Kleene the Platonist?

More from Kleene. I don't know that I've read enough mathematical epistemology, nor do I actually know that much about Kleene, but I find it interesting that he takes this digression in a book about formal systems to plead for meaning.

But in the case of arithmetic and analysis, theories culminating in set theory, mathematicians prior to the current epoch of criticism generally supposed that they were dealing with systems of objects, set up genetically, by definitions purporting to establish their structure completely. The theorems were thought of as expressing truths about these systems, rather than as propositions applying hypothetically to whatever systems of objects (if any) satisfy the axioms. But then how could contradictions have arisen in these subjects, unless there is some defect in the logic, some error in the methods of constructing and reasoning about mathematical objects, which we had hitherto trusted?

To say that now these subjects should instead be established on an axiomatic basis does not of itself dispose of the problem. After axiomatization, there must still be some level at which we have truth and falsity. If the axiomatics is informal, the axioms must be true. If the axiomatics is formal, at least we must believe that the theorems do follow from the axioms; and also there must be some relationship between these results and some actuality outside the axiomatic theory, if the mathematicians' activity is not to reduce to nonsense. The formally axiomatized propositions of mathematics cannot constitute the whole of mathematics; there must also be an intuitively understood mathematics. If we must give up our former belief that it comprises all of arithmetic, analysis and set theory, we shall not be wholly satisfied unless we learn wherein that belief was mistaken, and where now instead to draw a line of separation.