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Full Idea
If we take mathematics at its word, there are too many mathematical objects for it to be plausible that they are all mental or physical objects.
Gist of Idea
There are too many mathematical objects for them all to be mental or physical
Source
Michael D. Resnik (Maths as a Science of Patterns [1997], One.1)
Book Ref
Resnik,Michael D.: 'Mathematics as a Science of Patterns' [OUP 1999], p.3
A Reaction
No one, of course, has ever claimed that they are, but this is a good starting point for assessing the ontology of mathematics. We are going to need 'rules', which can deduce the multitudinous mathematical objects from a small ontology.
6295 | There are too many mathematical objects for them all to be mental or physical [Resnik] |
6296 | Maths is pattern recognition and representation, and its truth and proofs are based on these [Resnik] |
6299 | Axioms are often affirmed simply because they produce results which have been accepted [Resnik] |
6300 | Mathematical constants and quantifiers only exist as locations within structures or patterns [Resnik] |
6301 | Congruence is the strongest relationship of patterns, equivalence comes next, and mutual occurrence is the weakest [Resnik] |
6302 | Structuralism must explain why a triangle is a whole, and not a random set of points [Resnik] |
6303 | Sets are positions in patterns [Resnik] |
6304 | Mathematical realism says that maths exists, is largely true, and is independent of proofs [Resnik] |