| Since the late1960's, there has been widespread agreement among both philosophers and mathematicians that the only conception of a set is that of the iterative conception as embodied by Zermelo-Fraenkel (ZF) set theory and its related systems. In contrast, W.V. Quine's set theory with a universal set, New Foundations (NF), has been widely neglected if not outright dismissed as a going option for a theory of sets. I argue that there are no grounds for such dismissals and that many of the reasons offered in favor of ZF can also be offered for NF, and that these reasons are largely pragmatic in justification. From this I draw two main conclusions. The first, a historical point about Quine, is that much of his naturalistic and broadly pragmatic outlook, as exemplified in his major philosophical work Word and Object, emerged from his early engagement with Russell's foundational work in mathematics. As such, we can see Quine's philosophical work as intimately intertwined with his early technical endeavors. The second, a more general philosophical claim, is that mathematics in its foundations develops in ways much more akin to the natural sciences and as such is not the paradigm of a priori knowledge philosophers have often thought it to be. |
