I’m unsure if open sets (or whatever generalization) are a good formal underpinning of what we call concepts, but I’m in agreement that there seems needed at least a careful reconsideration of intuitions one takes for granted when working with a concept, when you’re actually working with a negation-of-concept. And “believing in” might be one of those things that you can’t really do with negation-of-concepts.
Also, I think a typo: you said “logical complement”, I’m imagining you meant “set-theoretic complement”. (This seems important to point out since in topological semantics for intuitionistic logic, the “logical complement” is in fact defined to be the interior of the set-theoretic complement, which guarantees an open.)
I should have written “algebraic complement”, which becomes logical negation or set-theoretic complement depending on the model of the theory.
Anyway, my intuition on why open sets are an interesting model for concepts is this: “I know when I see it” seems to describe a lot of the way we think about concepts. Often we don’t have a precise definition that could argue all the edge case, but we pretty much have a strong intuition when a concept does apply. This is what happens to recursively enumerable sets: if a number belongs to a R.E. set, you will find out, but if it doesn’t, you need to wait an infinite amount of time. Systems that take seriously the idea that confirmation of truth is easy falls under the banner of “geometric logic”, whose algebraic model are frames, and topologies are just frames of subsets. So I see the relation between “facts” and “concepts” a little bit like the relation between “points” and “open sets”, but more in a “internal language of a topos” or “pointless topology” fashion: we don’t have access to points per se, only to open sets, and we imagine that points are infinite chains of ever precise open sets
I’m unsure if open sets (or whatever generalization) are a good formal underpinning of what we call concepts, but I’m in agreement that there seems needed at least a careful reconsideration of intuitions one takes for granted when working with a concept, when you’re actually working with a negation-of-concept. And “believing in” might be one of those things that you can’t really do with negation-of-concepts.
Also, I think a typo: you said “logical complement”, I’m imagining you meant “set-theoretic complement”. (This seems important to point out since in topological semantics for intuitionistic logic, the “logical complement” is in fact defined to be the interior of the set-theoretic complement, which guarantees an open.)
I should have written “algebraic complement”, which becomes logical negation or set-theoretic complement depending on the model of the theory.
Anyway, my intuition on why open sets are an interesting model for concepts is this: “I know when I see it” seems to describe a lot of the way we think about concepts. Often we don’t have a precise definition that could argue all the edge case, but we pretty much have a strong intuition when a concept does apply. This is what happens to recursively enumerable sets: if a number belongs to a R.E. set, you will find out, but if it doesn’t, you need to wait an infinite amount of time. Systems that take seriously the idea that confirmation of truth is easy falls under the banner of “geometric logic”, whose algebraic model are frames, and topologies are just frames of subsets. So I see the relation between “facts” and “concepts” a little bit like the relation between “points” and “open sets”, but more in a “internal language of a topos” or “pointless topology” fashion: we don’t have access to points per se, only to open sets, and we imagine that points are infinite chains of ever precise open sets