Kati Kish Bar-On

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.

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.

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.

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.