 Let us first look again at non-eliminative versions of structuralism. Some positions falling under that label introduce abstract structures by means of a basic theory. This includes Shapiro’s structure theory, but also, e.g., Nodelman &amp; Zalta’s object theory and Leitgeb’s adaptation of graph theory. All of them are forms of ante rem structuralism, but significantly different ones. In addition, there are forms of non-eliminative structuralism that are based on abstraction principles instead, which we called “abstractionist forms of structuralism” above. We distinguished two variants of those, the Russellian and the Dedekindian variant, which involve different kinds of structures as the outcome of the abstraction process.  (If we reconstruct such abstraction as mathematical operations or functions, their arguments are the same but their values are different.)  And as this shows, there are abstractionist and non-abstractionist versions of non-eliminative structuralism.