Abstract: Develops a variant of the strategy of the preceding chapter, drawing on ideas of so-called mathematical structuralism, according to which statements about ‘the’ system of real numbers are really statements about all systems of objects and relations of the right kind. A structuralist reading of standard mathematics is no more nominalistically acceptable than a straightforward reading because there generally are not any systems objects and relations of the right kind among the concrete. But the structuralist reading can be modified to make it nominalistically acceptable by bringing in modal logic, and combining it with Leniewski's mereology and Boolos's plural quantification or both. The end result is a kind of partial vindication of logicism, in that all relevant mathematical assumptions are reduced to assumptions of a grand logic, including modality, mereology, and plural quantification.

Keywords: Boolos, Leniewski, logicism, mereology, modality, modal logic, plural quantification, structuralism,