Logic & Philosophy of Science Colloquium

In the first half of the nineteenth century, freedom became an important motif among foundational thinkers. Mathematicians were seen as being free to create mathematics to suit their own purposes, provided only that their creations were consistent and fruitful (i.e. they served the purpose for which they were created). In the late nineteenth and early twentieth centuries, Hilbert and other leading thinkers (e.g. Dedekind) transferred this idea to an axiomatic setting. These axiomatic creativists departed from the views of the earlier creativists by adding conditions that went beyond the traditional constraints of consistency and fruitfulness. One such constraint was a completeness condition, which Hilbert introduced as a kind of ideal or norm for creativist theorizing. This proposal is our focal concern as we consider the general question of fit between a completeness condition and the creativist outlook. Our general finding is that the fit is not very good.