Now, other philosophers have endorsed the idea that we possess a faculty of mathematical intuition, but Gödel's version of this view — and he seems to be alone in this — involves the idea that the mind is non-physical in some sense and that we are capable of forging contact with, and acquiring information from, non-physical mathematical objects.[16] This view has been almost universally rejected.
Thus, given that sheer postulation is (basically) enough in mathematics, mathematical objects have no epistemic burdens.
And indeed, when the nominalist claims that the truth-values of sentences of LM are fixed in a way which doesn't appeal to mathematical objects, it is precisely this sort of non-homophonic translation she has in mind.
Most of the debate with respect to this argument has concentrated on the particular application of the argument to the case of mathematical objects (for more on this debate see the entry on Platonism in metaphysics and Burgess and Rosen 1997, pp. 35 60).
It seems to have been Kepler who first introduced the idea, which was later to become a reigning principle in geometry, of continuous change of a mathematical object, in this case, of a geometric figure.
Mathematical objects and concepts are as objective as physical objects and properties.
Then it follows from Classical Semantics that many sentences of mathematics are ontologically committed to mathematical objects.
In his view, it is possible to explain successful applications of mathematics with no commitment to mathematical objects.
This does not prevent other mathematical objects from behaving like infinitesimals, and mathematical logicians of the 1920s and '30s actually showed how such objects could be constructed.
Although Benacerraf focuses on arithmetic, the objection naturally generalizes to most pure mathematical objects.
