It depends how you define intuition. Maybe we could define intuition as your personal "model" for making guesses on how things work. Let's claim that writing a

It depends how you define intuition. Maybe we could define intuition as your personal "model" for making guesses on how things work. Let's claim that writing a proof is similar to writing a program. You start from a state, your premise, and have your assumptions, input, which you process through a series of steps, maybe taking cases in between (branching) and return an output (hopefully True). Then considering this correspondence, it will be fine to say that you, as a programmer, can guess the output/behavior/semantics of any program without running it. Similarly, a mathematician would be able to reach the conclusion that the theorem holds, without going through all the details of the proof. Sure, in most cases you can, and as you get more and more experience and knowledge as a programmer you can say you improve your guesses. But this ends up backfiring with a good probability, when our model skipped a step, e.g. the interaction of two concurrent data structures. Thus I don't think it is right to say "as you are able to know all axioms and rules of inference, you should be able to achieve perfect intuition". You can know the fundamentals of each layer on your stack and you would find it insane if one asked you to guess the output of any non-trivial terminating program. Note also that the way of inference (logical consequences) can and does change (e.g. ZF vs ZFC). () I may be misusing the word model here -- it is usually defined (at least to things I work with) as the interpretation of a theory that satisfies all theorems (inferences). () constructive proofs yes. No flame wars please :)