We use impressions to get an idea of what the question is about but the sense that the proof must be true is not based on any particular impression. It is a sense that, for all those triangles we have never seen, it must be so.
Ok I think I understand what you mean. 'That sense' is some kind of intuition that probably has some evolutionary advantage. It is in a similar sort of sense to the way in which an animal has some intuition to avoid certain predators as you mentioned. So, if Hume would agree that that sort of 'sense' is knowledge then I think it is fair to assume that he was wrong as that impression is not directly had by that individual.
The problem I have with our brains containing patterns for every impression is that there are uncountably many impressions one could have. The experience of viewing an object is different at every moment of time and from every direction. I find it hard to imagine that all that information is already there. Why would it just not be the case that instead of it already being there, there are essentially uncountable many ways in which the brain's matter can be rearranged to store those impressions?
I see "knowing what they mean" in the same sort of way as I do with the brain containing patterns of impressions. It is not that we know what every impression means, but that there are some general rules which our brain uses to processes said information. One such rule might be that our brain treats objects as discrete entities. This isn't because we know what each entity means but because treating them as such has been useful.
The brain then learns from others that these discrete entities have labels that can be attached to convey information to others. I don't see how this is "knowing what those objects mean" it is more our brain has a certain way of processing incoming information. And this way of processing doesn't reflect how the world actually is, it reflects how to act in accordance with survival.
I'm sure I will have plenty more thoughts as I read though the rest of the posts, but I would ask what material you recommend reading on this. Who do you think are the people who most influenced your thinking on this outside of Leibniz? Before I got ME/CFS I wasn't studying philosophy and only recently started exploring these ideas.