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Full Idea
The mathematical object-theorist says a number is an object that represents a cardinality quantifier, with the representation relation as the entire essence of the nature of such objects as cardinal numbers like 4.
Gist of Idea
It is claimed that numbers are objects which essentially represent cardinality quantifiers
Source
Harold Hodes (Logicism and Ontological Commits. of Arithmetic [1984])
Book Ref
-: 'Journal of Philosophy' [-], p.139
A Reaction
[compressed] This a classic case of a theory beginning to look dubious once you spell it our precisely. The obvious thought is to make do with the numerical quantifiers, and dispense with the objects. Do other quantifiers need objects to support them?
| 10011 | Identity is a level one relation with a second-order definition [Hodes] |
| 10027 | Mathematics is higher-order modal logic [Hodes] |
| 10021 | It is claimed that numbers are objects which essentially represent cardinality quantifiers [Hodes] |
| 10016 | When an 'interpretation' creates a model based on truth, this doesn't include Fregean 'sense' [Hodes] |
| 10017 | Truth in a model is more tractable than the general notion of truth [Hodes] |
| 10015 | Higher-order logic may be unintelligible, but it isn't set theory [Hodes] |
| 10018 | Truth is quite different in interpreted set theory and in the skeleton of its language [Hodes] |
| 10022 | Numerical terms can't really stand for quantifiers, because that would make them first-level [Hodes] |
| 10023 | Talk of mirror images is 'encoded fictions' about real facts [Hodes] |
| 10026 | Arithmetic must allow for the possibility of only a finite total of objects [Hodes] |