether the axioms do not contain common components which must be removed so that the axioms are independent of each other(1). >Independence. 1. D. Hilbert: Mathematische Probleme, in: Ders. Gesammelte Abhandlungen (1935), Vol. III, pp. 290-329 (gekürzter Nachdruck v. S 299-301). --- Thiel I 262 We consider the first three axioms of Hilbert: 1. There are exactly two straight lines at each of two distinct points P, Q, which indicate(2) with P and Q. 2. For every line g and to any point P, which does not indicate with it, there is exactly one line that is indicated with P, but with no point of g. 3. There are three points which do not indicate with one and the same straight line. In Hilbert's original text, instead of points one speaks of "objects of the first kind" instead of straight lines of "objects of the second kind" and instead of the incidence of "basic relation". Thus, the first axiom is now: For each of two different objects of the first kind, there is precisely one object of the second kind, which is in a basic relation with the first two. Thiel I 263 If the axioms are transformed quantifier-logically, then only the schematic sign "π" (for the basic relation) is free for substitutions, the others are bound by quantifiers, and can no longer be replaced by individual names of points or lines. >Quantification, >Quantifiers. They are thus "forms of statements" with "π" as an empty space. >Propositional functions. They are not statements like those before Hilbert's axioms, whose truth or falsehood is fixed by the meanings of their constituents. >Truth values. In the Hilbert axiom concept (usually used today), axioms are forms of statements or propositional schemata, the components of which must be given a meaning only by interpretation by specifying the variability domains and the basic relation. The fact that this can happen in various ways, shows that the axioms cannot determine the meaning of their components (not their characteristics, as Hilbert sometimes says) themselves by their co-operation in an axiom system. Thiel I 264 Multiple interpretations are possible: e.g. points lying on a straight line, e.g. the occurrence of characters in character strings, e.g. numbers. Thiel I 265 All three interpretations are true statements. The formed triples of education regulations are models of our axiom system. The first is an infinite, the two other finite models. >Models, >Infinity. Thiel I 266 The axioms can be combined by conjunction to form an axiom system. >Conjunction. Through the relationships, the objects lying in the subject areas are interwoven with each other in the manner determined by the combined axioms. The regions V .. are thereby "structured" (concrete and abstract structures). >Domains, >Structures (Mathematics). One and the same structure can be described by different axiom systems. Not only are logically equivalent axiom systems used, but also those whose basic concepts and relations differ, but which can be defined on the basis of two systems of explicit definitions. Thiel I 267 Already the two original axiom systems are equivalent without the assumption of reciprocal definitions, i.e. they are logically equivalent. This equivalence relation allows an abstraction step to the fine structures. In the previous sense the same structures, are now differentiated: the axiom systems describing them are not immediately logically equivalent, but their concepts prove to be mutually definable. For example, "vector space" "group" and "body" are designations not for fine structures, but for general abstract structures. However, we cannot say now that an axiom system makes a structure unambiguous. A structure has several structures, not anymore "the" structure. Thiel I 268 E.g. body: the structure Q has a body structure described by axioms in terms of addition and multiplication. E.g. group: the previous statement also implies that Q is also e.g. a group with respect to the addition. Because the group axioms for addition form part of the body axioms. Modern mathematics is more interested in the statements about structures than in their carriers. From this point of view, structures which are of the same structure are completely equivalent. >Indistinguishability. Thiel: in algebra it is probably the most common to talk of structures. Here, there is often a single set of carriers with several links, which can be regarded as a relation. Thiel I 269 E.g. relation: sum formation: x + y = z relation: s (x, y, z). In addition to link structures, the subject areas often still carry order structures or topological structures. Thiel I 270 Bourbaki speaks of a reordering of the total area of mathematics according to "mother structures". In modern mathematics, abstractions, especially structures, are understood as equivalence classes and thus as sets. >N. Bourbaki, >Equivalence classes. 2. Indicate = belong together, i.e. intersect, pass through the point, lie on it. |
Berka I Karel Berka Lothar Kreiser Logik Texte Berlin 1983 T I Chr. Thiel Philosophie und Mathematik Darmstadt 1995 |
| Functions | Kauffman | I 407 Function/Kauffman: is simply put, a string of signs. Strings of signs affect other strings and produce new strings. They can be models for economy, molecules, chemicals, goods and services, etc. >Self-organization, >Character strings, >Molecules. |
Kau II Stuart Kauffman At Home in the Universe: The Search for the Laws of Self-Organization and Complexity New York 1995 Kauffman I St. Kauffman At Home in the Universe, New York 1995 German Edition: Der Öltropfen im Wasser. Chaos, Komplexität, Selbstorganisation in Natur und Gesellschaft München 1998 |
| Individuation | Individuation, philosophy: the picking out of an object by a determination by means of additional information which is not to be derived from a single statement which contains this object. For example, beliefs are individualized by content, not e.g. by the length of the character strings with which they are expressed. The contents of a belief are, in turn, not individuated by their repetition, but by other contents. |
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| Individuation | Schiffer | I 122 Individuation/sentence/Schiffer: sentences are individuated by the ordered pair. ((s) Phrases are individuated by meanings, assuming character strings, but not vice versa.) >Ordered pair. Schiffer: but only if one assumes propositions as meanings. >Proposition, >Meaning. Problem: how are we to individuate them differently? Paratactic analysis/Davidson/solution: "On Saying That" 1968(1)). >Paratactic analysis. 1. Donald Davidson (1968). On saying that. Synthese 19 (1-2):130-146 |
Schi I St. Schiffer Remnants of Meaning Cambridge 1987 |
| Models | Kauffman | I 415 Model/economy/Kauffman: in economics and other systems there are an enormous number of niches. What gives rise to these? According to which rules do workstations, tasks, functions and products connect to networks? >Niches, >Rules, >Progress, >Society. Thesis: we can view goods and services as sign strings that affect other strings. Hammer acts on nails and two boards. >Character strings, >Functions/Kauffman. I 416 Model/Kauffman: what use are models if we do not know the true laws of complementarity and substitutability? >Substitutability. Their benefit is that we can recognize the kind of things we would expect in the real world if our model is in the same "universality class". ((s) cf. >Brandom on singular terms, predicates in relation to the degree of generality). Definition Universality class/physics/Kauffman: Class of models that show the same robust behavioral patterns. >Models, cf. >Model theory. Lambda Calculus/Church/Kauffman: System for performing universal calculations. Also Emil Post. Universal system and Turing machine, all these systems are equivalent. >Lambda calculus, >Turing machine. I 417 Model/Post/Kauffman: For example, a system where the left-hand list of sign strings represents the "grammar", each pair of sign strings specifies a substitution. I 419 The sign strings can then interact with each other, like enzymes on substrates. Arbitrary rules can lead to non arbitrary ones! >Arbitraryness, >Contingency, >Necessity. The number of possible grammars is infinite. >Grammar, >Infinity, >Countability, >Overcountable. Complexity: if the right links of the sign strings are shorter than the left ones, the "soup" will react inert, because all the chains become shorter, and no longer fit on an "enzymatic digit". The different regions form universality classes. |
Kau II Stuart Kauffman At Home in the Universe: The Search for the Laws of Self-Organization and Complexity New York 1995 Kauffman I St. Kauffman At Home in the Universe, New York 1995 German Edition: Der Öltropfen im Wasser. Chaos, Komplexität, Selbstorganisation in Natur und Gesellschaft München 1998 |
| Proper Names | Tarski | Berka I 451 Def quotation name/Tarski: any name of a statement (or even meaningless expression) consisting of quotes and the expression, and which is precisely the signified through the considered name. E.g. the name ""it snows"". ((s) Quotation marks twice) N.B.: identical configured expressions must not be identified. - Therefore quotation names are general, not individual names (classes of character strings). >Description levels, >Quotation marks, cf. >Names of sentences. I 453 Syntactically simple expressions - such as letters - have no independent meaning. I 451 Def structural-descriptive name/Tarski: (different category than the quotation names): they describe, of what words the expression, designated by the name, consists and of which characters each individual word consists and in what order they follow one another. - This goes without quotation marks. Method: introduce single names for all letters and other characters (No quotation names). E.g. for letters f, j, P, etc.: Ef, Jay, Pee, ex - E.g. to the quotation name ""snow"" (quotation marks twice) corresponds the structural-descriptive name: word that consists of the six consecutive letters Es, En, O, double-u - (letter names without quotation marks). I 451 Semantically ambiguous/Russell/Tarski: E.g. name, designating: a) with respect to items b) to classes, relations, etc. I 464 Name/translation/metalanguage/object language/Tarski: difference: an expression of the object language in the metalanguage may a) be given a name, or b) a translation. >Object language, >Metalanguage. I 496 Names/variables/constants/Tarski: variables represent names constants are names. >Representation, >Proxy. For each constant and each variable of the object language (except for the logical constants of propositional calculus) can form a fundamental function that contains this character (the statement variables neither occur into the fundamental functions as functors nor as arguments). Statement variable: any ((s) individual) of them is regarded as an independent fundamental function.(1) >Constants/Tarski, >Functions/Tarski. 1. A.Tarski, Der Wahrheitsbegriff in den formalisierten Sprachen, Commentarii Societatis philosophicae Polonorum. Vol 1, Lemberg 1935 |
Tarski I A. Tarski Logic, Semantics, Metamathematics: Papers from 1923-38 Indianapolis 1983 Berka I Karel Berka Lothar Kreiser Logik Texte Berlin 1983 |
| Syntax | Quine | VII (a) 15 Syntax/Quine: their rules are meaningful in contrast to their notation. VI 69 Syntax/translation/indeterminacy/Quine: many of my readers have mistakenly assumed that uncertainty also extends to syntax. There was a subtle reason for this: in word and object(1) (pp. 107, 129 136) it says: VI 70 that also the specific apparatus of reification and object reference, which we make use of, is subject to indeterminacy. To this apparatus belong the pronomina, the "=", (equal sign) the plural endings and whatever performs the tasks of the logical quantifiers. But it is wrong to assume that these mechanisms belonged to syntax! >Equal sign, >Quantifiers, >Pronouns, >Indeterminacy. VI 97 Spelling/Quine: resolves the syntax and lexicon of each content sentence and merges it with the interpreter's language. It then has no more complicated syntax than the addition sign. 1. Quine, W. V. (1960). Word and Object. MIT Press VII (a) 15 Syntax/Quine/Goodman: their rules are meaningful as opposed to the notation itself. XI 114 Language/Syntax/Lauener: Language cannot be regarded purely syntactically as the set of all correctly formed expressions, because an uninterpreted system is a mere formalism. ((s) This is not truthful). XI 116 Lauener: it is a mistake to think that the language contributes the syntax but the theory contributes the empirical content. Therefore, one cannot say that an absolute theory can be formulated in different languages, or vice versa, that different (even contradictory) theories can be expressed in one language. XI 136 Mathematics/QuineVsHilbert/Lauener: Mathematics is more than just syntax. Quine reluctantly professes Platonism. XII 58 The problem of the inscrutability of the reference reaches much deeper than that of the indeterminacy of the translation: e.g. protosyntax. >Inscrutability. Protosyntax/Uncertainty/Quine: the language here is a formalized system of proof theory of the first level, whose subject area consists only of expressions, i.e. of character strings of a certain alphabet. Expressions: are types here, not tokens! (no occurrences). Each expression is the set of all its occurrences. (Summarized due to similarity of inscriptions). For example, the concatenation x^y is the set of all inscriptions that consist of two parts. These parts are tokens of x and y. Problem: it can happen that x^y is the empty set ((s) the combination does not occur) although both x and y are not empty. XII 59 The probability of this problem increases with increasing length of x and y! N.B.: this violates a law of protosyntax that says: x = z, if x^y = z^y. Solution: then you will not understand the objects as sets of inscriptions. But then you can still consider its atoms, the single characters as a set of inscriptions. Then there is no danger that the set is empty. ((s) Because the atoms have to be there, even if not every combination). N.B.: instead of interpreting the strings as sets of inscriptions, they can be regarded as a (mathematical) sequence (of characters). Character String/Expression: is then a finite set of pairs of a sign and a number. Vs: this is very artificial and complicated. Simpler: Goedel numbers themselves (the characters disappear). Problem: Question: How clear is it here that we have just started to talk about numbers instead of expressions? The only thing that is reasonably clear is that we want to fulfill laws with artificial models that are supposed to fulfill expressions in a non-explicit sense. XIII 199 Syntax/Quine: "glamour" and "grammar" were originally one and the same word. XIII 200 Later, the meaning also included magic. Grammar: (in the narrower sense) said which chains of words or phonemes were coherent and which were not. Always related to a particular language. Grammar: (wider sense): "The art of speaking" (in relation to the established use). >Grammar. Syntax/Quine: for the narrower sense we do not really need the word "grammar", but "syntax". It is about which character strings belong to the language and which do not. Problem: this is indefinite in two ways: 1. How the individuals are specified (formally, by components or phonemes) and 2. What qualifies them for the specification XIII 201 Recognizability is too indeterminate (liberal). Problem: ungrammatical forms are used by many people and are not incomprehensible. A language that excludes these forms would be the dialect of a very small elite. Problem: merely possible utterances in imaginable but not actual situations that are not themselves linguistic in nature. Solution: Def ungrammatic/William Haas/Quine: a form that would not make sense in any imaginable fictitious situation. Rules/Syntax/syntactic rules/Quine: are abstractions of the syntactic from long practice. They are the fulfillment of the first task (see above) to recognize which chains are grammatical. XIII 202 Solution: this is mainly done by recursion, similar to family trees. It starts with words that are the simplest chains and then moves on to more complex constructions. It divides the growing repertoire into categories. Parts of speech/Quine: there are eight: Nouns, pronouns, verb, adjective, adverb, preposition, conjunction, sentence. Further subdivisions: transitive/intransitive, gender, etc. But this is hardly a beginning. Nomina: even abstract ones like cognizance (of) and exception (to) are syntactically quite different, they stand with different prepositions. Recursion/syntax/Quine: if we wanted to win the whole syntax by recursion, it would have to be so narrow that two chains would never be counted as belonging to the same speech part, unless they could be replaced in all contexts salva congruitate. >Recursion. Def Replaceability salva congruitate/Geach/Quine: preserves grammaticality, never returns ungrammatical forms. VsRecursion/Problem: if speech parts were so narrowly defined, e.g. Nomina, which stand with different prepositions, they would then have to be counted among different kinds of speech parts. And these prepositions e.g. of and to, should not fall into the same category either! Then there would be too many kinds of speech parts, perhaps hundreds. Of which some would also be singletons ((s) singletons = categories with only one element). Solution: to give up recursion after having the roughest divisions. |
Quine I W.V.O. Quine Word and Object, Cambridge/MA 1960 German Edition: Wort und Gegenstand Stuttgart 1980 Quine II W.V.O. Quine Theories and Things, Cambridge/MA 1986 German Edition: Theorien und Dinge Frankfurt 1985 Quine III W.V.O. Quine Methods of Logic, 4th edition Cambridge/MA 1982 German Edition: Grundzüge der Logik Frankfurt 1978 Quine V W.V.O. Quine The Roots of Reference, La Salle/Illinois 1974 German Edition: Die Wurzeln der Referenz Frankfurt 1989 Quine VI W.V.O. Quine Pursuit of Truth, Cambridge/MA 1992 German Edition: Unterwegs zur Wahrheit Paderborn 1995 Quine VII W.V.O. Quine From a logical point of view Cambridge, Mass. 1953 Quine VII (a) W. V. A. Quine On what there is In From a Logical Point of View, Cambridge, MA 1953 Quine VII (b) W. V. A. Quine Two dogmas of empiricism In From a Logical Point of View, Cambridge, MA 1953 Quine VII (c) W. V. A. Quine The problem of meaning in linguistics In From a Logical Point of View, Cambridge, MA 1953 Quine VII (d) W. V. A. Quine Identity, ostension and hypostasis In From a Logical Point of View, Cambridge, MA 1953 Quine VII (e) W. V. A. Quine New foundations for mathematical logic In From a Logical Point of View, Cambridge, MA 1953 Quine VII (f) W. V. A. Quine Logic and the reification of universals In From a Logical Point of View, Cambridge, MA 1953 Quine VII (g) W. V. A. Quine Notes on the theory of reference In From a Logical Point of View, Cambridge, MA 1953 Quine VII (h) W. V. A. Quine Reference and modality In From a Logical Point of View, Cambridge, MA 1953 Quine VII (i) W. V. A. Quine Meaning and existential inference In From a Logical Point of View, Cambridge, MA 1953 Quine VIII W.V.O. Quine Designation and Existence, in: The Journal of Philosophy 36 (1939) German Edition: Bezeichnung und Referenz In Zur Philosophie der idealen Sprache, J. Sinnreich (Hg) München 1982 Quine IX W.V.O. Quine Set Theory and its Logic, Cambridge/MA 1963 German Edition: Mengenlehre und ihre Logik Wiesbaden 1967 Quine X W.V.O. Quine The Philosophy of Logic, Cambridge/MA 1970, 1986 German Edition: Philosophie der Logik Bamberg 2005 Quine XII W.V.O. Quine Ontological Relativity and Other Essays, New York 1969 German Edition: Ontologische Relativität Frankfurt 2003 Quine XIII Willard Van Orman Quine Quiddities Cambridge/London 1987 |
| Syntax | Schröter | Berka I 415 Character strings/Schröter: if they are alike, theay are not considered being different here. - Therefore we are always talking of a set of character strings that have the same design as a given sequence of characters.(1) - ((s)> Tarski: a quotationmark-name (quotation name, quotation-mark name) is always general: class of identically designed strings, not an individual name. >functionalism/Tarski) 1. K. Schröter, Was ist eine mathematische Theorie?, Jahresbericht der deutschen Mathematikervereinigung 53 (1943), 69-82 |
Berka I Karel Berka Lothar Kreiser Logik Texte Berlin 1983 |
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| Disputed term/author/ism | Author Vs Author |
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| Hilbert | Frege Vs Hilbert | Berka I 294 Consistency/Geometry/Hilbert: Proof through analogous relations between numbers. Concepts: if properties contradict each other, the concept does not exist. FregeVsHilbert: there is just nothing that falls under it. Real Numbers/Hilbert: here, the proof of consistency for the axioms is also the proof of existence of the continuum.(1) 1. D. Hilbert, „Mathematische Probleme“ in: Ders. Gesammelte Abhandlungen (1935) Bd. III S. 290-329 (gekürzter Nachdruck v. S 299-301) Thiel I 279 Hilbert: Used concepts like point, line, plane, "between", etc. in his Foundations of Geometry in 1899, but understood their sense in a hitherto unfamiliar way. They should not only enable the derivation of the usual sentences, but rather, in its entirety, specify the meaning of the concepts used in it in the first place! Thiel I 280 Later this was called a "definition by postulates", "implicit definition" >Definition. The designations point, line, etc. were to be nothing more than a convenient aid for mathematical considerations. FregeVsHilbert: clarifies the letter correspondence that his axioms are not statements, but rather statement forms. >Statement Form. He denied that by their interaction the concepts occurring in them might be given a meaning. It was rather a (in Frege’s terminology) "second stage concept" that was defined, today we would say a "structure". HilbertVsFrege: the point of the Hilbert’s proceeding is just that the meaning of "point", "line", etc. is left open. Frege and Hilbert might well have been able to agree on this, but they did not. Frege: Axiom should be in the classical sense a simple, sense-wise completely clear statement at the beginning of a system. Hilbert: statement forms that combined define a discipline. From this the "sloppy" figure of speech developed E.g. "straight" in spherical geometry was then a great circle. Thiel I 343 Formalism: 1) "older" formalism: second half of the 19th century, creators Hankel, Heine, Thomae, Stolz. "Formal arithmetic", "formal algebra". "Object of arithmetic are the signs on the paper itself, so that the existence of these numbers is not in question" (naive). Def "Permanence Principle": it had become customary to introduce new signs for numbers that had been added and to postulate then that the rules that applied to the numbers of the original are should also be valid for the extended area. Vs: that would have to be regarded as illegitimate as long as the consistency is not shown. Otherwise, you could introduce a new number, and E.g. simply postulate § + 1 = 2 und § + 2 = 1. This contradiction would show that these "new numbers" did not really exist. This explains Heine’s formulation that "existence is not in question". (> "tonk"). Thiel I 343/344 Thomae treated the problem as "rules of the game" in a somewhat more differentiated way. FregeVsThomae: he had not even precisely specified the basic rules of his game, namely the correlation to the rules, pieces and positions. This criticism of Frege was already a precursor of Hilbert’S proof theory, in which also mere character strings are considered without regard their possible content for their production and transformation according to the given rules. Thiel I 345 HilbertVsVs: Hilbert critics often overlook that, at least for Hilbert himself, the "finite core" should remain content-wise interpreted and only the "ideal", not finitely interpretable parts have no directly provable content. This important argument is of a methodical, not a philosophical nature. "Formalism" is the most commonly used expression for Hilbert’s program. Beyond that, the conception of formalism is also possible in a third sense: i.e. the conception of mathematics and logic as a system of action schemes for dealing with figures that are free of any content. HilbertVsFrege and Dedekind: the objects of the number theory are the signs themselves. Motto: "In the beginning was the sign." Thiel I 346 The designation formalism did not come from Hilbert or his school. Brouwer had hyped up the contrasts between his intuitionism and the formalism of Hilbert’s school to a landmark decision. |
F I G. Frege Die Grundlagen der Arithmetik Stuttgart 1987 F II G. Frege Funktion, Begriff, Bedeutung Göttingen 1994 F IV G. Frege Logische Untersuchungen Göttingen 1993 Berka I Karel Berka Lothar Kreiser Logik Texte Berlin 1983 T I Chr. Thiel Philosophie und Mathematik Darmstadt 1995 |
| Various Authors | Deutsch Vs Various Authors | DeutschVsinduction. I 36 Deutsch: induction needs no understanding, you could just explore all the character strings sequentially and randomly find a proper proof. ((s) but not randomly recognize it as correct! In addition, the evidence would not just happen to be right.) Deutsch: Hilbert’s rules could tell us almost nothing about reality. They would all be predicted, but not explained. Just like the "theory of everything". (DeutschVsTOE) I 220 Hilbert: "On the Infinite": scoffed at the idea that the demand for a "finite number of steps" was essential. DeutschVsHilbert: he was wrong. I 236 What is a "step" and what is "finite"? |
Deutsch I D. Deutsch Fabric of Reality, Harmondsworth 1997 German Edition: Die Physik der Welterkenntnis München 2000 |
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