Peirce’s c.1902 taxonomy of the sciences is briefly described, stressing its anti-foundationalist formulation. That taxonomy identified philosophy as a science of discovery and divided it into several empirical, albeit very general, inquiries. Of these, phaneroscopy (phenomenology) is the most basic. Phaneroscopy depends not only on empiricism’s expansion (Chapter 7) but also on a vocabulary of phaneroscopic description, free of metaphysical assumptions, drawn from the algebra of relations. The resulting three phaneroscopic categories are here developed systematically. Peirce’s 1903 reformulation of pragmatism (Chapter 4) mandated a phaneroscopic account of the meaning of modal and metaphysical ideas – of possibility, actuality, and, especially, law – by which to establish the meaningfulness of modal realism. This implies that some forms of lawfulness are directly perceived. But direct (i.e., non-inferential) perception of lawfulness does not prove law’s reality, which must remain a hypothesis supported, never proven, by common experience and especially by the progress of scientific inquiry.

This chapter discusses further types of multiverse hypotheses: the branching worlds of Everettian quantum theory, the totality of possible worlds according to David Lewis’s modal realism, and the totality of mathematical structures, which – according to Max Tegmark – are all physically realized. I argue that suggested solutions to the probability problem in Everettian quantum theory are unconvincing. A recent proposal by Sebens and Carroll to derive the Born rule in the Everettian framework using considerations about self-locating belief turns out to suffer from a circularity problem because the central principle on which the derivation is based has no independent motivation besides, possibly, appeal to the Born rule itself. I further argue that it is not possible to coherently, in full seriousness, believe David Lewis's modal realism. Nor is it possible to coherently believe Max Tegmark's thesis that there is a multiverse of all mathematical structures in which they are physically realized. Tegmark's claim that our universe is itself a mathematical universe hypothesis is found to be incoherent.