The great antinomy

Se plantea y discute una “gran antinomia” entre concepciones teoreticistas o fundacionistas, y concepciones pragmatistas, en relación con una amplia diversidad de enfoques científicos y/o filosóficos. Ilustramos este contraste de manera especial con el concepto de tiempo y algunas consideraciones en torno a los ‘negacionistas’ del tiempo, guiados por visiones teoreticistas.We formulate and discuss a “great antinomy” between theoreticist/foundationist conceptions and pragmatist conceptions, in relation to a wide diversity of scientific and/or philosophical approaches. The contrast is illustrated in particular with the concept of time, considering the ‘timelessness crowd’ that has been guided by a theoreticist vision

Dogmas and the changing images of foundations

Ofrecemos una revisión crítica de varias concepciones de la investigación sobre los fundamentos de la matemática, desde los tiempos de Gauss hasta el presente. Se trata de (1) la imagen tradicional, que guio a Gauss, Dedekind, Frege y otros, y que ve en la búsqueda de sistemas básicos más adecuados una excavación lógica de estructuras a priori, (2) el programa de encontrar sistemas formales correctos para la llamada matemática clásica que puedan demostrarse consistentes, habitualmente asociado al nombre de Hilbert, y (3) la alternativa historicista, que guio a Riemann, Poincaré, Weyl y otros, la cual busca perfeccionar los sistemas conceptuales disponibles a fin de evitar limitaciones conceptuales y ampliar el abanico de opciones teóricas. Defenderé que, en ocasiones, se encuentran supuestos acerca del trabajo sobre fundamentos que emergen de ciertos dogmas, frecuentemente heredados de imágenes previas ya superadas. Para completar la discusión, menciono algunos rasgos de un programa alternativo, que investiga la epistemología del conocimiento matemático.I offer a critical review of several different conceptions of the activity of foundational research, from the time of Gauss to the present. These are (1) the traditional image, guiding Gauss, Dedekind, Frege and others, that sees in the search for more adequate basic systems a logical excavation of a priori structures, (2) the program to find sound formal systems for so-called classical mathematics that can be proved consistent, usually associated with the name of Hilbert, and (3) the historicist alternative, guiding Riemann, Poincaré, Weyl and others, that seeks to perfect available conceptual systems with the aim to avoid conceptual limitations and expand the range of theoretical options. I shall contend that, at times, assumptions about the foundational enterprise emerge from certain dogmas that are frequently inherited from previous, outdated images. To round the discussion, I mention some traits of an alternative program that investigates the epistemology of mathematical knowledge

The road to modern logic - an interpretation

This paper aims to outline an analysis and interpretation of the process that led to First-Order Logic and its consolidation as a core system of modern logic. We begin with an historical overview of landmarks along the road to modern logic, and proceed to a philosophical discussion casting doubt on the possibility of a purely rational justification of the actual delimitation of First-Order Logic. On this basis, we advance the thesis that a certain historical tradition was essential to the emergence of modern logic; this traditional context is analyzed as consisting in some guiding principles and, particularly, a set of exemplars (i.e., paradigmatic instances). Then, we proceed to interpret the historical course of development reviewed in section 1, which can broadly be described as a two-phased movement of expansion and then restriction of the scope of logical theory. We shall try to pinpoint ambivalences in the process, and the main motives for subsequent changes. Among the latter, one may emphasize the spirit of modern axiomatic, the situation of foundational insecurity in the 1920s, the resulting desire to find systems well-behaved from a proof-theoretical point of view, and the metatheoretical results of the 1930s. Not surprisingly, the mathematical and, more specifically, the foundational context in which First-Order Logic matured will be seen to have played a primary role in its shaping

Kurt Gödel: revolución en los fundamentos de las matemáticas

Ofrecemos un repaso a las principales contribuciones de Kurt Gödel en el campo de Lógica y fundamentos de las matemáticas, analizando su impacto, que bien puede llamarse revolucionario. La pretensión es hacer comprensible la tendencia y orientación metodológica de los trabajos de Gödel, y considerar en algún detalle sus repercusiones filosóficas. Así, se ofrece una perspectiva de cómo cambió la filosofía de las matemáticas entre las fechas de nacimiento y muerte del genial lógico matemático.We offer a survey of Kurt Gödel’s main contributions in the field of Logic and foundations of mathematics, and we analyse their impact, which can well be called revolutionary. The aim is to contribute to an understanding of the aims and methodological orientation of Gödel’s work, and to deal in some detail with their philosophical consequences. Thus, a perspective is offered on how the philosophy of mathematics changed between the dates of birth and death of this great mathematical logician

On arbitrary sets and ZFC

Set theory deals with the most fundamental existence questions in mathematics– questions which affect other areas of mathematics, from the real numbers to structures of all kinds, but which are posed as dealing with the existence of sets. Especially noteworthy are principles establishing the existence of some infinite sets, the so-called “arbitrary sets.” This paper is devoted to an analysis of the motivating goal of studying arbitrary sets, usually referred to under the labels of quasi-combinatorialism or combinatorial maximalist. After explaining what is meant by definability and by “arbitrariness”, a first historical part discusses the strong motives why set theory was conceived as a theory of arbitrary sets, emphasizing connections with analysis and particularly with the continuum of real numbers. Judged from this perspective, the axiom of choice stands out as a most central and natural set-theoretic principle (in the sense of quasi-combinatorialism). A second part starts by considering the potential mismatch between the formal systems of mathematics and their motivating conceptions, and proceeds too fferan elementary discussion of how far the Zermelo–Fraenkel system goes in laying out principles that capture the idea of “arbitrary sets”. We argue that the theory is rather poor in this respect.Junta de Andalucía P07-HUM-02594Ministerio de Ciencia y Tecnología BFF2003-09579-C0

A double bind? Reflections on history, science and culture

El objetivo de este trabajo es explorar el doble vínculo entre ciencia y cultura a través de un repaso, desde la historia y la filosofía de la ciencia, de las actividades científicas, en particular las asociadas al conocimiento matemático, en tanto que conjuntos de saberes y prácticas pertenecientes a un contexto intelectual, social y político más amplio y complejo.The aim of this paper is to explore the double bind between science and culture through the analysis, from the history and philosophy of science perspective, of various scientific activities, in particular those linked to mathematical knowledge, described as knowledge-making practices belonging to a much wider and complex intellectual, social and political context