Forcing in Set-Theoretic Practice

This chapter examines the philosophical significance of the forcing method through the lens of contemporary set-theoretic practice. Building on the practice-based approach pioneered by Penelope Maddy, it situates forcing within the broader universe-multiverse debate and critically compares the universe view defended by Maddy with Joel D. Hamkins' multiverse pluralism. Although both accounts appeal to mathematical practice, they draw divergent conclusions, raising a methodological puzzle about how practice is to be interpreted. To address this puzzle, the chapter introduces a scope guideline for practice-based philosophy of set theory, according to which philosophical analyses must respect and explain the full diversity of set-theoretic practices rather than privileging a narrow subset. The guideline is applied in a detailed case study of forcing, drawing on qualitative interview data from professional set theorists representing both absolutist and pluralist orientations. The findings show that forcing is widely valued across the community but integrated into distinct research aims. The chapter argues that forcing is largely philosophically neutral, though not uniformly so, and that its methodological versatility reflects the coexistence of multiple epistemic ideals within set theory. More generally, the analysis highlights how practice-based philosophy must accommodate internal diversity if it is to yield credible philosophical conclusions about mathematics.