Domain extension in mathematics occurs whenever a given mathematical domain is augmented so as to include new elements. Manders argues that the advantages of important cases of domain extension are captured by the model-theoretic notions of existential closure and model completion. In the specific case of domain extension via ideal elements, I argue, Manders’s proposed explanation does not suffice. I then develop and formalize a different approach to domain extension based on Dedekind’s Habilitationsrede, to which Manders’s account (...) is compared. I conclude with an examination of three possible stances towards extensions via ideal elements. (shrink)

Review of W. J. Mander, The Volitional Theory of Causation: From Berkeley to the Twentieth Century. Oxford University Press, 2023.

The cognitive processing of spatial relations in Euclidean diagrams is central to the diagram-based geometric practice of Euclid's Elements. In this study, we investigate this processing through two dichotomies among spatial relations—metric vs topological and exact vs co-exact—introduced by Manders in his seminal epistemological analysis of Euclid's geometric practice. To this end, we carried out a two-part experiment where participants were asked to judge spatial relations in Euclidean diagrams in a visual half field task design. In the first part, (...) we tested whether the processing of metric vs topological relations yielded the same hemispheric specialization as the processing of coordinate vs categorical relations. In the second part, we investigated the specific performance patterns for the processing of five pairs of exact/co-exact relations, where stimuli for the co-exact relations were divided into three categories depending on their distance from the exact case. Regarding the processing of metric vs topological relations, hemispheric differences were found for only a few of the stimuli used, which may indicate that other processing mechanisms might be at play. Regarding the processing of exact vs co-exact relations, results show that the level of agreement among participants in judging co-exact relations decreases with the distance from the exact case, and this for the five pairs of exact/co-exact relations tested. The philosophical implications of these empirical findings for the epistemological analysis of Euclid's diagram-based geometric practice are spelled out and discussed. (shrink)

Cuốn sách dày hơn 120 trang, vừa được tác giả hoàn thành vào đầu năm nay, có những bài viết đề cập những vấn đề vẫn còn đang được bàn luận nhiều như ChatGPT (ChatGPT, help!). Tổng cộng cuốn sách gồm khoảng 40 bài viết ngắn, dựa trên bối cảnh và hình thái xã hội Việt Nam, được chia thành ba phần: Science Sobriety (tạm dịch: Tỉnh thức trong khoa học); Business Intelligence (Tri thức kinh doanh); và Mandering Thought (Suy (...) nghĩ vẩn vơ). (shrink)