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...

A general introduction to the celebrated foundational crisis, discussing how the characteristic traits of modern mathematics (acceptance of the notion of an “arbitrary” function proposed by Dirichlet; wholehearted acceptance of infinite sets and the higher infinite; a preference “to put thoughts in the place of calculations” and to concentrate on “structures” characterized axiomatically; a reliance on “purely existential” methods of proof) provoked extensive polemics and alternative approaches. Going beyond exclusive concentration on the paradoxes, it also discusses the role of the (...) axiom of Choice, issues of predicativity, and goes on to present the ideas of intuitionism and Hilbert's program. The essay closes with Gödel's contributions and the aftermath. (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)

Review of: José Ferreirós. 2016. Mathematical knowledge and the interplay of practices. Princeton: Princeton University Press.

Information can be public among a group. Whether or not information is public matters, for example, for accounts of interdependent rational choice, of communication, and of joint intention. A standard analysis of public information identifies it with (some variant of) common belief. The latter notion is stipulatively defined as an infinite conjunction: for p to be commonly believed is for it to believed by all members of a group, for all members to believe that all members believe it, and so (...) forth. This analysis is often presupposed without much argument in philosophy. Theoretical entrenchment or intuitions about cases might give some traction on the question, but give little insight about why the identification holds, if it does. The strategy of this paper is to characterize a practical-normative role for information being public, and show that the only things that play that role are (variants of) common belief as stipulatively characterized. In more detail: a functional role for “taking a proposition for granted” in non-isolated decision making is characterized. I then present some minimal conditions under which such an attitude is correctly held. The key assumption links this attitude to beliefs about what is public. From minimal a priori principles, we can argue that a proposition being public among a group entails common commitment to believe among that group. Later sections explore partial converses to this result, the factivity of publicity and publicity from the perspective of outsiders to the group, and objections to the aprioricity of the result deriving from a posteriori existential presuppositions. (shrink)

Delusions are one of the most classical symptoms described in schizophrenia. However, despite delusions are often emotionally charged, they have been investigated using tasks involving non-affective material, such as the Beads task. In this study we compared 30 patients with schizophrenia experiencing delusions with 32 matched controls in their pattern of responses to two versions of the Beads task within a Bayesian framework. The two versions of the Beads task consisted of one emotional and one neutral, both with ratios of (...) beads of 60:40 and 80:20, considered, respectively, as the “difficult” and “easy” variants of the task. Results indicate that patients showed a greater deviation from the normative model, especially in the 60:40 ratio, suggesting that more inaccurate probability estimations are more likely to occur under uncertainty conditions. Additionally, both patients and controls showed a greater deviation in the emotional version of the task, providing evidence of a reasoning bias modulated by the content of the stimuli. Finally, a positive correlation between patients’ deviation and delusional symptomatology was found. Impairments in the 60:40 ratio with emotional content was related to the amount of disruption in life caused by delusions. These results contribute to the understanding of how cognitive mechanisms interact with characteristics of the task in the context of delusional thinking. These findings might be used to inform improved intervention programs in the domain of inferential reasoning. (shrink)

Hegel ofrece dos formas -en su filosofía en último análisis complementarias- de caracterizar al idealismo: La primera es próxima a la concepción habitual de idealismo previa a Kant, a saber: como una posición que considera que aquello que conocemos es una modificación de la propia conciencia. La segunda descansa en la tesis de la nulidad de lo finito; según esto, ?idealista? es la filosofía que considera que lo finito no tiene en cuanto tal realidad; ?realista?, a la inversa, es la (...) filosofía que toma el ser de las cosas determinadas como algo fijo y definitivo. Hegel denomina a este idealismo ?absoluto?.El idealismo absoluto es el modelo teórico que interrelaciona la tesis del carácter intrínsecamente ideal de los objetos de conocimiento y la tesis de que las cosas singulares que constituyen el mundo real surgen como una autodeterminación del ser. La premisa que permite tal vínculo es la tesis defendida por Hegel de que ser y pensar nunca se han separado y que su separación es el resultado de una concepción posterior errónea de la mente humana sobre la naturaleza de su actividad cognitiva. La hipostasiación de la disociación que la forma cognitiva de la representación introduce en la unidad indiferenciada de ser y pensar de la que parte la actividad de conocimiento en el sentimiento y la intuición sensibles es para Hegel la que provoca la ilusión retroactiva de que ambos -ser y pensar, mundo conocido y mente que conoce- se diferencian y contraponen. Por el contrario, desde el momento que la sensación es también ella misma un modo particular de la actividad continua e ininterrumpida del pensamiento, en el itinerario que va de la percepción sensible a la comprensión del objeto no hay para Hegel un punto en donde pueda introducirse una fractura que disuelva la unidad primigenia entre el ser y el pensar. En esta precisa medida, el momento final de la actividad de conocimiento, es decir, la actividad de comprender el mundo, es a un tiempo la singularización de los contenidos universales que constituyen la teoría del sujeto sobre el mundo y la autodeterminación de la unidad ser-pensar de la que parte y en todo momento permanece el sujeto que elabora esa teoría. Arribados al punto de vista del concepto, la dicotomía realismo-idealismo queda develada como un corolario de la autoconcepción representacionalista de la subjetividad cognoscente. Esta concepción define al paradigma general del realismo, el cual incluye para Hegel como el anverso y reverso de un mismo modelo teórico tanto lo que tradicionalmente ha sido entendido como realismo como al idealismo que resulta de una universalización de la idealidad de la conciencia singular, ya sea en la variante propuesta por Berkeley, ya sea en la del idealismo trascendental de Kant. A ojos de Hegel, por el contrario, el idealismo no puede ser sino absoluto, es decir, debe incluir en sí mismo el momento del ser y, con él, la realidad de las cosas que el sujeto conoce: la tarea esencial de la filosofía no consiste para Hegel más que en tomar conciencia de la co-pertenencia y unidad intrínseca de ambos. (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 paper examines the Aristotelian roots of the mechanics of naval architecture, beginning with Mechanical Problems, through its various interpretations by Renaissance mathematicians including Vettor Fausto and Galileo at the Venice Arsenal, and culminating in the first synthetic works of naval architecture by the French navy professor Paul Hoste at the end of the seventeenth century.

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)

Liberados el romanticismo y el idealismo del lastre secular de su vínculo imaginario con la “catástrofe alemana” deviene una vez más posible recurrir a ellos para repensar los problemas teóricos y prácticos del presente y localizar nuevos instrumentos conceptuales para su solución. En este contexto recobran interés numerosos temas que fueron objetos privilegiados de reflexión por parte de los pensadores románticos e idealistas. El romanticismo exaltó la Revolución Francesa como un acontecimiento epocal de reivindicación de los derechos de los individuos (...) contra la opresión de las formas anquilosadas de la religión y la organización estatal. Los pensadores románticos se rebelaron contra el orden burgués en todas sus formas, contra la pauta ubicua de la razón económico-instrumental. Para Kant, lo que es útil e intercambiable no posee dignidad, sino precio. El rechazo romántico de la exhaustiva reducción del mundo a mercancía conduce a la recuperación del valor infinito de la naturaleza, en esa medida, a su “reencantamiento”. En una misma línea, el romanticismo preconiza tendencialmente todos los fenómenos no cuantificables, aquéllos que no permiten ser calculados en términos mercantiles: la corporeidad, el erotismo, la pasión, la imaginación y la fantasía, el arte y la poesía, lo telúrico, la diversidad individual y cultural. El idealismo, por su parte, busca una síntesis entre la concepción ilustrada del mundo material y la visión pre-moderna del mismo. El resultado de esa mediación es la explicitación de las estructuras del propio espíritu humano presentes en su comprensión del mundo, toda vez que para el idealismo el mundo es siempre la teoría de la mente humana sobre el mundo. El pensamiento ilustrado pudo por ello confundir la filosofía de la naturaleza de los idealistas con una visión de lo material viciada de antropomorfismo, más próxima a la mística y a los bestiarios medievales que a las ciencias naturales de la Edad Moderna. El idealismo, por el contrario, encuentra en esta suerte de principio antrópico de toda teoría un factor de liberación moral y política. La toma de conciencia de que es el sujeto el que crea sistemas de sentido y que, por tanto, éstos deben reflejar en sí mismos su origen humano, implica el corolario de que con las teorías –sea cual sea el ámbito de la realidad que ellas busquen explicar– se trata en verdad de dispositivos de ordenamiento para el conocimiento y la acción construidas por la propia subjetividad humana; en esta medida, en su origen no se encuentran datos brutos de un supuesto orden natural que debe, por tanto, ser respetado sacralmente, sino la propia comunidad de sujetos vivientes. Aunque haya dejado tras de sí y olvidado el proceso de mediaciones que le dio origen, aunque se presente, pues, como replegado sobre sí mismo y, en esa medida, como la suma de re-presentaciones de un orden encontrado en el mundo, cada sistema de ideas contiene el conjunto de prácticas subjetivas inferenciales que lo constituyeron y que fueron realizadas por múltiples sujetos en plexos históricos concretos. El idealismo consiste en explicitar a la subjetividad actuando en el objeto y, con ello, en conminar el peligro de la mutación de las teorías en ideologías. El fundamento último de la libertad descansa para el idealismo menos en el orden determinado que funciona como condición de posibilidad del ejercicio de la libertad que en el modo de autoconcepción de la propia subjetividad. A partir del mundo de las autoconcepciones del sujeto –para Hegel se trata del ámbito específico del sujeto que ha devenido por ello mismo absoluto– derivan luego, como sus corolarios prácticos, las formas concretas de ordenamiento social y político. Si bien asume en sí numerosos motivos románticos, el idealismo aspira también a una mediación de esos elementos con las contribuciones teóricas de la Ilustración, con la que guarda entonces una relación menos crítica que el romanticismo. Fruto de este espíritu más ecuménico y menos orientado a la ruptura, defiende una concepción más pacífica de la racionalidad capitalista y el mundo burgués. Hegel reivindica así el rol positivo del mercado con su división social del trabajo y una concepción menos revolucionaria y más reformista de la vida política. El neoliberalismo contemporáneo se concibe a sí mismo como el sistema que resulta naturalmente del principio de la libertad negativa. Una tesis central del idealismo alemán es que para que sea posible el ejercicio de la capacidad de decidirse autónomamente por diferentes metas es necesario el marco efectivo de la libertad positiva. Sin la presencia de un orden racional con poder coercitivo sobre los miembros de la sociedad para regular su conducta, la libertad negativa de esos miembros, a primera vista disminuida y parcialmente cercenada por aquel orden, no es en la práctica posible. (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)

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)

Como parte de las sucesivas modificaciones al marco jurídico que regula el sistema educativo mexicano, la rendición de cuentas basada en evidencias se ha convertido en una nueva responsabilidad para los docentes del nivel básico. No obstante, se sabe poco acerca de qué significa para los docentes, cómo la han asumido, más aún, qué efectos ha provocado en la práctica cotidiana. Por tal motivo, se realizó un estudio cuyo propósito fue analizar los efectos de la rendición de cuentas en el (...) trabajo docente, con particular atención a la dimensión pedagógica. Mediante entrevistas semiestructuradas a diez docentes de escuelas públicas de la Ciudad de México, se encontró lo que denominamos una autonomía relativa, limitada y mediada por múltiples disposiciones normativas; la invención de realidades como estrategia para cumplir con la entrega de informes y toda clase de evidencias requeridas por diversas autoridades; una idoneidad multitarea, reflejada en una multiplicidad de actividades más allá de la enseñanza; finalmente una profesionalización mecanicista, es decir, una búsqueda constante de cursos y talleres por cuenta propia pero siempre sobre temas y contenidos instrumentales. (shrink)

La presente comunicación tiene como principal objetivo poner sobre la mesa algunos de los problemas que latían de fondo en algunas de las charlas del anterior VII Congreso sobre Pensamiento Filosófico Contemporáneo de la Sociedad Asturiana de Filosofía. No pretendo, sin embargo, exponer tesis alguna, sino que procuraré dialogar con las ideas del pasado Congreso de la SAF, y así poder ganar distancia para orientar mejor la mirada o proponer una actitud diferente ante las cuestiones. Comenzaré centrándome en una de (...) las comunicaciones, concretamente la de Virginia Ballesteros. A continuación, procuraré señalar los presupuestos fundamentales que latían de fondo a partir de las investigaciones de Felipe Martínez Marzoa, con el que intentaré mostrar qué elementos se encuentran a la base de la violencia ontológica en nuestras sociedades. Finalmente, y sólo para señalar posibles líneas de investigación, indicaré brevemente una probable relación alternativa entre fenomenología y medicina. (shrink)

Telling a patient that the outcome of his illness is not good, or even hopeless, requires sensitivity and the ability to communicate with him in the setting of a hospital which is an unnatural environment divorced from family and friends. It is a task which must be taught and learned by doctors and nurses.

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 achievements of the textile industry in Roman Britain are often underestimated as a result of the meagreness of our available evidence. The Edict on maximum prices issued by Diocletian in A.D. 301 shows that British capes commanded high prices on the markets of the Empire, and that in the late third century A.D. British rugs were the best in the world. In view of the competition from the traditional centres of rug manufacture in the East, this is an astonishing (...) achievement. (shrink)

The received translation and interpretation of many of the technical terms current in the textile industry of the Roman Empire are inaccurate, because lexicographers have either fought shy of being precise, or have thought that they recognized in the ancient world technical processes which originated at a much later date. The evidence is often equivocal or insufficient, but may still yield details that have been overlooked. The textile expression scutulatus, to take an example, deserves more attention than Blümner has devoted (...) to it. When it refers to the appearance of a garment, Lewis and Short, s.v. scutulatus, translate ‘diamond or lozenge-shaped, checkered’. It seems worth while to attempt a more precise translation, and, if possible, to estimate the significance in context of the goods so designated. (shrink)

Living out of silence, out of a fully functioning, lovingly attentive mind, and not just out of thought, out of a partially functioning mind, is requisite for depth or profundity in living or relating. A fully attentive, truly silent or meditative mind sees that there is real dualism of time and the timeless and that time and the timeless each has its own unique value. The timeless, or real silence, that which alone can make for depth in one's living and (...) relating, can be there only when, in one's awareness, time and all that is related to time, have come to an end. An examination of a wide variety of statements made by writers, both past and present, shows that these understandings are seldom there in contemporary writings, just as they were not there in writings that have come down to us from the past. 'Living beyond the One and the Many' invites readers to consider the possibility that what is presented in it can bring deep meaning to one's way of living in the world"--Publisher's description, back cover. (shrink)

If the world itself is metaphysically indeterminate in a specified respect, what follows? In this paper, we develop a theory of metaphysical indeterminacy answering this question.

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)

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)

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)

The thesis that existence is radically different from the determinacy of an actually existing thing –Kant considers this thesis to be the unsurmountable objection against the cosmological and the ontological argument– is the same thesis that demands a specific explanation of the existence of that actually existing thing. The notion of existence that results from its complete exclusion from the realm of the contents thoroughgoing determined by real predicates requires, thus, precisely because this thoroughness excludes it, a specific reason that (...) explains that in the actually existing thing its own actual existence is in fact united with its own determinacy; that notion of existence, however, cancels at the same time the requirement of a sufficient reason of the existence of what actually exists, and again for the same reason, namely, because since existence is as such excluded from the determinacy of the existing things, it radically differs from the specific realm where determinacy is constituted (that is why existing things are thoroughgoing determined as possible), and, therefore, it invalidates any application to itself of the logical laws that are valid only for the merely possible domain of determinacy. (shrink)