Abstract

In this essay Javier de Lorenzo reconstructs the so‐called crisis of the foundations of mathematics, a crucial scientific debate of the early twentieth century whose larger significance is still in need of much research. This is not an introductory text, as some background knowledge of the positions of the main actors is taken for granted. Rather, we are offered a historical interpretation of the emergence of this debate that connects it with more general changes in contemporary mathematical practice.The author presents his interpretation in opposition to what he identifies as the “canonical‐orthodox” historiographical approach to the debate. According to this approach, around 1900 the foundations of mathematics were shattered by the discovery of unforeseen antinomies and paradoxes, anomalies that seemed to reveal major logical problems within certain basic assumptions. The “foundational programs” were designed to offer mathematical practice new and solid grounds.By contrast, de Lorenzo turns his attention to the actual practice of the mathematicians who were involved in the turn‐of‐the‐century debate. He describes the foundational crisis as originating through a series of conceptual and technical changes that were reshaping the field in the late nineteenth century. In general terms, these changes were related to the gradual shift from a practice still linked to geometrical intuition to one based on purely structural considerations of the most abstract kind . To this end, rather than referring to the usual case of non‐Euclidean geometries, de Lorenzo sketches the process of generalization and formalization in fields such as projective geometry and the theory of functions. He focuses on the changing meaning of the notion of “demonstration” and on its new epistemological and ontological implications. Similarly, the notion of “axiomatic method” is scrutinized, and the controversy between Frege and Hilbert is employed to clarify the passage from the traditional to the modern conception.Against this background of changing practices and shifting concepts, the “discovery” of antinomies loses much of its catastrophic flavor. The set paradoxes and the teratology of curves should be seen not as revealing the logical shortcomings of a preexisting and monolithic edifice of mathematics but, rather, as signs of the early development of a different form of mathematical practice. In this sense, paradoxes functioned as fruitful conceptual spaces in which mathematicians explored, negotiated, and legitimated new demonstrative methods, new definitions of basic notions, and the new meaning of existential proofs.From this analysis, a view of mathematics as simply the product of a certain kind of human praxis emerges. The “foundational” enterprise cannot refer to any transcendental principle but, rather, should be viewed as a critical activity of conceptual clarification. In this sense, the book contributes to the study of logical and mathematical knowledge by focusing on the practices and the purposes of working mathematicians, in contrast to the rational reconstructions provided by much philosophy of mathematics. However, its significance would have been increased had de Lorenzo related his argument more explicitly to the body of recent historiography that has been dealing with these processes of conceptual and technical change and with their wider cultural meaning