Hamkins’ multiverse view is a prominent position on the nature of set theory. It is posited against the universe view and proposed as a philosophical theory explaining current set-theoretic practice. This paper confronts the multiverse view with the results of an interview study investigating current set-theoretic practice. The study reveals a heterogeneity of set-theoretic research practices. The multiverse view is found to align well with pluralist research practices but not with absolutist practices. The generalisation claim of the multiverse view fails (...) because of this heterogeneity; only a reduced interpretation of the multiverse view might hold generally. Furthermore, Hamkins’ prediction that the community as a whole will not adopt axioms deciding the continuum hypothesis is found probably true, but for different reasons: although part of the community has already accepted axioms beyond ZFC, another part of the community will withdraw from doing so. In conclusion, I argue that Hamkins’ multiverse view is best interpreted as a valuable perspective on pluralist set-theoretic practices. (shrink)

This paper analyses the hidden use of new axioms in set-theoretic practice with a focus on large cardinal axioms and presents a general overview of set-theoretic practices using large cardinal axioms. The hidden use of a new axiom provides extrinsic reasons in support of this axiom via the idea of verifiable consequences, which is especially relevant for set-theoretic practitioners with an absolutist view. Besides that, the hidden use has pragmatic significance for further important sub-groups of the set-theoretic community---set-theoretic practitioners with (...) a pluralist view and set-theoretic practitioners who aim for ZFC-proofs. By describing this, the paper gives a more complete picture of new axioms in set-theoretic practice. These observations, for instance, show that set-theoretic practitioners interested in ZFC-proofs use tools that go beyond ZFC. The analysis is based on empirical data that was collected in an extensive interview study with set-theoretic practitioners. (shrink)

Frege’s full comprehension scheme, formulated in "Grundgesetze der Arithmetik", was central to his logicist project. In its unrestricted form, however, it led directly to Russell’s paradox and was therefore rejected as a foundation for mathematics. This chapter revisits Frege’s comprehension principle from the perspective of modern set theory. Drawing on historical analysis and contemporary mathematical practice, I argue that although full comprehension cannot be sustained as a formal axiom, an informal version continues to guide set-theoretic reasoning. By examining the blurred (...) boundary between sets and proper classes in current mathematical work, I suggest that Frege’s original idea survives as an informal working principle, applied competently by practitioners without generating contradictions. (shrink)

English translation of Kant's Beantwortung der Frage: Was ist Aufklärung? (Königsberg in Prussia, 30 September 1784).

There are different narratives on mathematics as part of our world, some of which are more appropriate than others. Such narratives might be of the form ‘Mathematics is useful’, ‘Mathematics is beautiful’, or ‘Mathematicians aim at theorem-credit’. These narratives play a crucial role in mathematics education and in society as they are influencing people’s willingness to engage with the subject or the way they interpret mathematical results in relation to real-world questions; the latter yielding important normative considerations. Our strategy is (...) to frame current narratives of mathematics from a virtue-theoretic perspective. We identify the practice of mathematizing, put forward by Freudenthal’s ‘Realistic mathematics education’, as virtuous and use it to evaluate different narratives. We show that this can help to render the narratives more adequately, and to provide implications for societal organization. (shrink)

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. (shrink)

There can sometimes be weariness surrounding coordinated efforts to induce inclusion. Detractors of certain such efforts may cite coercion or a loss of autonomy as a concern. In their book Nudge, Thaler and Sunstein presented the idea of the “nudge.” Nudges change the context in which a decision is made to increase the likelihood that a certain choice is made by the decision maker. However, nudges are designed to be easy to avoid. Thaler and Sunstein laid out criteria such that (...) nudges can remain ethical. In this paper, I show that certain nudges that greatly increased inclusion in their respective communities do follow these criteria. In fact, subjected even to the relatively tighter constraints imposed by a prominent critic of Thaler and Sunstein’s philosophy, Wilkinson, these nudges remain unscathed. Thus, inclusion-promoting nudges are not necessarily unethical from the standpoint of manipulation or autonomy. (shrink)