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I read that platonists believe there are abstract mathematical objects.

How do they think about formal systems? For example,

  1. Do they discriminate the set of natural numbers N on ZF and that on ZFC?

  2. Do they see the sets in a formal system as itself abstract mathematical objects, or as something gives expressions of the abstract objects?

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  • In a nutshell, for a platonist the (string of) symbol "one" refers to (it is the "name" of) an (abstract) object: the number 1. – Mauro ALLEGRANZA Jan 23 '16 at 10:01
  • What do you mean with "the sets in a formal system as itself abstract mathematical objects" ? – Mauro ALLEGRANZA Jan 23 '16 at 10:02
  • To Mauro ALLEGRANZA. I mean that when it is said that 'We call something which satisfies the following axioms "sets" ', do platonists call these objects "sets" in their meaning? – user14830 Jan 23 '16 at 10:28
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    For a "pure" platonist : yes, sets exists and are the objects referred to by the (only) true theory of sets. Of course, the current "proliferation" of set theories (see Alternative Axiomatic Set Theories) is a non trivial issue. – Mauro ALLEGRANZA Jan 23 '16 at 10:41
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I am currently reading Ian Hacking’s Why is there Philosophy of Mathematics at all, and it is mostly about the contemporary dabate platonism/ nominalism, so I would recommend it as a good place to look for an answer to this question. A crude copy-paste is given below.

Hacking asserts that it is Paul Bernays who introduced the modern idea of 'platonism' and further he writes about the different brands of platonism, noting (p228):

After Bernays had introduced the word 'platonism' to the philosophy of mathematics, it should have been clear that one should not speak of platonism, but about platonism restricted to some domain of objects, such as the class of integers, or any Zermelo-Fraenkel set, or whatever. Perhaps even something like the class of all ZF sets, in a van Neumann-Godel-Bernays set theory.

So a good point to remember is that according to him

Absolute platonism, asserting the existence of all definable mathematical objects and relations, is untenable. What remain are relative platonisms. The weakest interesting platonism described by Bernays asserts the existence of a totality of whole numbers...a twenty-first-century platonist will say: positive integers are abstract objects. Kronecker said that God created them. So he thought they exist. That's a species of platonism - about numbers. He was a very modest platonist, but a platonist all the same.

Hacking comments:

Such a thought seems never to have crossed Bernays' mind, for he thought of platonism in terms of totalities rather than 'abstract objects'.

So Bernays turns out to be a restricted platonist while

In Bernays' vocabulary Boolos is a cautious platonist. He has no problem about the totality of whole numbers, but he has many qualms about sets whose existence is proven within Zermelo-Fraenkel set theory with the axiom of choice.(229)

Perhaps there is something of an answer in this heap of quotes

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