An idea related to this helped me get a grasp on writing proofs when I was first struggling to teach myself mathematics. The idea was roughly: A particular fo

An idea related to this helped me get a grasp on writing proofs when I was first struggling to teach myself mathematics. The idea was roughly: A particular formalized proof (or lemma) is one of many ways of expressing/representing a more general concept that grounds some mathematical idea. It seems to mirror the way you could have the general concept of "having gone to the store and purchased bananas" in your head but express it many different ways through natural language. If humans were doing mathematics by thinking purely in terms of the theorem space of some particular formal system, then 'collapses' should be expected after minor errors: if you take one wrong turn, your error should be compounded in any further progress. But that does not appear to be how we do mathematics. Instead we're operating in a more general space of concepts, and we do not do math by proceeding linearly in our thoughts from the beginning of a proof to the final conclusion--we assemble the conception piecemeal until we get some feel for the sufficient overall integrity/coherence of a general conception, and then secure it by building up a kind of formal carapace around it. But if part of the proof/carapace is wrong, it's like a malformed section of armor that needs to be replaced: the whole structure isn't going to collapse because of it (not that that couldn't happen). Thinking about things that way gave me a better view (I think) on how to use and think about formalizations when doing mathematics.