I'm reading up on Hilbert, and wondering if there's actually anything fundamental to his distinction between finitary and infinitary mathematics. His system seems to be an attempt to avoid too much ontological commitment (though from what I understand, he is still ontologically committed to finitary numbers as abstract objects, though he would protest otherwise - correct me if I'm wrong). However, given that his plans famously failed, might it be better for us to just have platonism with ZFC - for us to accept that we in some way intuit the truth of the relevant axioms (as we must at some point in our reasoning regardless) and deny any real epistemological distinction between finitary and infinitary mathematics - i.e. our claims about both are just as true.
I get the feeling I'm missing something epistemologically troubling about infinitary mathematics, so if I am please tell me!