We propose a logic of interactive proofs as a framework for an intuitionistic foundation for interactive computation, which we construct via an interactive analog of the Goedel-McKinsey-Tarski-Artemov definition of Intuitionistic Logic as embedded into a classical modal logic of proofs, and of the Curry-Howard isomorphism between intuitionistic proofs and typed programs. Our interactive proofs effectuate a persistent epistemic impact in their intended communities of peer reviewers that consists in the induction of the (propositional) knowledge of their proof goal by means of the (individual) knowledge of the proof with the interpreting reviewer. That is, interactive proofs effectuate a transfer of propositional knowledge (knowable facts) via the transmission of certain individual knowledge (knowable proofs) in multi-agent distributed systems. In other words, we as a community can have the formal common knowledge that a proof is that which if known to one of our peer members would induce the knowledge of its proof goal with that member. Last but not least, we prove non-trivial interactive computation as definable within our simply typed interactive Combinatory Logic to be nonetheless equipotent to non-interactive computation as defined by simply typed Combinatory Logic.