Reflections on the role of statements on statements in mathematics

Mathematicians reason about abstract objects, or at least they seem to do so. In the course of doing so, they routinely make statements on mathematical statements. In fact, every inference, such as “From equations (5.17) and (6.2) one obtains the following equation.”, is a statement on mathematical statements. Other examples of statements on mathematical statements are: “With the obvious modifications of the proof of Theorem A, one gets a proof of Theorem B.”, “This proof is really nice.”, “A statement in the language of rings is true in every algebraically closed field of characteristic zero if and only if there are infinitely many prime numbers p such that the statement is true in some algebraically closed field of characteristic p.”, “All of our proofs can be formulated in ZFC; for convenience of the reader, as usual we present them informally.” By the above examples, one notices immediately that there are fundamentally different kinds of such statements. An important special case of statements on statements are statements on the existence of proofs and in particular formal proofs, such as the last statements above. The last statement is the kind of statement we are most concerned with in this note. What kind of statement is actually made here? Shall the formal proof “in ZFC” be a mathematical object in the same vain as, say, finite geometries or groups? If so, is the claim then a claim in mathematics? Shall the claim not rather be a claim on mathematics? But what shall then the meaning of the claim be? Clearly, we would want that a claim that there is a proof, in this case a formal proof, be of a different, “more real” nature than a mere existence claim in the usual set-theoretic mathematics. But how is it possible to give such a “more real” interpretation if the alleged formal proof is not written down, in all likelihood will not be written down and maybe also cannot be