 Let us take stock. In response to worries that infinities in mathematics are suspect (section   2), rigorous mathematical theories of infinity have been developed (this section). But one might worry that, even if we can talk with mathematical rigor about infinities, they do not correspond to or apply to anything in the real world (as we think finite quantities do, however they do). Infinities might just be castles in the sky. Furthermore, one might suspect that we can, by some further mathematical developments, remove any reference to infinities in any practically important mathematics. The following section places this dialectic in the context of general questions about mathematical ontology, canvassing some important historical attempts to expunge the infinite from mathematics. It then explains the very difficult—perhaps insurmountable—challenges that any such attempt faces.