Unknowable Truths without Objects

I believe in mathematical objects, but I think the following appeal to them is dead wrong:

"The existence of mathematical objects is what allows there to be unknowable mathematical truths, whereas there are no unknowable logical or `conceptual' truths."

Corresponding to every unknowable AxFx statement in arithmetic, there's a purely modal statement, that's not ontologically commital, but would let you infer the arithmetical statement and hence must be equally unknowable, namely:

"It is impossible for there to be a machine on an infinite tape which a) acts in such and such such-and-such a physically specified way (here we have we list physical correlates of rules for some Turing machine program that checks every instance of the AxFx statement), and b) stops."