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As far as I understand it the definitions are:

Non-eliminative structuralists believe that talk of structures is ontologically committed to the existence of abstract structures.

In re structuralism is the belief that structures are ontologically posterior to the structures that instantiate them. So that strucutures only exist in the systems that exemplify them.

Intuitively I would think that 'in re' structuralism is a non-eliminative theory, but I cannot find any confirmation of this in any literature.

Any insights would be appreciated.

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    What would count as non-eliminativist? I guess you could define your terms so that in re structuralism falls on either side. What hangs on this?
    – Schiphol
    Dec 18, 2019 at 17:22
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    I think you are right. IEP's Mathematical Structuralism states:"Another, perhaps ontologically cleaner, option is to reject the existence of structures, in any sense of “existence”... The view is sometimes dubbed eliminative structuralism". Shapiro's ante rem vs in re terminology derives from medieval scholastics where ante rem structuralism corresponded to platonism, in re structuralism to Aristotelian hylomorphism about universals, still a form of realism, and eliminative structuralism to outright nominalism.
    – Conifold
    Dec 19, 2019 at 0:47

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