 Parsons (1990), Resnik (1997), and Shapiro (1997) contend that, in the case of mathematical theories, coherence suffices for the existence of the objects mentioned in those  theories.[17]  They do not offer an explicit criterion for distinguishing abstract and concrete objects. Nevertheless, their proposals implicitly draw the distinction; abstract objects are those objects that exist in virtue of the truth of certain modal claims. In particular, the existence of mathematical objects is “grounded in” pure modal truths. For example, numbers exist “in virtue of” the fact that there could have been an \(\omega\)-sequence of objects; sets exist because there might be entities that satisfy the axioms of one or another set theory, etc. Since these pure modal truths are necessary, this explains why pure abstract objects exist necessarily. It also explains a sense in which they are insubstantial: their existence is grounded in truths that do not (on the face of it) require the actual existence of anything at  all.[18] 