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bogus comments on What do professional philosophers believe, and why? - Less Wrong Discussion
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I think it might be best if you read the above-linked SEP article and some of the related pieces. But the short form.
C: We should believe in the existence of the abstract objects in our best scientific theories.
One and two seem uncontroversial. 3 can certainly be quibbled with and I spent a few years as a nominalist trying to think of ways to paraphrase out or find reasons to ignore the abstract objects among science's ontic commitments. Lots of people have done this and have occasionally demonstrated a bit of success. A guy named Hartry Field wrote a pretty cool book in which he axiomatizes Newtonian mechanics without reference to numbers or functions. But he was still incredibly far away from getting rid of abstract objects all together (lots of second order logic) and the resulting theory is totally unwieldy. At some point, personally, I just stopped seeing any reason to deny the existence of abstract objects. Letting them exists costs me nothing. It doesn't lead to false beliefs and requires far less philosophizing.
The concrete-abstract distinction still gets debated but a good first approximation is that concrete objects can be part of causal chains and are spatio-temporal while abstract objects are not. As for common-or-garden numbers: I see no reason to exclude them.
Doesn't this imply that equivalent scientific theories may have quite different implications wrt. what abstract objects exist, depending on how exactly they are formulated (i.e. the extent to which they rely on quantifying over variables)?
Also, given the context, it's not clear that rejecting theories which rely on second-order and higher-order logics makes sense. The usual justification for dismissing higher-order logics is that you can always translate such theories to first-order logic, and doing so is a way of "staying honest" wrt. their expressiveness. But any such translation is going to affect how variables are quantified over in the theory, hence what 'commitments' are made.
I'm not sure what you mean by "equivalent" here. If you mean "makes the same predictions" then yes-- but that isn't really an interesting fact. There are empirically equivalent theories that quantify over different concrete objects too. Usually we can and do adjudicate between empirically equivalent theories using additional criteria: generality, parsimony, ease of calculation etc.