Find counter arguments by entering NameVs… or …VsName.
The author or concept searched is found in the following 13 entries.
| Disputed term/author/ism | Author |
Entry |
Reference |
|---|---|---|---|
| Calculus | Wittgenstein | Hintikka I 26 Calculus/Wittgenstein/Hintikka: if language is calculus, (WittgensteinVs) you can use formalism to describe those parts of language, which are subject to variation. >Formalism. II 83 Calculus/Description/Border/Wittgenstein: a calculus cannot be described without using it, and language cannot be described without specifying its meaning. >Language, >Description, >Meaning. II 212 Mental Acts/Wittgenstein: mental acts are not used in addition to calculating or speaking - instead: the calculus is precisely speaking as such. Calculating: is one step at a time. There is no mental act which anticipates the whole. Even meaning is not a mental process that would accompany the words. II 426 Calculus/Wittgenstein: two different calculi can result in e.g. 3 - but they are still two different results. II 427 The meaning of such a question is determined by the solution method. The question corresponds to a general law to find an answer. II 428 Rational Numbers/Wittgenstein: here we are talking about cuts with right and left classes. Hardy gives concrete examples. II 429 Question: are the examples essential? What is the meaning of the symbol "P", which denotes all rational numbers of the property, if no examples are given? What is the property of rationality (of numbers) as opposed to what? Calculus/Term/Wittgenstein: the general expressions L (left class) and R (right class) do not expand the area, but form a new expression type. A new calculation. And that is not the discovery of a wider area. Here we have a new area. VI 120 Mathematics/WittgensteinVsHilbert/Schulte: the demand for consistency disrupts peace! VI 121 Instead: "verificationist" approach (intuitionism). Search and find. >Verificationism, >Intuitionism. Search: in mathematics different from the material object. The calculus tells me where to look. Only the method teaches what you have actually asked for. >Method. The meaning of the sentence is the method of its verification. cf. >Sentence meaning. |
W II L. Wittgenstein Wittgenstein’s Lectures 1930-32, from the notes of John King and Desmond Lee, Oxford 1980 German Edition: Vorlesungen 1930-35 Frankfurt 1989 W III L. Wittgenstein The Blue and Brown Books (BB), Oxford 1958 German Edition: Das Blaue Buch - Eine Philosophische Betrachtung Frankfurt 1984 W IV L. Wittgenstein Tractatus Logico-Philosophicus (TLP), 1922, C.K. Ogden (trans.), London: Routledge & Kegan Paul. Originally published as “Logisch-Philosophische Abhandlung”, in Annalen der Naturphilosophische, XIV (3/4), 1921. German Edition: Tractatus logico-philosophicus Frankfurt/M 1960 Hintikka I Jaakko Hintikka Merrill B. Hintikka Investigating Wittgenstein German Edition: Untersuchungen zu Wittgenstein Frankfurt 1996 Hintikka II Jaakko Hintikka Merrill B. Hintikka The Logic of Epistemology and the Epistemology of Logic Dordrecht 1989 |
| Consistency | Hilbert | Berka I 413 Hilbert/Lecture: "Mathematical Problems" (1900)(1): the second problem of the mathematical problems is to prove the consistency of the arithmetic axioms. Consistency/arithmetics/problem/Schröter: at first, there is no way to see, since a proof by specifying a model is self-banning, since arithmetic is the simplest area on whose consistency all consistency proofs should be returned in other areas. So a new path must be taken. Consistency proof/Schröter: for the arithmetic axioms: the consistency requires the proof that an arithmetical statement cannot also be used to derive the contradictory negation of this statement from the axioms. To do this, it suffices to prove the non-derivability of any statement e.g. 0 unequal 0. If this is to be successful, it must be shown that all the deductions from the arithmetic axioms have a certain property which come off the statement that states 0 unequal 0. I 414 Problem: the amount of the consequences is completely unpredictable. Solution/Hilbert: the process of infering (logical inference) has to be formalized itself. With this however, the concluding/infering is deprived of all content. Problem: now, one can no longer say that a theory, e.g. is about the natural numbers. Formalism/Schröter: according to formalism, mathematics is no longer concerned with objects which refer to a real or an ideal world, but only by certain signs, or their transformations, which are made according to certain rules. WeylVsHilbert: that would require a reinterpretation of all the mathematics so far. 1. David Hilbert: Mathematische Probleme, in: Nachrichten der Königlichen Gesellschaft der Wissenschaften zu Göttingen, mathematisch-physikalische Klasse, issue 3, 1900, pp. 253–297. |
Berka I Karel Berka Lothar Kreiser Logik Texte Berlin 1983 |
| Consistency | Bigelow | I 182 Consistency/Bigelow/Pargetter: a way to guarantee that a description is consistent is to show that something meets this description. Definition Principle of instantiation/Bigelow/Pargetter: we can call this the principle of instantiation (instantiation principle). Contradiction-free/Bigelow/Pargetter: is essential for mathematics, for other areas it is more like housekeeping. Consistency/Hilbert: precedes existence. A mathematical proof exists only if it is non-contradictory. Consistency/FregeVsFormalism/FregeVsHilbert/Bigelow/Pargetter: Existence precedes the consistency. Consistency requires the existence of a consistently described thing. If it exists, the corresponding description is consistent. If it does not exist, how do we guarantee consistency? --- I 183 Frege/Bigelow/Pargetter: thinks here epistemically, in terms of "guarantees". But his view can be extended: if there is no object, there is no difference between a consistent and a contradictory description. Frege/Bigelow/Pargetter: pro Frege: this is the basis for modern mathematics. This is also the reason why quantum theory is so important: it provides examples of everything that mathematicians wish to investigate (at least until recently). |
Big I J. Bigelow, R. Pargetter Science and Necessity Cambridge 1990 |
| Definitions | Frege | I 15 Definition/Frege: you cannot define: "The number one is a thing" because there is a definite on the one side of the equation and an indefinite article on the other. >Equations, >Articles, >Definability. I 78 Definition/Frege: specifying a mode of operation is not a definition. I 99 Definition/Object/Introduction/Frege: the way in which an object was introduced is not a property of the object. >Introduction. The definition of an object only specifies the use of a sign, it says nothing about the object. ((s) Here: introduction of an object in the speech = definition) Introduction/Frege: after the introduction, the definition turns into a judgment about the object. I 130 FregeVsFormalism: formalism only gives instructions for definitions, not definitions as such. >Formalism. I 131 E.g. Number i/Frege: you have to re-explain the meaning of "sum". FregeVsFormalism/FregeVsHilbert: it is not enough to demand only one meaning. >Foundation. IV 100ff Definition/Object/Frege: the definite article must be on both sides here. Defining an object only specifying the use of a sign. More interesting are definitions of properties. IV 100ff Indefinable/Frege: truth and identity are indefinable as simple basic concepts. Other AuthorsVs. > truth theories, > theories of meaning. >Truth theories, >Meaning theories. |
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 |
| Existence | Hilbert | Berka I 294 Existence/consistency/concept/Hilbert: if one assigns features to a concept which contradicts itself, I say: the concept does not exist mathematically. FregeVsHilbert/(s): Hilbert would say the concept can exist, but there is no object for it. Existence/number/Hilbert: the existence of a concept is proved if it can be shown that there are never contradictions in the application of a finite number of logical conclusions. This would prove the existence of a number or a function. Berka I 294/295 Real numbers/existence/axioms/Hilbert: here, the consistency is a proof for the axioms and it is also the proof for the existence of the continuum(1). 1. D. Hilbert: Mathematische Probleme, in: Ders. Gesammelte Abhandlungen (1935), Vol. III, pp. 290-329 (gekürzter Nachdruck v. p. 299-301). |
Berka I Karel Berka Lothar Kreiser Logik Texte Berlin 1983 |
| Formalism | Frege | I 127 Sign/FregeVsFormalism: blank signs are only a blackening of the paper. Their use would be a logical error. Blank signs do not solve any task, e.g. x + b = c: if b > c, there is no natural number x, which can be used. To accept the difference (c - b) as an artificial new sign is no solution. Sign/Frege: where a solution is possible, it is not the sign that is the solution, but the meaning of the sign. I 130 FregeVsFormalism: formalism only offers instructions for definitions - not the definition itself. I 131 E.g. Number i: one has to re-explain the meaning of "sum" - FregeVsHilbert: it is not enough just to call for a sense. Cf. >Foundation, >Content, >Sense, >Signs, >Symbols, >Equations, >Definitions, >Formalization, cf. >Introduction, >"tonk"/Belnap-Prior debate. |
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 |
| Formalism | Geach | I 173 Characters/Geach: a mere coloring of the paper (Graphic occurrence) can not be true or false. >Blackening of the paper, >Formalism, >Foundation, >Truth value, >FregeVsHilbert. |
Gea I P.T. Geach Logic Matters Oxford 1972 |
| Formalism | Wittgenstein | VI 119 Formalism/Substitute/Sign/Symbol/WittgensteinVsFrege: Frege: characters are either mere blackening or signs of something - then what they represent, is their meaning - Wittgenstein: false alternative - E.g. chess pieces: represent nothing. Solution: use like in the game instead of representation of something. - ((s) Use is more than mere blackening of the paper and less than representation of an object). Wittgenstein: formalism is not entirely unjustified. >Blackening of the paper, >Mathematics, >Representation, >Rules; cf. >FregeVsHilbert. |
W II L. Wittgenstein Wittgenstein’s Lectures 1930-32, from the notes of John King and Desmond Lee, Oxford 1980 German Edition: Vorlesungen 1930-35 Frankfurt 1989 W III L. Wittgenstein The Blue and Brown Books (BB), Oxford 1958 German Edition: Das Blaue Buch - Eine Philosophische Betrachtung Frankfurt 1984 W IV L. Wittgenstein Tractatus Logico-Philosophicus (TLP), 1922, C.K. Ogden (trans.), London: Routledge & Kegan Paul. Originally published as “Logisch-Philosophische Abhandlung”, in Annalen der Naturphilosophische, XIV (3/4), 1921. German Edition: Tractatus logico-philosophicus Frankfurt/M 1960 |
| Formalism | Thiel | I 20 Formalism/Thiel: Carries out, so to speak, the "linguistic turn" in mathematics. It is now asked what the object of the mathematician's work is. Rules for actions. Symbols are replaced by others. The formalist does not ask for the "meaning". Mathematics: Theory of formalisms or formal systems. >Formalism. In addition to this "calculus-theoretical variant" of formalism, there is the "structure-theoretical variant". (>Hilbert). Different formal systems can be interpreted as valid from exactly the same mathematical object domains. We can call this their "description" by the formal systems. >Mathematical entities. I 279 Formalism/Geometry/Hilbert/Thiel: In 1899 Hilbert had used terms such as point, straight line, plane, "between", etc. in his foundations of geometry, but had understood their meaning in a previously unfamiliar way. It should not only enable the derivation of the usual sentences, but in its entirety should also determine the meaning of the terms used in them. I 280 Later this was called "definition by postulates", "implicit definition". >Definitions, >Definability, The terms point, straight line, etc. should at most be a convenient aid for mathematical understanding. FregeVsHilbert: clarifies in his correspondence that his axioms are not statements but forms of statements. >Statement form. He contested the fact that their combination gave meaning to the terms appearing in them. Rather a (in Frege's terminology) "second level term" is defined, today one would also say a "structure". HilbertVsFrege: N.B.: Hilbert's approach is precisely that the meaning of "point", "straight line" etc. is left open. Frege and Hilbert could have agreed on it, but did not. Axioms/Frege/Thiel: an axiom should be a simple statement at the beginning of a system. Axioms/Hilbert: forms of statement that together define a discipline. This has developed into the "sloppy" way of speaking, e.g. "straight line" in sphere geometry is a great circle. Thiel I 342 Intuitionism and formalism are often presented as alternatives to logicism. The three differ so strongly that a comparison is even difficult. I 343 Formalism/Thiel: 1. "older" formalism: second half 19th century creators Hankel, Heine, Thomae, Stolz. "formal arithmetic," "formal algebra". "The subject of arithmetic are the signs on the paper itself, so that the existence of these numbers is not in question" (naively). Def "principle of permanence": it had become customary to introduce new signs for additional numbers and then to postulate that the rules valid for the numbers of the initial range should also be valid for the extended range. Vs: this should be considered illegitimate as long as the consistency is not shown. Otherwise a new figure could be introduced, and one could simply postulate e.g. § + 1 = 2 and § + 2 = 1. This contradiction would show that the "new numbers" do not really exist. This explains Heine's formulation that the "existence is not at all in question". I 343/'344 Thomae treated the problem as "rules of the game" in a more differentiated way. FregeVsThomae: he did not even specify the basic rules of his game, namely the correspondences to the rules, figures, and positions. This criticism of Frege was already a forerunner of Hilbert's theory of proof, in which mere series of signs are also considered with disregard for their possible content on their creation and transformation according to given rules. I 345 HilbertVsVs: Critics of Hilbert often overlook the fact that, at least for Hilbert himself, the "finite core" should remain interpreted in terms of content and only the "ideal" parts that cannot be interpreted in a finite way have no content that can be directly displayed. This note is methodical, not philosophical. For Hilbert's program, "formalism" is also the most frequently used term. Beyond that, the concept of formalism has a third sense: namely, the concept of mathematics and logic as a system of schemes of action for dealing with figures free of any content. HilbertVsFrege and Dedekind: the objects of number theory are the signs themselves. Motto: "In the beginning was the sign." I 346 The term formalism did not originate from Hilbert or his school. Brouwer had stylized the contrasts between his intuitionism and the formalism of the Hilbert School into a fundamental decision. Brouwer: his revision of the classical set and function concept brings another "Species of Mathematics". Instead of the function as assignment of function values to arguments of the function, sequences of election actions of a fictitious "ideal mathematician" who chooses a natural number at every point of the infinitely conceived process take place, whereby this number may be limited by the most different determinations for the election action, although in the individual case the election action is not predictable. |
T I Chr. Thiel Philosophie und Mathematik Darmstadt 1995 |
| Induction | Poincaré | Waismann I 70 Induction/Brouwer/Poincaré/Waismann: the power of induction: it is not a conclusion that carries to infinity. The sentence a + b = b + a is not an abbreviation for infinitely many individual equations, as well as 0.333 ... is not an abbreviation, and the inductive proof is not the abbreviation for infinitely many syllogisms (VsPoincaré). In fact, with the formulation of the formulas we begin a+b = b+a a+(b+c) = (a+b)+c a whole new calculus, which cannot be inferred from the calculations of arithmetic in any way. But: Principle/Induction/Calculus/Definition/Poincaré/Waismann: ... this is the correct thing in Poincaré's assertion: the principle of induction cannot be proved logically. VsPoincaré: But he does not represent, as he thought, a synthetic judgment a priori; it is not a truth at all, but a determination: If the formula f(x) applies for x = 1, and f(c + 1) follows from f(c), let us say that "the formula f(x) is proved for all natural numbers". --- A. d'Abro Die Kontroversen über das Wesen der Mathematik 1939 in Kursbuch 8 Mathematik 1967 46 Induction/PoincaréVsHilbert: in some of his demonstrations, the principle of induction is used and he asserts that this principle is the expression of an extra-logical view of the human mind. Poincaré concludes that the geometry cannot be derived in a purely logical manner from a group of postulates. --- 46 Induction is continually applied in mathematics, inter alia also in Euclid's proof of the infinity of the prime numbers. Induction principle/Poincaré: it cannot be a law of logic, for it is quite possible to construct a mathematics in which the principle of induction is denied. Hilbert, too, does not postulate it among his postulates, so he also seems to be of the opinion that it is not a pure postulate. |
Waismann I F. Waismann Einführung in das mathematische Denken Darmstadt 1996 Waismann II F. Waismann Logik, Sprache, Philosophie Stuttgart 1976 |
| Platonism | Quine | XII 44 Platonic Idea/Quine: is not the same as a mental idea. >Ideas/Quine. XI 136 Mathematics/QuineVsHilbert/Lauener: is more than just syntax. Quine reluctantly professes Platonism. XI 155 CarnapVsPlatonism/CarnapVsNominalism: is a metaphysical pseudo discussion. Solution: it is about choosing a language. >Language/Quine. VII(f) 125 Conceptualism VsPlatonism/Quine: treats classes as constructions, not as discoveries. Problem: Poincaré's impredicative definition: Def Impredicative Definition/Poincaré/Quine: the specification of a class by a realm of objects within which this class is located. >Classes/Quine. VII (f) 126 Classes/Platonism/Quine: when classes are considered pre-existing, there is no objection to picking one of them by a move that presupposes their existence. Classes/Conceptualism/Quine: for him, however, classes only exist if they originate from an ordered origin. Of course, this should not be interpreted in terms of time. VII (f) 127 Platonism/Conceptualism/Quine: both allow universals and classes as irreducible. Conceptualism: allows fewer classes. But rests on a rather metaphorical reason: "Origin". >Conceptualism/Quine. V 126 Platonism/Quine: is opened by form words, not by color words! Reason: a union of color spots has the same color, but a union of spots of a certain shape does not necessarily have the same shape. |
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 |
| Signs | Frege | II 31 Signs/Frege: as long as e.g. the plus sign is used only between integers ("a + b"), it only needs to be explained for this purpose. If other objects are to be linked, e.g. "sun" with something else, the plus sign must be redefined. >Definition, >Definability, >Connectives, >Equal sign, >Copula. II 41 Frege: a sign is a proxy. >Proxy. II 88 Numeral/Frege: e.g. "2" is saturated. In contrast: the functional character, e.g. "sin" (sine, sinus) is unsaturated. >Unsaturated. II 91 Sign/Frege: signs are the requirements for conceptual thinking - they no longer refer to the individual thing, but to what several things have in common. I 127 Sign/FregeVsFormalism: empty signs are only black spots on paper. Their use would be a logical error. Empty signs do not solve any task. E.g. x + b = c: if b > c, there is no natural number x that can be inserted - nor to accept the difference (c - b) as an artificial new sign. Sign/Frege: and where a solution is possible, the sign is not the solution, but the meaning of the sign. Husted V 130 FregeVsFormalism: formalism only gives instructions for definitions, not definitions themselves. >Formalism. Frege I 131 E.g. Number i: the meaning of "total" must be re-explained. FregeVsHilbert: it is not enough just to call for a sense. |
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 Husted I Jörgen Husted "Searle" In Philosophie im 20. Jahrhundert, A. Hügli/P. Lübcke Reinbek 1993 Husted II Jörgen Husted "Austin" In Philosophie im 20. Jahrhundert, A. Hügli/P. Lübcke Reinbek 1993 Husted III Jörgen Husted "John Langshaw Austin" In Philosophie im 20. Jahrhundert, A. Hügli/P. Lübcke Reinbek 1993 Husted IV Jörgen Husted "M.A. E. Dummett. Realismus und Antirealismus In Philosophie im 20. Jahrhundert, A. Hügli/P. Lübcke (Hg) Hamburg 1993 Husted V J. Husted "Gottlob Frege: Der Stille Logiker" In Philosophie im 20. Jahrhundert, A. Hügli/P. Lübcke (Hg) Reinbek 1993 |
| 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 |
| |||
| Disputed term/author/ism | Author Vs Author |
Entry |
Reference |
|---|---|---|---|
| Constructivism | Verschiedene Vs Constructivism | Barrow I 65/66 Constructivism: Founder Leopold Kronecker: "The whole numbers were made by God, everything else is human work." The meaning of a mathematical formula lies only in the chain of operations with which it is constructed. Constructivism introduces a third status: undecided! A statement that cannot be decided in a finite number of steps comes into the junk chamber of undecidedness. I 67 VsConstructivism: before constructivism, mathematics had developed all possible methods of proof which are not feasible in a finite number of steps. Def Reductio ad absurdum/raa: evidence which assumes that something is wrong in order to prove its indispensability, in that a contradiction arises from the very assumption of falsity. I 68 BrouwerVsHilbert: (Einstein: the "war of frogs and mice" also >"frog mouse war") Hilbert prevailed: The board of editors of the joint newspaper "Mathematische Annalen" was dissolved and refounded without Brouwer. I 69 Constructivism: strange anthropocentrism: BarrowVsConstructivism: the idea of a universal human intuition of the natural numbers cannot be kept historically (see above). A constructivist cannot say whether the intuition of a human being is the same as that of another, nor whether such an intuition will develop further in the future. |
B I John D. Barrow Warum die Welt mathematisch ist Frankfurt/M. 1996 B II John D. Barrow The World Within the World, Oxford/New York 1988 German Edition: Die Natur der Natur: Wissen an den Grenzen von Raum und Zeit Heidelberg 1993 B III John D. Barrow Impossibility. The Limits of Science and the Science of Limits, Oxford/New York 1998 German Edition: Die Entdeckung des Unmöglichen. Forschung an den Grenzen des Wissens Heidelberg 2001 |
| Formalism | Frege Vs Formalism | Brandom I 606 FregeVsFormalists: How can evidence be provided that something falls under a concept? Frege uses the concept of necessity to prove the existence of an object. Brandom I 609 Free Logic: "Pegasus is a winged horse" is regarded as true, although the object does not exist physically. It can serve as substituent. FregeVs. (>Read). Brandom I 620 Frege: Pegasus has "sense" but no "meaning". FregeVsFormalism: Important argument: it is not enough merely to refer to the Peano axioms, identities such as "1 = successor to the number 0" are trivial. They do not combine two different ways of picking out an object. Solution: Abstraction: it is necessary to connect the use of the expressions of the successor numbers with the already common expressions. Frege I 130 Equation/Frege: you must not put the definite article on one side of an equation and the indefinite article on the other. FregeVsFormalism: a purely formal theory is sufficient. It’s only an instruction for the definitions, not a definition as such. I 131 Number System/Expansion/Frege: in the expansion, the meaning cannot be fixed arbitrarily. E.g. the meaning of the square root is not already unchangeable before the definitions, but it is determined by these. ((s) Contradiction? Anyway, Frege is getting at meaning as use). Number i/Frege: it does not matter whether a second, a millimeter or something else is to play a role in this. I 132 It is only important that the additions and multiplication sentences apply. By the way, i falls out of the equation again. But, E.g. with "a ´bi" you have to explain what meaning "total" has in this case. It is not enough to call for a sense. That would be just ink on paper. (FregeVsHilbert). Bigelow I 182 Consistency/FregeVsFormalism/FregeVsHilbert/Bigelow/Pargetter: Existence precedes consistency. For consistency presupposes the existence of a consistently described object. If it exists, the corresponding description is consistent. If it does not exist, how can we guarantee consistency? Frege I 125 Concept/Frege: How can you prove that it does not contain a contradiction? Not by the determination of the definition. I 126 E.g. ledger lines in a triangle: it is not sufficient for proof of their existence that no contradiction is discovered in on their concept. Proof of the disambiguity of a concept can strictly only be carried out by something falling under it. The reverse would be a mistake. E.g. Hankel: equation x + b = c: if b is > c, there is no natural number x which solves the problem. I 127 Hankel: but nothing keeps us from considering the difference (c - b) as a sign that solves the problem! Sign/FregeVsHankel/FregeVsFormalism: there is something that hinders us: E.g. considering (2 - 3) readily as a sign that solves the problem: an empty sign does not solve the problem, but is only ink on paper. Its use as such would then be a logical error. Even in cases where the solution is possible, it is not the sign that is the solution, but the content. Wittgenstein I 27 Frege/Earlier Wittgenstein/Hintikka: ((FregeVsFormalism) in the philosophy of logic and mathematics). Frege dispensed with any attempt to attribute a semantic content to his logical axioms and rules of evidence. Likewise, Wittgenstein: "In logical syntax, the meaning of a sign must never play a role, it may only require the description of the expressions." Therefore, it is incorrect to assert that the Tractatus represents the view of the inexpressibility of language par excellence. The inexpressibility of semantics is merely limited to semantics, I 28 syntax can certainly be linguistically expressed! In a letter to Schlick, Wittgenstein makes the accusation that Carnap had taken his ideas, without pointing this out (08.08.32)! |
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 Bra I R. Brandom Making it exlicit. Reasoning, Representing, and Discursive Commitment, Cambridge/MA 1994 German Edition: Expressive Vernunft Frankfurt 2000 Bra II R. Brandom Articulating reasons. An Introduction to Inferentialism, Cambridge/MA 2001 German Edition: Begründen und Begreifen Frankfurt 2001 Big I J. Bigelow, R. Pargetter Science and Necessity Cambridge 1990 W II L. Wittgenstein Wittgenstein’s Lectures 1930-32, from the notes of John King and Desmond Lee, Oxford 1980 German Edition: Vorlesungen 1930-35 Frankfurt 1989 W III L. Wittgenstein The Blue and Brown Books (BB), Oxford 1958 German Edition: Das Blaue Buch - Eine Philosophische Betrachtung Frankfurt 1984 W IV L. Wittgenstein Tractatus Logico-Philosophicus (TLP), 1922, C.K. Ogden (trans.), London: Routledge & Kegan Paul. Originally published as “Logisch-Philosophische Abhandlung”, in Annalen der Naturphilosophische, XIV (3/4), 1921. German Edition: Tractatus logico-philosophicus Frankfurt/M 1960 |
| Frege, G. | Waismann Vs Frege, G. | Waismann I 77 Frege: Definition of the number in two steps a) when two sets are equal. b) Definition of the term "number": it is equal if each element of one set corresponds to one element of the other set. Unique relation. Under Def "Number of a Set"/Frege: he understands the set of all sets equal to it. Example: the number 5 is the totality of all classes of five in the world. VsFrege: how shall we determine that two sets are equal? Apparently by showing such a relation. For example, if you have to distribute spoons on cups, then the relation did not exist before. As long as the spoons were not on the cups, the sets were not equal. However, this does not correspond to the sense in which the word equal is used. So it is about whether you can put the spoons on the cups. But what does "can" mean? I 78 That the same number of copies are available. Not the assignment determines the equivalence, but vice versa. The proposed definition gives a necessary, but not sufficient condition for equal numbers and defines the expression "equal number" too narrowly. Class: List ("school class") logical or term (mammals) empirical. With two lists it is neither emopirical nor logical to say that they can be assigned to each other. Example 1. Are there as many people in this room as in the next room? An experiment provides the answer. 2. Are 3x4 cups equal to 12 spoons? You can answer this by drawing lines, which is not an experiment, but a process in a calculus. According to Frege, two sets are not equal if the relation is not established. You have defined something, but not the term "equal numbered". You can extend the definition by saying that they can be assigned. But again this is not correct. For if the two sets are given by their properties, it always makes sense to assert their "being-assignment", (but this has a different meaning, depending on the criterion by which one recognizes the possibility of assignment: that the two are equal, or that it should make sense to speak of an assignment! In fact, we use the word "equal" according to different criteria: of which Frege emphasizes only one and makes it a paradigm. Example 1. If there are 3 cups and 3 spoons on the table, you can see at a glance how they can be assigned. I 79 2. If the number cannot be overlooked, but it is arranged in a clear form, e.g. square or diamond, the equal numbers are obvious again. 3. The case is different, if we notice something of two pentagons, that they have the same number of diagonals. Here we no longer understand the grouping directly, it is rather a theorem of geometry. 4. Equal numbers with unambiguous assignability 5. The normal criterion of equality of numbers is counting (which must not be understood as the representation of two sets by a relation). WaismannVsFrege: Frege's definition does not reflect this different and flexible use. I 80 This leads to strange consequences: According to Frege, two sets must necessarily be equal or not for logical reasons. For example, suppose the starlit sky: Someone says: "I don't know how many I've seen, but it must have been a certain number". How do I distinguish this statement from "I have seen many stars"? (It is about the number of stars seen, not the number of stars present). If I could go back to the situation, I could recount it. But that is not possible. There is no way to determine the number, and thus the number loses its meaning. For example, you could also see things differently: you can still count a small number of stars, about 5. Here we have a new series of numbers: 1,2,3,4,5, many. This is a series that some primitive peoples really use. It is not at all incomplete, and we are not in possession of a more complete one, but only a more complicated one, beside which the primitive one rightly exists. You can also add and multiply in this row and do so with full rigor. Assuming that the things of the world would float like drops to us, then this series of numbers would be quite appropriate. For example, suppose we should count things that disappear again during counting or others emerge. Such experiences would steer our concept formation in completely different ways. Perhaps words such as "much", "little", etc. would take the place of our number words. I 80/81 VsFrege: his definition misses all that. According to it, two sets are logically necessary and equal in number, without knowledge, or they are not. In the same way, Einstein had argued that two events are simultaneous, independent of observation. But this is not the case, but the sense of a statement is exhausted in the way of its verification (also Dummett) Waismann: So you have to pay attention to the procedure for establishing equality in numbers, and that's much more complicated than Frege said. Frege: second part of the definition of numbers: Def Number/Frege: is a class of classes. ((s) Elsewhere: so not by Frege! FregeVs!). Example: the term "apple lying on the table comes to the number 3". Or: the class of apples lying on the table is an element of class 3. This has the great advantage of evidence: namely that the number is not expressed by things, but by the term. WaismannVsFrege: But does this do justice to the actual use of the number words? Example: in the command "3 apples!" the number word certainly has no other meaning, but after Frege this command can no longer be interpreted according to the same scheme. It does not mean that the class of apples to be fetched is an element of class 3. Because this is a statement, and our language does not know it. WaismannVsFrege: its definition ties the concept of numbers unnecessarily to the subject predicate form of our sentences. In fact, it results the meaning of the word "3" from the way it is used (Wittgenstein). RussellVsFrege: E.g. assuming there were exactly 9 individuals in the world. Then we could define the cardinal numbers from 0 to 9, but the 10, defined as 9+1, would be the zero class. Consequently, the 10 and all subsequent natural numbers will be identical, all = 0. To avoid this, an additional axiom would have to be introduced, the Def "infinity axiom"/Russell: means that there is a type to which infinitely many individuals belong. This is a statement about the world, and the structure of all arithmetic depends essentially on the truth of this axiom. Everyone will now be eager to know if the infinity axiom is true. We must reply: we do not know. It is constructed in a way that it eludes any examination. But then we must admit that its acceptance has no meaning. I 82 Nor does it help that one takes the "axiom of infinity" as a condition of mathematics, because in this way one does not win mathematics as it actually exists: The set of fractions is dense everywhere, but not: The set of fractions is dense everywhere if the infinity axiom applies. That would be an artificial reinterpretation, only conceived to uphold the doctrine that numbers are made up of real classes in the world (VsFrege: but only conditionally, because Frege does not speak of classes in the world). Waismann I 85 The error of logic was that it believed it had firmly underpinned arithmetic. Frege: "The foundation stones, fixed in an eternal ground, are floodable by our thinking, but not movable." WaismannVsFrege: only the expression "justify" the arithmetic gives us a wrong picture, I 86 as if its building were built on basic truths, while she is a calculus that proceeds only from certain determinations, free-floating, like the solar system that rests on nothing. We can only describe arithmetic, i.e. give its rules, not justify them. Waismann I 163 The individual numerical terms form a family. There are family similarities. Question: are they invented or discovered? We reject the notion that the rules follow from the meaning of the signs. Let us look at Frege's arguments. (WaismannVsFrege) II 164 1. Arithmetic can be seen as a game with signs, but then the real meaning of the whole is lost. If I set up calculation rules, did I then communicate the "sense" of the "="? Or just a mechanical instruction to use the sign? But probably the latter. But then the most important thing of arithmetic is lost, the meaning that is expressed in the signs. (VsHilbert) Waismann: Assuming this is the case, why do we not describe the mental process right away? But I will answer with an explanation of the signs and not with a description of my mental state, if one asks me what 1+1 = 2 means. If one says, I know what the sign of equality means, e.g. in addition, square equations, etc. then one has given several answers. The justified core of Frege's critique: if one considers only the formulaic side of arithmetic and disregards the application, one gets a mere game. But what is missing here is not the process of understanding, but interpretation! I 165 For example, if I teach a child not only the formulas but also the translations into the word-language, does it only make mechanical use? Certainly not. 2. Argument: So it is the application that distinguishes arithmetic from a mere game. Frege: "Without a content of thought an application will not be possible either. WaismannVsFrege: Suppose you found a game that looks exactly like arithmetic, but is for pleasure only. Would it not express a thought anymore? Why cannot one make use of a chess position? Because it does not express thoughts. WaismannVsFrege: Let us say you find a game that looks exactly like arithmetic, but is just for fun. Would it notexpress a thought anymore? Chess: it is premature to say that a chess position does not express thoughts. Waismann brings. For example figures stand for troops. But that could just mean that the pieces first have to be turned into signs of something. I 166 Only if one has proved that there is one and only one object of the property, one is entitled to occupy it with the proper name "zero". It is impossible to create zero. A >sign must designate something, otherwise it is only printer's ink. WaismannVsFrege: we do not want to deny or admit the latter. But what is the point of this assertion? It is clear that numbers are not the same as signs we write on paper. They only become what they are through use. But Frege rather means: that the numbers are already there somehow before, that the discovery of the imaginary numbers is similar to that of a distant continent. I 167 Meaning/Frege: in order not to be ink blotches, the characters must have a meaning. And this exists independently of the characters. WaismannVsFrege: the meaning is the use, and what we command. |
Waismann I F. Waismann Einführung in das mathematische Denken Darmstadt 1996 Waismann II F. Waismann Logik, Sprache, Philosophie Stuttgart 1976 |
| Hilbert | Deutsch Vs Hilbert | I 236 Hilbert: "Über das Unendliche": spottete über den Gedanken, dass die Forderung nach der"endlichen Anzahl von Schritten" wesentlich ist. DeutschVsHilbert: aber er irrte sich. DeutschVsGödel: Zumindest eine von Gödels Einsichten in Beweise stellte sich als fehlerhaft heraus. I 237 Diesem Gedanken zufolge ist ein Beweis etwas besonderes, eine Reihe von Aussagen, die Beweisregeln gehorchen. Wir haben schon gesehen, dass ein Beweis besser nicht als ein Ding, sondern als ein Vorgang (Programm) gesehen werden sollte. Eine Art von Berechnung. Im klassischen Fall ist also die Umwandlung von Beweisvorgängen in Beweisdinge immer durchführbar. Wenn wir aber eine klassisch nicht auszuführende mathematische Berechnung, die ein Quantencomputer leicht machen kann betrachten: hier gibt es keine Möglichkeit all das aufzuzeichnen, was im Beweisprozess abläuft, weil das Meiste in anderen Universen passiert. Auf diese Weise kann man keinen Beweis alter Art führen. |
Deutsch I D. Deutsch Fabric of Reality, Harmondsworth 1997 German Edition: Die Physik der Welterkenntnis München 2000 |
| 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 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 |
| Hilbert | Quine Vs Hilbert | IX 187 Notation/Set Theory/Terminology/Hilbert/Ackermann: (1938, 1949): still lean to the old theory of Russell's statements functions (AF): for classes and relations: "F", "G", etc. with repressible indices, in stead of "x ε a" and "xRy" (Russell: "φx" and "ψ(x,y)" Hilbert "F(x)" and "G(x,y)". Quine: the similarity is misleading: The values of "F", "G", etc. are not AF, but classes and extensional relations and so for the only criterion that those are identical with the same extension. QuineVsHilbert: disadvantage that the attention is drawn away from essential differences between ML and logic. IX 188 It encourages us (incorrectly) to just consider the theory of classes and relations as a continuation of the QL, in which the thus far schematic predicate letters are re-registered in quantifiers and other places which were previously reserved for "x", "y" I.e. "F", "EG", "H(F,G)". The existence assumptions become too inconspicuous, although they are far-reaching! Just implicitly by quantification. Therefore every comprehension assertion, e.g. EF∀x(FX >> ... x ...) by such insertions simply follows from "G EF ∀x (Fx Gx) which in turn follows from "∀x(Gx Gx)". This had escaped Hilbert and Ackermann, they also took on comprehension axioms, they realized that they could have taken a primitive concept of abstraction instead (like Russell). Predicate Calculus/Functions Calculus/Church/Quine: (nth order): type theory breaking of after n types, fusion of set theory and logic (QuineVs). E.g. PK 2nd Stage: Theory of individuals and classes of individuals. It was simply seen as a QL where predicate letters are approved quantifiers. The actual QL then became a first stage PK. This trend also contained an erroneous distinction between TT and ML, as if one did not contain as good as assumptions about the other. On the other, hand he nourished the idea that the Ql itself already contained a theory of classes or attributes and relations in its "F" and "G". QuineVs: the vital distinction between schematic letters and quantifiable variables is neglected. X 96 Logic 2nd Stage/Hilbert's Successor/Quine: "higher-level PK": the values of these variables are in fact sets. This type of introduction makes them deceptively similar to logic. But it is wrong that only a few quantifiers are applied to existing predicate letters. E.g. the hypothesis "(Ey)(x)((x ε y) Fx)": here the existence of a set is asserted: {x:Fx}. This must be restricted to avoid antinomies. QuineVsHilbert: in the so-called higher order PK this assumption moves out of sight. The assumption is: "(EG) (x) (Gx Fx)" and follows from the purely logical triviality (x)(Fx Fx)" As long as we keep the scope of the values of "x" and "G" apart there is no risk of an antinomy. Nevertheless, a large piece of set theory has crept in unnoticed. XI 136 Mathematics/QuineVsHilbert/Lauener: more than mere syntax. Quine reluctantly professes Platonism. |
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 |
| Hilbert | Tarski Vs Hilbert | Horwich I 127 Truth/Philosophy/Mathematics/HilbertVsTarski: (only "philosophical" objection at all, from a mathematician!): the truht definition would have nothing to do with the "philosophical problem". But that should not be a criticism. Term/TarskiVsHilbert: I never understood what the "essential" of a term should be. ((s) >Frege: terms have features that can be regarded as necessary, because otherwise it is another term, in contrast to objects that can also turn out to have other properties, but are still the object under consideration.) Truth/Tarski: I do not think there is any "philosophical problem" here.(1) 1. A. Tarski, The semantic Conceptions of Truth, Philosophy and Phenomenological Research 4, pp. 341-75 |
Tarski I A. Tarski Logic, Semantics, Metamathematics: Papers from 1923-38 Indianapolis 1983 |
| Hilbert | Wittgenstein Vs Hilbert | VI 120 Mathematics/WittgensteinVsHilbert/Schulte: the demand for consistency disturbs the peace. --- VI 121 Instead: "verificationist" approach (intuitionism). Search and find. Search: in mathematics, unlike the material object. The calculus shows me, where I have to look. Only the method teaches for what one actually asked. "The sense of a proposition is the method of its verification." --- VI 122 Contradiction/WittgensteinVsHilbert/Schulte: if one is searching for a contradiction, one actually does not know for what one is looking for. Because the question is not connected with any known technique. Where no verification process is known, our statements have no meaning. Contradiction: misconception, as if the contradiction was hidden from the very beginning in the axioms, such as tuberculosis. No "hidden contradiction". |
W II L. Wittgenstein Wittgenstein’s Lectures 1930-32, from the notes of John King and Desmond Lee, Oxford 1980 German Edition: Vorlesungen 1930-35 Frankfurt 1989 W IV L. Wittgenstein Tractatus Logico-Philosophicus (TLP), 1922, C.K. Ogden (trans.), London: Routledge & Kegan Paul. Originally published as “Logisch-Philosophische Abhandlung”, in Annalen der Naturphilosophische, XIV (3/4), 1921. German Edition: Tractatus logico-philosophicus Frankfurt/M 1960 |
| Hilbert | Verschiedene Vs Hilbert | Berka I 414 Problem: the number of conclusions is completely incalculable. Solution/Hilbert: the process of following (logical inference) has to be formalized itself. This, however, removes all content from the closing process. Problem: now one can no longer say that a theory is about natural numbers, for example. Formalism/Schröter: after that mathematics is no longer about objects, which refer to a real or an ideal world, but only about certain signs, resp. their transformations, which are done according to certain rules. WeylVsHilbert: this makes a reinterpretation of the whole mathematics necessary. Klaus von Heusinger, Eselssätze und ihre Pferdefüsse Uni Konstanz Fachgruppe Sprachwissenschaft Arbeitspapier 64; 1994 Heusinger I 29 Donkey Sentences/Epsilon Analysis/Heusinger: Thesis: that certain and indefinite nominal phrases are context-dependent. I 30 The epsilon operator EO represents NP and anaphora as context-dependent selection functions - classic: by Hilbert. VsHilbert: too inflexible - modified: represents the progress of information - modified EO: selects a certain object in a certain situation. I 36 Modified Epsilon Operator/situation/Egli/Heusinger: (Egli 1991, Heusinger 1992,1993), Van der Does 1993) the epsilon operator receives a parameter for the situation. Selection function/VsHilbert/Heusinger: Problem: the selection principle does not say which element is selected. ((s) it means only afterwards: "the selected element"). Problem: with an ordered range like the numbers this can be the smallest. The linguistic range lacks such an order. Order/Language/Linguistic/Lewis: Solution: Def "Salience Hierarchy"/Lewis: (Lewis 1979) (s): contextual or situational outline of a given linguistic area. (salient. = outstanding). Selection function/Heusinger: so we have to assume a whole family of selection functions. I.e. not from a selection function defined by the model M. Salience Hierarchy/Epsilon operator/Egli/Heusinger: the salience hierarchy is represented by modified epsilon expressions. Index i/Spelling mode/Heusinger: represents the respective selection function here: For example, eix Fx refers to the most salient (outstanding) object in context i that has the property F. Unambiguity/Situation/Heusinger: the modified epsilon operator always specifies a certain object. Context/Sincerity/Heusinger: in changing contexts different objects can be selected. Solution/Heusinger: 1. the individual range of a model M must be extended by the range of indices I. The individual range of a model M must be extended by the range of indices I. 2. the function F is added to model M itself. |
Berka I Karel Berka Lothar Kreiser Logik Texte Berlin 1983 |
| Hilbert | Poincaré Vs Hilbert | A. d'Abro Die Kontroversen über das Wesen der Mathematik in Kursbuch IV S 46 Frankfurt 1967 PoincaréVsHilbert: in some of his demonstrations, the principle of induction is used, and it is claimed that this principle is the expression of an extra-logical perspective of the human mind. Poincaré concludes that geometry cannot be derived in a purely logical way from a group of postulates. (>Induction). Induction is applied continually in mathematics, among others even in Euclid's proof of the infinity of primes. Induction Principle/Poincaré: this cannot be a law of logic, since it is quite possible to construct a mathematics in which the induction principle is denied. Even Hilbert does not mention it among his postulates, he therefore also appears to be of the opinion that it is not a pure postulate! |
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| Russell, B. | Turing Vs Russell, B. | Hofstadter II 522/523 Turing: worked on the question of whether such undecidable assertions should be isolated, or whether they "grew through" the whole of mathematics. He found out that one cannot build a machine that can recognize infallibly undecidable sentences. (Reason: Gödel). Universal Turing machine: could act indistinguishably from other machines. TuringVsHilbert,VsRussell: assuming there was such a universal Turing machine, it would lead to a contradiction with itself. This universal machine could accept the symbolic number of any other machine and simulate it. Problem: what does it do with its own number? Contradiction. This is the reason why undecidable problems run through all mathematics and cannot be isolated. Can also be applied to humans: Example: "Will you answer this question with "no"? This shows: no matter how conscious one is of one's own mind, one cannot fully calculate one's own complexity if one tries to understand each other. So even machines cannot predict their own behaviour (in such cases). |
Hofstadter I Douglas Hofstadter Gödel, Escher, Bach: An Eternal Golden Braid German Edition: Gödel, Escher, Bach - ein Endloses Geflochtenes Band Stuttgart 2017 Hofstadter II Douglas Hofstadter Metamagical Themas: Questing for the Essence of Mind and Pattern German Edition: Metamagicum München 1994 |
| Tarski, A. | Hilbert Vs Tarski, A. | Horwich I 127 Truth/Philosophy/Mathematics/HilbertVsTarski: (only "philosophical" objection at all, from a mathematician!): the truth definition would have nothing to do with the "philosophical problem". But that should not be a criticism. Term/TarskiVsHilbert: I never understood what the "essential" of a term should be. ((s) >Frege: Concepts have characteristics that can be considered necessary, because otherwise it is another concept, in contrast to objects that can turn out to be something else, but are still the "object considered". Truth/Tarski: I don't think there is a "philosophical problem" here. |
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| 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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| Disputed term/author/ism | Pro/Versus |
Entry |
Reference |
|---|---|---|---|
| Platonism | Pro | Quine Lauener XI 136 Platonism: Quine reluctantly, but QuineVsFormalism/QuineVsHilbert |
Q XI H. Lauener Willard Van Orman Quine München 1982 |
| Formalism | Versus | Quine Lauener XI 136 Platonism: Quine reluctantly, but QuineVsFormalism/QuineVsHilbert |
Q XI H. Lauener Willard Van Orman Quine München 1982 |
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