Mathematical Platonism, formally defined, is the view that (a) there exist abstract objects—objects that are wholly nonspatiotemporal, nonphysical, and nonmental and (b) there are true mathematical sentences that provide true descriptions of such objects.
So there exist abstract mathematical objects that these expressions refer to and quantify over.
Platonism is the view that there exist abstract (that is, non-spatial, non-temporal) objects.
Let anti-nominalism be the negation of nominalism, that is, the claim that there exist abstract objects.
Anti-nominalism says there exist abstract mathematical objects, whereas platonism adds Independence, which says that mathematical objects are independent of intelligent agents and their language, thought, and practices.
According to Balaguer, the so-called "story of mathematics" consists in the thesis that there actually exist abstract mathematical objects of the kinds that platonists have in mind, i.e., the kinds that our mathematical theories purport to be about.
And this is where fictionalism comes in: it grants the platonistic view of the truth conditions of mathematical sentences but still denies the platonist's ontological thesis that there exist abstract objects.
Platonism is the view that (a) there exist abstract mathematical objects (i.e., nonspatiotemporal mathematical objects), and (b) our mathematical sentences and theories provide true descriptions of such objects.
(2) If there exist any abstract objects, then they exist entirely outside of space-time.
But if they exist as abstract entities, then Benacerraf's epistemological problem reappears.
