Mathochist, it remains completely unclear to me what genuine alternative Structuralism offers as you’ve presented it. As with any abstract concept, there are various alternatives that have been neatly wrapped up in the so-called “Problem of Universals”, about which three general positions have been formulated: realism, nominalism, and conceptualism. Personally, I don’t think conceptualism is a genuine third option to what is otherwise a binary state (do universals exist or dont they?), but Ive never seen a very good defense of it, either, so I’ll admit that I might be missing something there.
For mathematics in particular, as seems to be common agreement, two positions have seemed to surface, which are platonism and formalism, which themselves have various sub-categories and nuances which aren’t necessarily worth elaborating on.
You have suggested several times that the platonic response is that “2 exists” and that’s that, and that structuralism offers an alternative by positing structures and not numbers per se, but I think it sells the position of platonism short to suggest that “2 exists” is the end of the platonist position. I almost feel like I’ve said, “Pyramids exist” and you respond, “All I see are squares and triangles, but I have this structure here that I’ll call a ‘shape holder’ where a pyramid-like thing fits…” :smack:
When we come to the rather definitive question of, “Do these mathematical structures exist”, you seem to suggest that we are stepping into a realist/anti-realist portion, and apparently structuralism finds a home for both positions. That being the case, it is not an alternative to either, but in fact a meta-theory, like relativism. Epistemological relativism, for example, makes no comment about whether objective knowledge exists, and so is just as happy in objectivism as it is in subjectivism. But that is simply because relativism is not an alternative to those ideas. Generally speaking, it seems that structuralism is a definitional/semantic framework which couches itself in some metaphysical terms but, upon inspection, takes no stance on them (i.e., it does not suggest that mathematical structures exist). But if it doesn’t answer that question, then it doesn’t resolve any issue about soundness of the reification of abstract objects like mathematical entities any more than some set theory resolved the issue of platonism by defining all natural numbers through zero and a successor function (is “2” real, or is the structure real, or is zero and a successor function real, etc, etc).
That is what I am not getting from you. How it is an alternative. There is nothing about platonism that demands “2” is some kind of object and is not the amalgam or consequence of other objects, though it is convenient to avoid more complicated maths and deal with arithmetic when discussing platonism. Nothing about platonism requires that “2” be a singular object, and nothing about platonism forbids that it be a composite one.