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.

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)

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)

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".

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.

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.

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.

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)

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)

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)

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)

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)

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)

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)

Review of M. Giaquinto, The Search for Certainty: A Philosophical Account of Foundations of Mathematics (Osford, 2002).

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)

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)

Some transhumanist authors make the prophecy of immortality thanks to the transfer of the human mind to a superintelligent computer that would guarantee the survival of the person. That immortality would mean a happy life. In this article we try to show that this supposed indefinite survival is not exactly what is usually understood by immortality. In addition, we try to think about what immortality is based on the theological understanding of eternity and personal communion in which the life of (...) God consists. The decisive questions in this dialogue with the postures of transhumanism are the meaning of the body for the human person and what happiness is. (shrink)

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.

This chapter can be considered as made up of two parts, a general discussion of the notion of mathematical practice and the limits of its use, comprised by the first three sections, and a particular case study that is presented in order to exemplify the idea of the web of practices, which occupies the remaining three. The presentation of my approach to the notion of mathematical practice is brief and synthetic but more articulated theoretically than in a previous book (Ferreirós (...) 2016). Considerations in Sect. 3 about mathematical cognition and ethnomathematics have to do with the “limits” of mathematical practice, i.e., with practices that do not fall under that rubric.The case study concerns the evolution of the function concept, which offers a complex example of the web-of-practices approach. We start with early Modern developments that led to the eighteenth-century idea of function (as “analytical expression”) and the nineteenth-century “logical” idea of function, showing how complex is the spiderweb of interrelated mathematical and scientific practices knotted around the function concept. In the last section, I consider the difficult question of the most basic practices and cognitive experiences that form the basis of the idea of function. I propose that two conceptions – a dynamic and an algebraic or operational one – coalesced to give rise to the early Modern idea of function. (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)

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.

This is a contribution to the philosophy of experimental work, engaging with questions posed by Hacking, Franklin, Pickering, Schaffer and Collins. It focuses on the dynamics of experimentation and offers a detailed argument that one finds no "regress" of the kind posited by Collins. In particular, we reanalyze the celebrated series of experimental investigations by Newton on optical phenomena, taking into account Schaffer's partial reconstruction, and we show how it must be supplemented to obtain a more complete picture.

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)

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...

De manera inmediata y tradicional pensamos que hablar sobre política exige forzosamente ligarla a la idea de conflicto. Si bien es cierto que el carácter político marca ciertos modos de habitar el mundo, también es relevante aclarar que, a partir del concepto que tengamos de política, podremos darnos cuenta de si la idea de conflicto es inherente a esta o si la política puede ser pensada desde otro horizonte. Hans-Georg Gadamer nos ofrece una serie de reflexiones hermenéuticas en torno al (...) lenguaje y al diálogo; a partir de allí, intentamos atisbar un concepto de política no solo inherente al conflicto, sino también a su parte medular como lo es el diálogo prudente. Pensar en una política dentro del marco de un horizonte dialógico nos permite arrojar luz a problemas particulares que nos aquejan hoy, los cuales nos interpelan a pensar y repensarlos para comprender por qué hemos llegado a ser lo que somos. (shrink)

This is my introduction to a bilingual Spanish-German edition of selected writings by Bernhard Riemann. Published in Madrid: CSIC, 2000.

info:eu-repo/semantics/publishedVersion.

El objetivo de este artículo es el de explorar las principales características del ensayo como genus dicendi filosófico y cómo se cumplen en las obras de Ortega. En consecuencia, analizo en primer lugar las cuatro características principales del búsqueda de la verdad, valor cognitivo de las opiniones, rechazo del argumento de autoridad y tolerancia intelectual. Y, en segundo lugar, demuestro cómo se han tenido en cuenta en los escritos filosóficos de Ortega. ENGLISH: The aim of this paper is to explore (...) the main characteristics of the essay as philosophical genus dicendi and how they are fulfilled in Ortega’s works. Consequently, I firstly analyze the main four characteristics of the essay, namely, search for truth, cognitive value of opinions, rejection of the argument from authority, and intellectual tolerance. And secondly, I demonstrate how they have been taken into account by Ortega in his philosophical writing. (shrink)

El artículo explora la crítica de Santiago Ramírez a la clasificación de la analogía de Cayetano. Ramírez argumenta que la división tradicional de la analogía no refleja completamente la complejidad de las nociones de santo Tomás de Aquino. En particular, se introduce la "analogía de atribución intrínseca" como una categoría adicional dentro de la analogía de atribución. A través de un análisis detallado, el texto examina cómo esta forma de analogía mantiene elementos de la proporcionalidad y la atribución extrínseca, pero (...) se distingue por ser una atribución "según el ser" y "según la intención". Se destacan las implicaciones teológicas y filosóficas de esta distinción, especialmente en relación con la predicación del concepto de verdad respecto a Dios, las criaturas y los juicios humanos. (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)

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)

The COVID-19 pandemic has changed the lives of millions of people around the globe and some of the unprecedent emerged disruptions, are likely to have been particularly challenging for young children. Studying the impact of such extraordinary circumstances on their well-being is crucial to identify processes leading to risk and resilience. To better understand how Spanish children have adapted to the stressful disruptions resulting from the pandemic outbreak, we examined the effects of child coping and its interactions with contextual stressors (...) on child adjustment, incorporating in our analysis a developmental perspective. Data was collected in April 2020, through parent-reports, during the acute phase of the pandemic and, temporarily coinciding with the mandatory national quarantine period imposed by the Spanish Government. A sample of 1,123 Spanish children aged 3 to 12 participated in the study. Results showed differences in the use of specific strategies by children in different age groups. Despite the uncontrollable nature of the pandemic-related stressors, child disengagement coping was distinctively associated to negative outcomes, whereas engagement coping predicted psychosocial adjustment across all age groups. Moreover, interactively with child coping, parent fear of the future and parent dispositional resilience appear as relevant contextual factors to predict both negative and positive outcomes, but their effects seem to be age dependent, suggesting a higher contextual vulnerability for younger children. These findings might have implications for identifying individual and contextual risk and informing potential preventive interventions aimed to reduce the impact of future pandemic outbreaks on children of different ages. (shrink)