Hellman Nominalism

Hellman uses the standard Russellian argument against absolutely everything as an argument for nominalism.

He considers whether the "Carnapian" view that there are lots of "factually equivalent" ontologies cuts against absolutely everything. Against Putnam, he suggests that it does, with the examples from last time.

He suggests that the denial of the existence of absolutely everything can be coherently expressed using semantic ascent and that ordinary general claims (no talking donkeys) can be expressed using restricted quantifiers.

Many of the articles we have read have tried to show that the assumption that there is quantification over everything leads to paradox by invoking various mathematical procedures justified by intuition or the like. Hellman gives a general argument underpinning such moves:

The open-endedness of mathematics. It has always been a bad idea in mathematics to reject a consistent proposal. Gathering things in a collection is the characteristic proposal of set theory, which is probably the most successful new piece of mathematics invented in historical times. Thus, while there is no logical inconsistency in denying that the domain of the quantifiers over absolutely everything form a set, it goes against everything we have learned about good mathematical procedure.

What leads to the problems is not the absolutely infinite but the unlimited in the kind of contrast between limited and unlimited that we see in sets vs classes and similarly for functions, ordinals, and categories. In ordinary language, Carnap made a distinction between Allwörter and limited predicates. Allwörter include 'thing,' 'entity,' and though this leads to difficulties, 'event,' 'process,' and others. Hellman talks a bit about a Carnapian resolution of the problem of unlimited quantification: A framework just presupposes its domain, and so if one takes, for example, the theory of sets to be a Carnapian framework, ….

-- ShaughanLavine - 16 Apr 2009