 The idea of implicit definition is not in conflict, then, with the traditional account. Where conflict arises is in the philosophical applications of the idea. The failure of strict reductionist programs of the late-nineteenth and early-twentieth century prompted philosophers to explore looser kinds of reductionism. For instance, Frege’s definition of number proved to be inconsistent, and thus incapable of sustaining the logicist thesis that the principles of arithmetic are analytic. It turns out, however, that the principles of arithmetic can be derived without Frege’s definition. All that is needed is one consequence of it, namely, Hume’s Principle: