Abstract

In his Philosophical Investigations, Wittgenstein said that “turning our whole investigation around” is the only way to shake the illusion of a “preconceived idea of crystalline purity.” Commentators have built sweeping descriptions of Wittgenstein’s general approach to philosophy out of their interpretations of this slogan. For his own part, Wittgenstein specified a particular subject. “For,” he wrote, “the crystalline purity of logic was…not a result of investigation: it was a requirement.” In asking what Wittgenstein could have meant by reversing the direction of the investigation of logic and of what he called its apparent “hardness,” I propose to follow his own advice: to look around before thinking. Two episodes from logic’s history serve as illustrative examples of concepts being dislodged from their original position as theorems or discoveries and relocated as starting points of an investigation (and vice-versa). The details of the cases showcase how “turning our whole investigation around” has proven its practical utility in the growth and development of logic in the last century. Further, it suggests that Wittgenstein might have intended something at once humbler and, for logic, more important than a grand reconceptualization of the nature of necessity: the science of logic itself relies on a continuous inversion of discoveries and requirements.