My personal take here is linked to my broader view of the world. When we invent concepts, we can invent them only for ourselves: we can then share them, but that necessitates a translation of these abstract concepts into some representation. I see no reason to believe that the concepts, once transferred, are identical; as such, I think we should generally not expect questions like "is this an X?" or "is this part of this X?" (equivalently: "what is the boundary of this X?") to have answers we will all agree on. For me, numbers do not fall into a particularly different category than other objects; they only happen in our head, but then, so does every other form of categorisation. Hence, I am not inclined to say that mathematical objects have any more independent existence than any other notion.
Moreover, since anything we say about the world is necessarily said about some approximation of it that is currently available to us, I expect the answers to be continuous. As such, any mathematics that cannot handle non-binary truth values falls short of capturing how I see the world. More practically, it also falls short of capturing how we discover mathematics itself, which is a process of proofs that we know is not symmetric (that is, "provable" and "unproveable" are very different.)
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