Is there a text that tries to formalize natural deduction and proofs in natural deduction, as purely mathematical objects? I know that in a Hilbert-style proof system, a proof is a certain kind of mathematical object, namely, a sequence of well-formed formulas. Is there some similar formalism for natural deduction?
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Have you come across Natural Deduction: A Proof-Theoretical Study by Dag Prawitz? That would be my first recommendation for a formal proof-theoretic study of natural deduction; its first chapter gives a very good definiton of proofs in natural deduction as formal objects.
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