Inconsistency is in the Eye of the Mathematician

Consistency appears to be an essential feature of mainstream mathematics: practices deemed inconsistent are either rejected as incorrect, or treated as outsiders. In this paper I analyze the role of consistency in mathematics by looking at deviations from consistency: I argue that attempts to ground the distinction between mainstream and inconsistent mathematics cannot succeed without incorporating some kind of practice-level commitment to inconsistency, and propose a notion of inconsistent practice which clarifies, by way of contrast, how consistency functions in the mainstream. This exemplifies a general methodology for better understanding core features of mainstream practices through contrast with deviating practices.