20th Century Philosophy of Mathematics

This paper describes both an exegetical puzzle that lies at the heart of Frege’s writings—how to reconcile his logicism with his definitions and claims about his definitions—and two interpretations that try to resolve that puzzle, what I call the “explicative interpretation” and the “analysis interpretation.” This paper defends the explicative interpretation primarily by criticizing the most careful and sophisticated defenses of the analysis interpretation, those given my Michael Dummett and Patricia Blanchette. Specifically, I argue that Frege’s text either are inconsistent (...) with the analysis interpretation or do not support it. I also defend the explicative interpretation from the recent charge that it cannot make sense of Frege’s logicism. While I do not provide the explicative interpretation’s full solution to the puzzle, I show that its main competitor is seriously problematic. (shrink)

On which philosophical foundations is the attribution of numerical magnitudes to qualitative phenomena based? That is, what is the philosophical basis for attributing, through measurement operations, numbers to empirical qualities that our senses perceive in the outside world? This question, nowadays rarely addressed in such a way, actually refers to an old debate about the quantification of qualities. A historical analysis reveals that it was a major issue in the “context of discovery” of the first attempts to mathematize new fields (...) of knowledge, whereas the “context of justification” leading our contemporary perspective on such episodes often overlooks such problems. As a consequence, little attention has so far been given to tracing its emergence back to its original context, leaving in the shadows an important debate about measurement. In this respect, the work of William Stanley Jevons is worth analysis. As a preliminary step toward his ambitious project to mathematize economics, he first attempted to understand the mechanism underlying measurement in physics, through a very innovative approach. Its main merit consisted in trying to overcome the Aristotelian distinction between quantities and qualities. In this article, I will strive to highlight the significance of his work for the history of philosophy of science. (shrink)

Suppose we are asked to draw up a list of things we take to exist. Certain items seem unproblematic choices, while others (such as God) are likely to spark controversy. The book sets the grand theological theme aside and asks a less dramatic question: should mathematical objects (numbers, sets, functions, etc.) be on this list? In philosophical jargon this is the ‘ontological’ question for mathematics; it asks whether we ought to include mathematicalia in our ontology. The goal of this work (...) is to answer this question in the affirmative, by drawing on considerations on the the applicability of mathematics to natural science. (shrink)

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. (shrink)

This collection of previously unpublished essays presents a new approach to the history of analytic philosophy--one that does not assume at the outset a general characterization of the distinguishing elements of the analytic tradition. Drawing together a venerable group of contributors, including John Rawls and Hilary Putnam, this volume explores the historical contexts in which analytic philosophers have worked, revealing multiple discontinuities and misunderstandings as well as a complex interaction between science and philosophical reflection.

A review of Hao Wang's Reflections on Kurt Goedel, emphasising Goedel's reaction against his Vienna Circle background.