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Full Idea
The essential properties of mathematical entities seem to be relational, ...so we make no progress unless we can pick out some mathematical entities wihout presupposing other entities already picked out.
Gist of Idea
Since mathematical objects are essentially relational, they can't be picked out on their own
Source
Michael Jubien (Ontology and Mathematical Truth [1977], p.112)
Book Ref
'Philosophy of Mathematics: anthology', ed/tr. Jacquette,Dale [Blackwell 2002], p.112
A Reaction
[compressed] Jubien is a good critic of platonism. He has identified the problem with Frege's metaphor of a 'borehole', where we discover delightful new properties of numbers simply by reaching them.
Related Idea
Idea 9910 Number-as-objects works wholesale, but fails utterly object by object [Benacerraf]
| 9963 | If we all intuited mathematical objects, platonism would be agreed [Jubien] |
| 9962 | How can pure abstract entities give models to serve as interpretations? [Jubien] |
| 9964 | Since mathematical objects are essentially relational, they can't be picked out on their own [Jubien] |
| 9965 | There couldn't just be one number, such as 17 [Jubien] |
| 9966 | The subject-matter of (pure) mathematics is abstract structure [Jubien] |
| 9967 | 'Impure' sets have a concrete member, while 'pure' (abstract) sets do not [Jubien] |
| 9968 | A model is 'fundamental' if it contains only concrete entities [Jubien] |
| 9969 | The empty set is the purest abstract object [Jubien] |