Which is the more fundamental concept: “knowing how” or “knowing that”? Is it possible to “know” that 37×43=1591, yet not “know” the prime factors of 1591? If so, what is the thing that you know in the one case but not in the other?
Do you know an infinite number of propositions?
Do statements about computational processes blur the distinction between analytic and synthetic knowledge? Is the statement that a quantum computer (say) will output a particular answer “more analytic” or “more synthetic”? Does the answer change if we instead consider the distinction between a priori and a posteriori knowledge?
Does every mathematical proof depend for its validity on implicit physical assumptions (for example, that a demon isn’t changing what’s written on the page as you read it), as Deutsch argues? Is the situation different for the modern “proof protocols” studied in theoretical computer science and cryptography (zero-knowledge proofs, quantum proofs, etc.), than for traditional proofs? Or do these proof protocols merely make more vivid a dependence on physics that was already present in traditional proofs?