I don’t think 2 is answered even if you say that the mathematical objects are themselves real. Consider a geometry that labels “true” everything that follows from its axioms. If this geometry is consistent, then we want to say that it is true, which implies that everything it labels as “true”, is. And the axioms themselves follow from the axioms, so the mathematical system says that they’re true. But you can also have another valid mathematical system, where one of those axioms is negated. This is a problem because it implies that something can be both true and not true.
Because of this, the sense in which mathematical propositions can be true can’t be the same sense in which “snow is white” can be true, even if the objects themselves are real. We have to be equivocating somewhere on “truth”.
It’s easy to overcome that simply by being a bit more precise—you are saying that such and such a proposition is true in geometry X. Meaning that the axioms of geometry X genuinely do imply the proposition. That this proposition may not be true in geometry Y has nothing to do with it.
It is a different sense of true in that it isn’t necessarily related to sensory experience—only to the interrelationships of ideas.
You are tacitly assuming that Platonists have to hold that what is formally true (proveable, derivable from axioms) is
actuallty true. But a significant part of the content of Platonism is that mathematical statements are only really
true if they correspond to the organisation of Plato’s heaven. Platonists can say, “I know you proved that, but
it isn’t actually true”. So there are indeed different notions of truth at play here.
Which is not to defend Platonism. The notion of a “real truth” which can’t be publically assessed or agreed upon in the way that formal proof can be is quite problematical.
I don’t think 2 is answered even if you say that the mathematical objects are themselves real. Consider a geometry that labels “true” everything that follows from its axioms. If this geometry is consistent, then we want to say that it is true, which implies that everything it labels as “true”, is. And the axioms themselves follow from the axioms, so the mathematical system says that they’re true. But you can also have another valid mathematical system, where one of those axioms is negated. This is a problem because it implies that something can be both true and not true.
Because of this, the sense in which mathematical propositions can be true can’t be the same sense in which “snow is white” can be true, even if the objects themselves are real. We have to be equivocating somewhere on “truth”.
It’s easy to overcome that simply by being a bit more precise—you are saying that such and such a proposition is true in geometry X. Meaning that the axioms of geometry X genuinely do imply the proposition. That this proposition may not be true in geometry Y has nothing to do with it.
It is a different sense of true in that it isn’t necessarily related to sensory experience—only to the interrelationships of ideas.
You are tacitly assuming that Platonists have to hold that what is formally true (proveable, derivable from axioms) is actuallty true. But a significant part of the content of Platonism is that mathematical statements are only really true if they correspond to the organisation of Plato’s heaven. Platonists can say, “I know you proved that, but it isn’t actually true”. So there are indeed different notions of truth at play here.
Which is not to defend Platonism. The notion of a “real truth” which can’t be publically assessed or agreed upon in the way that formal proof can be is quite problematical.