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Full Idea
I take the view that (agreeing with Aristotle) mathematics only requires the notion of a potential infinity, ...and that mathematics is higher-order modal logic.
Gist of Idea
Mathematics is higher-order modal logic
Source
Harold Hodes (Logicism and Ontological Commits. of Arithmetic [1984])
Book Ref
-: 'Journal of Philosophy' [-], p.149
A Reaction
Modern 'modal' accounts of mathematics I take to be heirs of 'if-thenism', which seems to have been Russell's development of Frege's original logicism. I'm beginning to think it is right. But what is the subject-matter of arithmetic?
| 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] |