This edited volume, aimed at both students and researchers in philosophy, mathematics and history of science, highlights leading developments in the overlapping areas of philosophy and the history of modern mathematics. It is a coherent, wide ranging account of how a number of topics in the philosophy of mathematics must be reconsidered in the light of the latest historical research and how a number of historical accounts can be deepened by embracing philosophical questions.

We offer an analysis of the disciplinary transformations underwent by mathematical or symbolic logic since its emergence in the late 19th century. Examined are its origins as a hybrid of philosophy and mathematics, the maturity and institutionalisation attained under the label �logic and foundations,� a second wave of institutionalisation in the Postwar period, and the institutional developments since 1975 in connection with computer science and with the study of language and informatics. Although some �internal history� is discussed, the main focus (...) is on the emergence, consolidation and convolutions of logic as a discipline, through various professional associations and journals, in centers such as Torino, Göttingen, Warsaw, Berkeley, Princeton, Carnegie Mellon, Stanford, and Amsterdam. (shrink)

Dedekind used to refer to Riemann as his main model concerning mathematical methodology, particularly regarding the use of abstract notions as a basis for mathematical theories. So, in passages written in 1876 and 1895 he compared his approach to ideal theory with Riemann’s theory of complex functions. In this paper, I try to make sense of those declarations, showing the role of abstract notions in Riemann’s function theory, its influence on Dedekind, and the importance of the methodological principle of avoiding (...) ‘forms of representation’ in shaping ideal theory. In order to emphasize the abstract viewpoint of Riemann and Dedekind, I compare their work with that of their great german contemporaries, Weierstrass and Kronecker; so, an influential ‘Göttingen group’ is confronted with the ‘Berlin school’ of mathematics. Some light is also thrown on the relation between Riemann and Dedekind, particularly with respect to Dedekind’s interest on Riemann’s ideas --in function theory, geometry and topology--, beginning in the 1850s and through later works. The abstract approach of the ‘Göttingen group’ was determinant for the turn to abstract mathematics in our century, and the direct influence of Riemann on Dedekind shows how this approach developed. (shrink)

Desde la didáctica de la Historia, el análisis de los libros de texto es una de las líneas de investigación más arraigadas en los últimos años. Este trabajo analiza la forma en que se ha enseñado la Transición democrática a través de libros de texto de Bachillerato correspondientes a la asignatura de Historia de España, de diferentes editoriales, tanto confesionales como laicas.

Este documento ofrece, con algunas explicaciones y añadidos, una conferencia pronunciada por su autor durante las 9ª Jornadas de Teología celebradas este presente año 2000 en el Centro de Estudios Teológicos de Sevilla, cuyo tema general fue "La Iglesia entre dos milenios: profecía e historia".

On knowledge and practices: a manifesto -- The web of practices -- Agents and frameworks -- Complementarity in mathematics -- Ancient Greek mathematics: a role for diagrams -- Advanced math: the hypothetical conception -- Arithmetic certainty -- Mathematics developed: the case of the reals -- Objectivity in mathematical knowledge -- The problem of conceptual understanding.

In 1887–1894, Richard Dedekind explored a number of ideas within the project of placing mappings at the very center of pure mathematics. We review two such initiatives: the introduction in 1894 of groups into Galois theory intrinsically via field automorphisms, and a new attempt to define the continuum via maps from ℕ to ℕ in 1891. These represented the culmination of Dedekind’s efforts to reconceive pure mathematics within a theory of sets and maps and throw new light onto the nature (...) of his structuralism and its specificity in relation to the work of other mathematicians. (shrink)

La carta Placuit Deo es un documento preciso e iluminador para presentar una comprensión cristiana de la salvación. Se destacan sólo algunos aspectos, pero se explican en un conjunto que ofrece el esquema fundamental para una soteriología. En el documento aparecen contenidos básicos y criterios orientativos para una reflexión sobre la salvación.

La iniciación cristiana implica la conversión del hombre a Cristo. Los sacramentos de la iniciación cristiana realizan la introducción del creyente en el misterio personal de Dios. El hombre recibe el don de la vida divina y la acoge conformándose a Cristo. Esa acogida conlleva una transformación de su existencia en el proceso de identificación con Cristo. En este artículo se intenta mostrar que el proceso de la conversión del hombre es la misma iniciación cristiana y se abordan algunas dificultades (...) para su vivencia en el contexto actual. (shrink)

This paper presents the epistemological proposal of Helen Longino, a philosopher of science who understands scientific knowledge as an unfinished result and who takes into account the context for its realization, the subjects who do science, the importance of criticism as a tool for the improvement of knowledge and science. The normative values that this author postulates are presented, and they are compared with the proposal of Thomas Kuhn, in order to make explicit a novel and alternative proposal to traditional (...) science, which takes into account other perspectives to improve the conduct of scientific activity, and the democratization of it. (shrink)

El análisis de los sesgos de género en ciencia ha dado lugar a una revisión crítica del conocimiento científico y a un cuestionamiento en profundidad del modelo de ciencia existente. Lo que, a su vez, da paso al interés por investigar en torno a las claves epistémicas que harían posible una ciencia no sexista, racista o clasista, fraguando un debate epistemológico de gran alcance.Las pretensiones de estas epistemologías son, principalmente, mostrar que los valores contextuales, es decir, los considerados no cognitivos, (...) influyen en la ciencia, en las metáforas científicas, el lenguaje utilizado, los procesos y métodos de investigación, y en los conocimientos. En esta comunicación se abordará principalmente el tema de la Ciencia Sostenible como alternativa a la denominada ciencia tradicional, y también como disyuntiva a la denominada ciencia feminista. Las posturas que confluyen son las de tres autoras: D. Haraway, S. Harding y Londa Schiebinger. Las cuales, en conjunto, dan lugar a una actividad práctica en ciencia, además de a una forma de mirar diferente, que intenta elaborar propuestas novedosas para la mejora del quehacer científico. (shrink)

Review by A. Kanamori, Boston University (author of The Higher Infinite), review in The Bulletin of Symbolic Logic: “Notwithstanding and braving the daunting complexities of this labyrinth, José Ferreirós has written a magisterial account of the history of set theory which is panoramic, balanced and engaging. Not only does this book synthesize much previous work and provide fresh insights and points of view, but it also features a major innovation, a full-fledged treatment of the emergence of the set-theoretic (...) approach in mathematics from the early nineteenth century. Much of history is in the details, and much of good history is in tone, emphasis, and the drawing together of different strands. Without being Whiggish, Ferreirós is never far from major synthetic themes, themes that in the main work toward balance and pull back from excesses of emphasis of some recent histories. The main criticism of the book would have to be on matters of broad interpretation. Yet, Ferreirós brings a considerable mathematical acuity and expertise to his enterprise, and this makes his viewpoints the more compelling.” . (shrink)

ABSTRACTWe discuss the thesis of universality of geometric notions and offer critical reflections on the concept of “natural geometry” employed by Spelke and others. Promoting interdisciplinary wor...

This paper defends a conceptualistic version of structuralism as the most convincing way of elaborating a philosophical understanding of structuralism in line with the classical tradition. The argument begins with a revision of the tradition of “conceptual mathematics”, incarnated in key figures of the period 1850 to 1940 like Riemann, Dedekind, Hilbert or Noether, showing how it led to a structuralist methodology. Then the tension between the ‘presuppositionless’ approach of those authors, and the platonism of some recent versions of philosophical (...) structuralism, is presented. In order to resolve this tension, we argue for the idea of ‘logical objects’ as a form of minimalist realism, again in the tradition of classical authors including Peirce and Cassirer, and we introduce the basic tenets of conceptual structuralism. The remainder of the paper is devoted to an open discussion of the assumptions behind conceptual structuralism, and—most importantly—an argument to show how the objectivity of mathematics can be explained from the adopted standpoint. This includes the idea that advanced mathematics builds on hypothetical assumptions (Riemann, Peirce, and others), which is presented and discussed in some detail. Finally, the ensuing notion of objectivity is interpreted as a form of particularly robust intersubjectivity, and it is distinguished from fictional or social ontology. (shrink)

Dedekind’s methodology, in his classic booklet on the foundations of arithmetic, has been the topic of some debate. While some authors make it closely analogue to Hilbert’s early axiomatics, others emphasize its idiosyncratic features, most importantly the fact that no axioms are stated and its careful deductive structure apparently rests on definitions alone. In particular, the so-called Dedekind “axioms” of arithmetic are presented by him as “characteristic conditions” in the _definition_ of the complex concept of a _simply infinite_ system. Making (...) sense of Dedekind’s method may be dependent on an analysis of the classical model of deductive science, as presented by authors from the eighteenth and early nineteenth centuries. Studying the modern reconstructions of Euclidean geometry, we show that they did not presuppose deductive independence of the axioms from the definitions. Authors like Wolff elaborated a mathematics _based on definitions_, and the Wolffian model of deductive science shows significant coincidences with Dedekind’s method, despite the great differences in content and approach. Wolff had a conception of definitions as _genetic_, which bears some similarities with Kant’s idea of synthetic definitions: they are understood as positing the content of mathematical concepts and introducing thought objects (_Gedankendinge_) that are the objects of mathematics. The emphasis on the spontaneity of the understanding, which can be found in this philosophical tradition, can also be fruitfully related with Dedekind’s idea of the “free creation” of mathematical objects. (shrink)

David Hilbert’s early foundational views, especially those corresponding to the 1890s, are analysed here. I consider strong evidence for the fact that Hilbert was a logicist at that time, following upon Dedekind’s footsteps in his understanding of pure mathematics. This insight makes it possible to throw new light on the evolution of Hilbert’s foundational ideas, including his early contributions to the foundations of geometry and the real number system. The context of Dedekind-style logicism makes it possible to offer a new (...) analysis of the emergence of Hilbert’s famous ideas on mathematical existence, now seen as a revision of basic principles of the “naive logic” of sets. At the same time, careful scrutiny of his published and unpublished work around the turn of the century uncovers deep differences between his ideas about consistency proofs before and after 1904. Along the way, we cover topics such as the role of sets and of the dichotomic conception of set theory in Hilbert’s early axiomatics, and offer detailed analyses of Hilbert’s paradox and of his completeness axiom (Vollständigkeitsaxiom). (shrink)

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 exemplares (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 ambivalencies in the process, and the main motives for subsequent changes. Among the latter, one may emphasize the spirit of modern axiomatics, 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 Firs-Order-Logic matured will be seen to have played a primary role in its shaping. (shrink)

A down-to-earth admission of abstract objects can be based on detailed explanation of where the objectivity of mathematics comes from, and how a ‘thin’ notion of object emerges from objective mathematical discourse or practices. We offer a sketch of arguments concerning both points, as a basis for critical scrutiny of the idea that mathematical and social objects are essentially of the same kind—which is criticized. Some authors have proposed that mathematical entities are indeed institutional objects, a product of our collective (...) imposition of function onto reality (the phrase comes from Searle) and of surrogation or hypostasis. Yet there are significant disanalogies between the typical social objects and mathemata, on which basis I argue that one should make a clear distinction between both. The comparison of mathematical with social objects helps understanding how non-physical objects can figure prominently in our explanations of reality. Yet mathematical objects have a different kind of cognitive grounding, and the more elementary of them emerge under relatively very simple sociocultural conditions. The differences are also reflected in the wide scope of use of mathematical concepts, and the much higher degree of variation found among social objects. On the basis of all of these features, I defend the thesis that one can significantly distinguish degrees of objectivity, and I use the distinction to articulate a graded ontology where one can locate the different kinds of mathematical and social objects. (shrink)

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 maximality. 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 to offer an 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. (shrink)

The place of Richard Dedekind in the history of logicism is a controversial matter. The conception of logic incorporated in his work is certainly old-fashioned, in spite of innovative elements that would play an important role in late 19th and early 20th century discussions. Yet his understanding of logic and logicism remains of interest for the light it throws upon the development of modern logic in general, and logicist views of the foundations of mathematics in particular. The paper clarifies Dedekind's (...) views and their relation to transcendental logic, makes explicit the role of the principle of comprehension (unrestricted) and offers a reconstrucion ot his dichotomy conception. The last section explores the differences with Frege (half epistemic, half metaphysical) making clear that there were different brands of logicism, and that logicism does not necessarily lead to singularism or “essentialistic objectualism”. (shrink)

La evolución del rol del profesor en las aulas en las dos últimas décadas se ha destacado por la preocupación de cumplir con las grandes exigencias de las instituciones educativas: adquirir habilidades digitales que les permita alcanzar un nivel “óptimo” de conocimiento en el uso de los recursos tecnológicos informáticos e incorporar estos recursos en sus actividades en el aula. No obstante, aún no se encuentra la fórmula para que los profesores modifiquen el modelo de enseñanza (transmisor-receptor) vigente en las (...) escuelas y en el salón de clase. Sin embargo, aunque las actividades que se realizan en el aula se encierran en una aparente caja negra, en este trabajo se presentan modelos tecno-pedagógicos que han implementado profesores de educación superior en el aula y que propone un nuevo rol del profesor, el docente innovador. Un profesor que combina el curriculum, contenidos, didácticas, actitudes, habilidades digitales, y estilos de aprendizaje para innovar en su método didáctico y que da cuenta de su habilidad didáctica, su conocimiento y preparación profesional así como de una cultura digital que va más allá de saber usar los dispositivos digitales y que conjunta los saberes humanos y tecnológicos en un modelo tecno-pedagógica para construir nuevos entornos de aprendizaje y modelos de enseñanza. (shrink)

The present paper is a contribution to the history of logic and its philosophy toward the mid-20th century. It examines the interplay between logic, type theory and set theory during the 1930s and 40s, before the reign of first-order logic, and the closely connected issue of the fate of logicism. After a brief presentation of the emergence of logicism, set theory, and type theory, Quine’s work is our central concern, since he was seemingly the most outstanding logicist around 1940, though (...) he would shortly abandon that viewpoint and promote first-order logic as all of logic. Quine’s class-theoretic systems NF and ML, and his farewell to logicism, are examined. The last section attempts to summarize the motives why set theory was preferred to other systems, and first orderlogic won its position as the paradigm logic system after the great War. (shrink)

The celebrated “creation” of transfinite set theory by Georg Cantor has been studied in detail by historians of mathematics. However, it has generally been overlooked that his research program cannot be adequately explained as an outgrowth of the mainstream mathematics of his day. We review the main extra-mathematical motivations behind Cantor's very novel research, giving particular attention to a key contribution, the Grundlagen (Foundations of a general theory of sets) of 1883, where those motives are articulated in some detail. Evidence (...) from other publications and correspondence is pulled out to provide clarification and a detailed interpretation of those ideas and their impact upon Cantor's research. Throughout the paper, a special effort is made to place Cantor's scientific undertakings within the context of developments in German science and philosophy at the time (philosophers such as Trendelenburg and Lotze, scientists like Weber, Riemann, Vogt, Haeckel), and to reflect on the German intellectual atmosphere during the nineteenth century. (shrink)

We discuss critically some recent theses about geometric cognition, namely claims of universality made by Dehaene et al., and the idea of a “natural geometry” employed by Spelke et al. We offer arguments for the need to distinguish visuo-spatial cognition from basic geometric knowledge, furthermore we claim that the latter cannot be identified with Euclidean geometry. The main aim of the paper is to advance toward a characterization of basic, practical geometry – which in our view requires a combination of (...) experiments on visuo-spatial cognition with studies in cognitive archaeology and comparative history. Examples from these fields are given, with special emphasis on the comparison of ancient Chinese and ancient Greek geometric ideas and procedures. (shrink)

We offer an analysis of the disciplinary transformations underwent by mathematical or symbolic logic since its emergence in the late 19 th century. Examined are its origins as a hybrid of philosophy and mathematics, the maturity and institutionalisation attained under the label “logic and foundations,” a second wave of institutionalisation in the Postwar period, and the institutional developments since 1975 in connection with computer science and with the study of language and informatics. Although some “internal history” is discussed, the main (...) focus is on the emergence, consolidation and convolutions of logic as a discipline, through various professional associations and journals, in centers such as Torino, Göttingen, Warsaw, Berkeley, Princeton, Carnegie Mellon, Stanford, and Amsterdam. (shrink)

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. (shrink)

Building on evidence for embodied representations, we investigated whether Spanish spatial terms map onto the NEAR/FAR perceptual division of space. Using a long horizontal display, we measured congruency effects during the processing of spatial terms presented in NEAR or FAR space. Across three experiments, we manipulated the task demands in order to investigate the role of endogenous attention in linguistic and perceptual space mapping. We predicted congruency effects only when spatial properties were relevant for the task but not when attention (...) was allocated to other features. Results showed faster responses for words presented in Near-space in all experiments. Consistent with our hypothesis, congruency effects were observed only when a reaching estimate was requested. Our results add important evidence for the role of top-down processing in congruency effects from embodied representations of spatial terms. (shrink)

Building on evidence for embodied representations, we investigated whether Spanish spatial terms map onto the NEAR/FAR perceptual division of space. Using a long horizontal display, we measured congruency effects during the processing of spatial terms presented in NEAR or FAR space. Across three experiments, we manipulated the task demands in order to investigate the role of endogenous attention in linguistic and perceptual space mapping. We predicted congruency effects only when spatial properties were relevant for the task but not when attention (...) was allocated to other features. Results showed faster responses for words presented in Near-space in all experiments. Consistent with our hypothesis, congruency effects were observed only when a reaching estimate was requested. Our results add important evidence for the role of top-down processing in congruency effects from embodied representations of spatial terms. (shrink)

A long-awaited edition of Zermelo’s works Content Type Journal Article Pages 1-4 DOI 10.1007/s11016-011-9548-y Authors José Ferreirós, Instituto de Filosofia, CCHS-CSIC, Albasanz, 26-28, 28037 Madrid, Spain Journal Metascience Online ISSN 1467-9981 Print ISSN 0815-0796.

Public decisions concerning large projects with detrimental environmental or heritage impacts involve value conflicts which stem from the diverse interests and variety of ways of evaluating the costs and benefits of such projects. They are also framed by institutionalised procedures and practices which favour certain concerns to the detriment of others. This paper aims to contribute towards a better understanding of how these procedures and practices, namely decision support tools such as the Environmental Impact Assessment (EIA), tend to shape public (...) decision-making processes in particular ways. It draws on a study of the public controversy surrounding the Foz Tua dam in Portugal, with a focus on the values upheld by the different parties in the controversy and their interplay in the production of justifications, specifically the actors’ positions on values and value conflicts and the restrictions posed by institutionalised public decision-making procedures on the expression and consideration of certain values. (shrink)

info:eu-repo/semantics/publishedVersion.

A book-length study of Riemann's multi-dimensional work (in Spanish), which considers his contributions to physics, philosophy and mathematics. Plus a bi-lingual edition (German-Spanish) of some of his landmark papers: the lecture on geometry, with Weyl's comments; the paper introducing the Riemann Conjecture, part of his 1857 paper on function theory; all of the philosophical fragments, etc. These different contributions, and their interconnections, are carefully studied in the introductory essay of 150 pages.