Disputed term/author/ism  Author 
Entry 
Reference 

Constructivism  Quine  XIII 33 Constructivism/Mathematics/Quine: (VsUniversals). Thesis: theorems that can be shown constructively should be preferred. Def Constructivism/Mathematics/Quine: Thesis: any abstract object is specifiable. Against: Predicative Set Theory: is too weak to prove that there must be unspecifiable classes and unspecifiable real numbers. Quantification/Variables/Quine: the quantification is different if it is certain that every object in the domain is specifiable. For example, the natural numbers are such a domain. This is because there is an Arabic number for each of them. Def Substitutional Quantification/sQ/Quine: (Universal Quantification (x)): the formula preceding the quantifier becomes true under any grammatically permissible substitution for the letter "x". Referential Quantification/refQ/Substitutional Quantification: For example, natural numbers: here both are the same. >Substitutional Quantification/Quine. On the other hand: If not all objects can be specified: If not all objects in a domain can be specified by singular terms of the language used, then the two types of quantification diverge. For example, if the universal quantifier is fulfilled by all specifiable objects, but not by the nonspecifiable ones, then the substitutional quantification is true and the referential quantification is false. Existential Quantification/EQu/Substitutional Quantification/refQ/Quine: behaves accordingly. XIII 35 Substitutional Quantification/Referential Quantification: diverge in the case of existential quantification if the formula is satisfied by some unspecifiable, but not by any specifiable one. Substitutional Quantification/Quine: is unrealistic for concrete objects. Specifiability/Name/Namability/namable/Quine: Question: is each concrete object individually specifiable? For example every past or future bee, every atom and every electron? Yes, by numerical coordinates with rational numbers. But unlimited referential quantification is simply more natural here. Predicative set theory: here substitutional quantification is more attractive and manageable because abstract objects are parasitic in relation to language, in a way that concrete objects are not. Abstract/Charles Parsons/Quine: abstract objects are parasitic in relation to language, concrete objects are less parasitic. Substitutional Quantification/Quine: does not simply eliminate abstract objects from ontology, but grants them a "thinner" kind of existence. Abstract/Quine: expressions themselves are abstract, but not as wild as the inhabitants of higher set theory. Substitutional Quantification/Quine: is a compromise with militant nominalism. Abstract Objects/Quine: are then classes, like those of predicative set theory (RussellVs). >Abstractness/Quine. Substitutional Quantification/Referential Quantification/Parsons: has shown how both go together (Lit). By using two kinds of variables. Then you can also link them together (intertwine). Problem/Russell: predicative set theory is inadequate for the classical mathematics of real numbers. XIII 36 Real Numbers/Russell/Quine: their theory leads to unspecifiable real numbers and other unspecifiable classes. Substitutional Quantification/Quine: this problem did not lead to the substitutional quantification by itself. Constructive Mathematics/Constructivism/QuineVsBrouwer: heated minds developed and still develop constructive mathematics that are suitable for all sciences. Problem: this leads to unattractive deviations from standard logic. Standard Logic/Constructivism/Quine: Experiments with standard logic: Weyl, Paul Lorenzen, Erret Bishop. Hao Wang, Sol Feferman. These are solutions with predicative set theory together with artistic circles. Problem: you do not know exactly how much mathematics scientists need. Nominalism/Quine: we probably do not need nominalism through and through, but an attractive approach to it. 
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 
Excluded Middle  d’Abro  A. d'Abro Die Kontroversen über das Wesen der Mathematik 1939 in Kursbuch 8 Mathematik 1967 53 The sentence of the excluded middle/VsBrouwer: the sentence of the excluded middle speaks for the fact that the antinomies occur not only in connection with infinite quantities. E.g. Liar Paradox. Here, too, the principle of the logically excluded middle did not seem to apply.  54 E.g. Russell's barber's paradox, with the barber shaving everyone in town except those who shave themselves. Here, too, the sentence of the excluded middle does not apply without infinite quantities. The intuitionists assert with Poincaré that antinomies without any infinities are lopish. Poincaré: The antinomies of certain logicians are simply circular. 

Disputed term/author/ism  Author Vs Author 
Entry 
Reference 

Brouwer, L.E.J.  Verschiedene Vs Brouwer, L.E.J.  A. d’Abro Die Kontroversen über das Wesen der Mathematik in Kursbuch IV S. 53 Frankfurt 1967 According to Weyl, the concept of irrational number must either be abandoned or thoroughly modified. Brouwer: in the treatment of infinite sets, the sentence from the excluded middle does not apply. The sentence of the excluded middle/VsBrouwer: speaks of the fact that antinomies do not only occur in connection with infinite sets. Example liar paradox. Here the principle of the logically excluded third party also did not seem to apply. IV 54 E.g. Russell's Barber Paradox, where the barber shaves everyone in town except those who shave themselves. Again, no sentence of the excluded middle without infinite sets. The intuitionists claim with Poincaré that antinomies are ridiculous without infinity. Poincaré: The antinomies of certain logicians are simply circular. 
