Lexicon of Arguments

 

XI 136
Intuitionism/Quine/Lauener: he compares it with ancient conceptualism: universals are created by the mind.
VII (f) 125
ConceptualismVsPlatonism/Quine: treats classes as constructions, not as discoveries - Problem: Poincaré's "impredicative" definition: Def impredicative/Def Poincaré: "impredicative" means the specification of a class through a realm of objects, within which that class is located.
VII (f) 126
Classes/Conceptualism/Quine: for him, classes only exist if they originate from an ordered origin.
Classes/Conceptualism/Quine: conceptualism does not require classes to exist beyond conditions of belonging to elements that can be expressed.
Cantor's proof: would entail something else: It appeals to a class h of those elements of class k which are not elements of the subclasses of k to which they refer.
VII (f) 127
But this is how the class h is specified impredicatively! h is itself one of the partial classes of k. >Classes/Quine.
Thus a theorem of classical mathematics goes overboard in conceptualism.
The same fate strikes Cantor's proof of the existence of supernumerary infinity.
QuineVsConceptualism: this is a welcome relief, but there are problems with much more fundamental and desirable theorems of mathematics: e.g. the proof that every limited sequence of numbers has an upper limit.
VII (a) 14
Universals Dispute/Middle Ages/Quine: the old groups reappear in modern mathematics:
Realism: Logicism
Conceptualism: Intuitionism
Nominalism: Formalism.
Conceptualism/Middle Ages/Quine: holds on to universals, but as mind-dependent.
ConceptualismVsReduceability Axiom: because the reduceability axiom reintroduces the whole platonistic class logic. >Universals/Quine.