 One of the technical advances of (1991a) and (1993d) was that they showed how a structuralist account of set theory was even possible. This part of the work was co-authored with John P. Burgess and A. P. Hazen. Given a large enough universe (i.e., that the cardinality of the mereological atoms is an  inaccessible cardinal),  and given plural quantification, we can say exactly what constraints a function must satisfy for it to do the work we want the singleton function to do. (By ‘the singleton function’ I mean the function that maps anything that has a singleton onto its singleton. Since proper classes don’t have singletons, and nor do fusions of sets and objects, this will be a partial function.) Given that, we can understand mathematical claims made in terms of sets/classes as quantifications over singleton functions. That is, we can understand any claim that would previous have used ‘the’ singleton function as a claim of the form for all s: …s…s…, where the terms s go where we would previously have referred to ‘the’ singleton function. It is provable that this translation won’t introduce any inconsistency into mathematics (since there are values for s), or any indeterminacy (since the embedded sentence …s…s… has the same truth value for any eligible value for s).