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Full Idea
The mathematically significant properties and relations of natural numbers arise from the successor function that orders them; the natural numbers are identified simply as the objects that answer to this basic function.
Gist of Idea
Numbers are identified by their main properties and relations, involving the successor function
Source
Fraser MacBride (Structuralism Reconsidered [2007], §1)
Book Ref
'Oxf Handbk of Philosophy of Maths and Logic', ed/tr. Shapiro,Stewart [OUP 2007], p.563
A Reaction
So Julius Caesar would be a number if he was the successor of Pompey the Great? I would have thought that counting should be mentioned - cardinality as well as ordinality. Presumably Peano's Axioms are being referred to.
| 17798 | Cantor presented the totality of natural numbers as finite, not infinite [Cantor, by Mayberry] |
| 17423 | The essence of natural numbers must reflect all the functions they perform [Sicha] |
| 9965 | There couldn't just be one number, such as 17 [Jubien] |
| 13676 | Only higher-order languages can specify that 0,1,2,... are all the natural numbers that there are [Shapiro] |
| 8923 | Numbers are identified by their main properties and relations, involving the successor function [MacBride] |
| 17877 | The number series is primitive, not the result of some set theoretic axioms [Almog] |