If Platonism holds, mathematical objects exist independently of our minds, and so the number two exists independently of our minds, but can there be multiple number 2's independent of our minds (e.g. one complex, and one real)?
Overall, is there a good place for some history/literature on this question? Has it been studied in detail?
The thing is that for Plato the number two cannot be confused with the symbol that represents the number. This symbol that you print on a paper or even that you think is not the number two, but an instatiation of that number, it participates in the nature of the number two.
When a mathematician construct the real numbers (as Dedekind cuts or equivalent classes) with some set theory as foundation, he is representing the real matematics with a symbolic system that is not the mathematical object in itself. Formally speaking, one could say that the complex numbers are a different set from the reals and with use a inclusion map to make an isomorphic embeding, and this is true in the symbolic sense, but for a platonist both numbers, the real number two and the complex number 2 are representations or participations in the entity number two.
A Platonist with respect to what?
If you're a platonist with respect to sets, the real number 2 and the complex number 2 are in fact different. For the reals, one proceeds via, say, Dedekind cuts, for the complex numbers we realize the underlying set as R x R, then proceed to define addition and multiplication. In fact, 2 as a natural, integer, rational, real, complex, etc... are all in fact different- although there is a natural (set)-embedding from each structure to its superstructures.
This may seem strange, but its a consequence of Platonism + using, say, ZFC as a foundation.
Some (say Beneceraff) find this strange. You'll want to think of structuralism for a different perspective on the matter.
EDIT:
(a): by platonism, I mean something along the lines of existence, abstraction, and independence holding for "the" mathematical ontology. This may not be what the historical Plato thought, see Landry for further discussion. Nonetheless, this is typically what is meant in the literature today.
(b): Of course, once can also be a platonist about structures (ante rem).
(c): for an alternate platonistic view that does not priviledge any one set theoretic universe, one can also consider Balagauers plenitudinuous platonism. It consciously draws on model theoretic concepts, so that both the real and complex number two "satisfy" being the number two. Thus, there can be multiple number twos that exist independently of us.
