Kati Kish Bar-On

This book provides a fresh perspective on how mathematicians interpreted and responded to Dutch mathematician L.E.J. Brouwer's ideas. It offers an original outlook on the dynamics between mathematicians, their commitments to their working frameworks, and the establishment of scientific norms. Expanding on the collective scientific work literature, it prompts a dialogue on applying insights from social epistemology to mathematics, exploring whether such an approach can offer new insights into individual and communal responses to Brouwer's intuitionistic pursuit. As a result, the book is of great value to those interested in STS studies, history and philosophy of science and sociology of science, delving into topics like the social construction of knowledge, epistemic cultures, and disagreements and trust in communities. Amid ongoing discussions about the marginalization and legitimization of knowledge, this book arrives just at the right time to offer a thorough exploration of the reasons behind the marginalization of a specific school in the history of mathematics.

Polarization poses a critical threat to the stability of nations around the world, as it impacts climate change, populism, democracy, and global health. This perspective examines the conceptual understanding, measurement challenges, and potential interventions for polarization. Our analysis highlights the distinction and interactions between the individual and collective levels of polarization, conceptually, methodologically, and in terms of interventions. We conclude by pointing out future directions for understanding polarization and highlighting the interrelations between polarization and other social phenomena.

Brouwer's philosophy of mathematics is usually regarded as an intra-subjective, even solipsistic approach, an approach that also underlies his mathematical intuitionism, as he strived to create a mathematics that develops out of something inner and a-linguistic. Thus, points of connection between Brouwer's mathematical views and his views about and the social world seem improbable and are rarely mentioned in the literature. The current paper aims to challenge and change that. The paper employs a socially oriented prism to examine Brouwer's views on the construction, use, and practice of mathematics. It focuses on Brouwer's views on language, his social interactions, and the importance of group context as they appear in the significs dialogues. It does so by exploring the establishment and dissolution of the significs movement, focusing on Gerrit Mannoury's influence and relationship with Brouwer and analyzing several fragments from the significs dialogues while emphasizing the role Brouwer ascribed to groups in forming and sharing new ideas. The paper concludes by raising two questions that challenge common historical and philosophical readings of intuitionism.

Society is a composite of interacting people and groups. These groups play a significant role in maintaining social status, establishing group identity and social identity, and enforcing norms. As such, groups are essential for understanding human behavior. Nevertheless, the study of groups in everyday group life yields many diverse and sometimes contradicting theories of group behavior, and researchers tend to agree that we have yet to understand the emergence of groups out of aggregates of individuals. The current paper aims to shed new light on the convoluted interrelation between groups and individuals by focusing on individuals’ social identities and group categorization. It does so by exploring the dynamic nature of the self and its implications on identity and group membership, and introducing a framework recognizing the fluidity of groups and group categorization. Incorporating historical insights with contemporary theories, this paper argues for a flexible understanding of group dynamics that surpasses rigid in-group and out-group classifications, proposing instead that group affiliations exist along a continuum that reflects the ever-changing social landscape.

Brouwer’s intuitionism was a far-reaching attempt to reform the foundations of mathematics. While the mathematical community was reluctant to accept Brouwer’s work, its response to later-developed brands of intuitionism, such as those presented by Hermann Weyl and Arend Heyting, was different. The paper accounts for this difference by analyzing the intuitionistic versions of Brouwer, Weyl, and Heyting in light of a two-tiered model of the body and image of mathematical knowledge. Such a perspective provides a richer account of each story and points to a possible connection between the community’s reaction and the changes each mathematician had proposed.

Heyes presents a compelling account of how cultural evolutionary processes shape and create “rules,” or norms, of social behavior. She suggested that normativity depends on implicit, genetically inherited, domain-general processes and explicit, culturally inherited, domain-specific processes. Her approach challenges the nativist point of view and provides supporting evidence that shows how social interactions are responsible for creating mental processes that assist in understanding and behaving according to rules or norms. We agree. In our commentary, we suggest that it is not only that mental processes for grasping norms are recreated in each generation but also that social interactions shape the kinds of social groups that are recognized (for a more extensive discussion, see Kish Bar-On & Lamm, 2023). We highlight evidence showing that accounts of norm psychology thus require a richer notion of human groups.

Brouwer’s intuitionistic program was an intriguing attempt to reform the foundations of mathematics that eventually did not prevail. The current paper offers a new perspective on the scientific community’s lack of reception to Brouwer’s intuitionism by considering it in light of Michael Friedman’s model of parallel transitions in philosophy and science, specifically focusing on Friedman’s story of Einstein’s theory of relativity. Such a juxtaposition raises onto the surface the differences between Brouwer’s and Einstein’s stories and suggests that contrary to Einstein’s story, the philosophical roots of Brouwer’s intuitionism cannot be traced to any previously established philosophical traditions. The paper concludes by showing how the intuitionistic inclinations of Hermann Weyl and Abraham Fraenkel serve as telling cases of how individuals are involved in setting in motion, adopting, and resisting framework transitions during periods of disagreement within a discipline.

People’s attitudes towards social norms play a crucial role in understanding group behavior. Norm psychology accounts focus on processes of norm internalization that influence people’s norm following attitudes but pay considerably less attention to social identity and group identification processes. Social identity theory in contrast studies group identity but works with a relatively thin and instrumental notion of social norms. We argue that to best understand both sets of phenomena, it is important to integrate the insights of both approaches. Social status, social identity, and social norms are considered separate phenomena in evolutionary accounts. We discuss assumptions and views that support this separation, and suggest an integrated view of our own. We argue that we should be open to the early origins of human social complexity, and conjecture that the longer that the human social world involved multi-level societies the more probable it is that norm psychology and social identity interacted in rich ways.

The paper explores Hermann Weyl’s turn to intuitionism through a philosophical prism of normative framework transitions. It focuses on three central themes that occupied Weyl’s thought: the notion of the continuum, logical existence, and the necessity of intuitionism, constructivism, and formalism to adequately address the foundational crisis of mathematics. The analysis of these themes reveals Weyl’s continuous endeavor to deal with such fundamental problems and suggests a view that provides a different perspective concerning Weyl’s wavering foundational positions. Building on a philosophical model of scientific framework transitions and the special role that normative indecision or ambivalence plays in the process, the paper examines Weyl’s motives for considering such a radical shift in the first place. It concludes by showing that Weyl’s shifting stances should be regarded as symptoms of a deep, convoluted intrapersonal process of self-deliberation induced by exposure to external criticism.