My purpose is to suggest and open a dialogue between Plato and Adorno, and for this I have chosen the concept of mimesis, a concept that plays a central role in both thinkers both epistemologically and aesthetically. While Plato uses the term in order to expel the poets from Kallipolis, to be more precise certain kinds of poetry such as tragedy and comedy, Adorno uses the term to show the relationship between art and natural beauty, the somatic dimension of knowledge, (...) and the importance subversive poets and artists. For Plato, poets are three times removed from the truth and for Adorno, by contrast, the work of art is a vehicle of truth. This dialogue between an ancient and a modern intends to show not only the presence of the old in the new, but the valuable present and the fruitful transformation that an old concept can have. (shrink)

Resumen Mi intención es sugerir y abrir un diálogo entre Platón y Adorno, para lo cual he elegido el concepto de mímesis, un concepto que juega un papel central en ambos pensadores tanto gnoseológicamente como estéticamente. Mientras que el término le sirve a Platón para expulsar a los poetas de Kallipolis, más exactamente ciertos tipos de poesía como la tragedia y la comedia, Adorno lo usa para mostrar el lazo entre el arte y la belleza natural, la dimensión somática del (...) conocimiento, y la importancia subversiva de los poetas y los artistas. Para Platón los poetas están tres veces alejados de la verdad y para Adorno, por el contrario, la obra de arte es un vehículo de la verdad. Este diálogo quiere mostrar no sólo la presencia de lo antiguo en lo moderno, sino la valiosa actualidad y la fructífera transformación que puede padecer un concepto antiguo.My purpose is to suggest and open a dialogue between Plato and Adorno, and for this I have chosen the concept of mimesis, a concept that plays a central role in both thinkers both epistemologically and aesthetically. While Plato uses the term in order to expel the poets from Kallipolis, to be more precise certain kinds of poetry such as tragedy and comedy, Adorno uses the term to show the relationship between art and natural beauty, the somatic dimension of knowledge, and the importance subversive poets and artists. For Plato, poets are three times removed from the truth and for Adorno, by contrast, the work of art is a vehicle of truth. This dialogue between an ancient and a modern intends to show not only the presence of the old in the new, but the valuable present and the fruitful transformation that an old concept can have. (shrink)

Resumen: En este trabajo me propongo exponer las relaciones entre racionalidad responsiva y extrañeza en la obra de Bernhard Waldenfels. Expondré esta relación a partir de seis tesis con el propósito de que ellas se complementen mutuamente y logren mostrar, de la manera más apropiada posible, la relación entre responsividad y la experiencia de lo extraño. Estos puntos son: 1. Una breve aclaración terminológica. Diferencia entre lo otro y lo extraño. 2. La relación entre discurso escrito y hablado. 3. La (...) concepción de orden, vinculada con mecanismos de inclusión y exclusión. 4. La relación entre pathos y respuesta, como los puntos focales de la racionalidad responsiva. 5. La relación entre el decir y lo dicho y la temporalidad que los vincula y separa. Y por último, 6. Me ocuparé del problema de cómo hablar sobre lo extraño y el problema del lenguaje indirecto.: The purpose of this paper is to show the relations between responsiveness and strangeness in the work of Bernhard Waldenfels. I will expose this relationship in six theses, so that they complement each other and show, in the most appropriate possible way, the relationship between responsiveness and the experience of the strange. These points are, 1. A brief clarification of terminology. Difference between the other and the strange. 2. The relationship between written and spoken speech. 3. The conception of order, linked to mechanisms of inclusion and exclusion. 4. The relationship between pathos and response, as focal points of responsive rationality. 5. The relationship between saying and what is said and the temporality that links and separates them. And finally, 6. I will deal with the problem of how to talk about the strange and the problem of indirect language. (shrink)

Nuestro punto de partida es la concepción de Merleau-Ponty del lenguaje como "habla". Sostenemos que esta, como experiencia corporal del lenguaje -como lenguaje "en" nosotros-, permite comprender el fenómeno de la /nter-culturalidad como un fenómeno esencialmente /ntra-cultural, en el sentido de una experiencia de la cultura "desde dentro", es decir, desde la experiencia que cada quien hace del lenguaje. Este carácter endógeno e individual de pensar la cultura pone en evidencia, sin embargo, un "entre-lazamiento" recíproco originario entre lo propio y (...) lo extraño que impide de entrada privilegiar ya sea una visión universalista o totalizante o bien individualista o regionalista de la cultura. En dos momentos doy cuenta de este planteamiento: 1. Mundo de la vida y diversidad cultural y 2. El habla como lugar de acceso a las culturas. Our starting point is the conception in Merleau Ponty of language as "speech". We support this concept as corporal experience of language -as language "in" us - allows to understand the phenomenon of ¿nter-culturality as an essentially ¿ntra-cultural phenomenon, in the sense of an experience of culture "from within", that is to say; from the experience that everyone does with the language. This endogenous and individual character of thinking about culture highlights, however, a reciprocal "entanglement" originated among one's own and the strange that initially prevents to privilege, even a universal or totalizing or either an individualistic or regionalist vision of culture. Two moments I take on this approach: 1. World of life and cultural diversity. 2. Speech as an access place to the cultures. (shrink)

CONTENIDO: Itinerarios nietzscheanos / Aníbal Córdoba M. / - Nietzscher y la historia / Luis Antonio Restrepo A. / - Lenguaje, genealogía e interpretación / Jairo Montoya G. / - El amor y la muerte en la vida y el pensamiento de Nietzsche / Carlos Mario González R. / - La caverna de la interpretación / Jorge Mario Mejía T. / - Genealogía y poder / Jean Paul Margot / - Nietzche y la escritura / Jorge Alberto Naranjo M. (...) / - Salud, enfermedad, música. Reflexiones sobre la filosofía de Friedrich Nietzsche / Javier Escobar I / - Nietzche y Deleuze: encuentros / Juan Gonzalo Moreno V. (shrink)

Este ensayo tiene como propósito la interpretación del pasaje 37d5-7 del Timeo de Platón, en el cual se dice que el tiempo es una imagen móvil eónica del aión que permanece en lo uno y que marcha según el número. La tesis central es que el tiempo es entendido por Platón desde la dimensión del lenguaje (lógos) y la dimensión de la necesidad (anánké), que se entrecruzan en el tiempo. Para ello se discuten brevemente estos conceptos, así como la manera (...) en que ambos participan en la determinación de aquello que llamamos tiempo. También se trata de pensar un concepto de aión como vida que esté libre de la connotación de eternidad introducida por la tradición cristiana. Se quiere además mostrar como los conceptos de lógos y anánké, de lenguaje y necesidad, son los conceptos que subyacen a toda su filosofía. (shrink)

Me propongo discutir la teoría del lenguaje que Platón sostiene en este diálogo. Su punto de vista busca evitar tanto los peligros del enfoque naturalista de Cratilo como el convencionalista de Hermógenes, aunque considero que su posición es más cercana a Hermógenes, quien destaca el carácter práctico del lenguaje (387c ss.) El horizonte que guía su indagación es la relación epistémica entre lógos y cosa (on y pragma), esto es, la pregunta sobre qué me permite conocer el lenguaje de las (...) cosas y del mundo. Las partes etimológicas del diálogo muestran que partir de la palabra atómica es erróneo, ya que una palabra desvinculada de la frase no crea conexión alguna entre lenguaje y mundo. Sólo podemos conocer o mostrar la esencia de algo mediante un entrelazamiento de palabras, en la que la oración es la unidad mínima significativa del lenguaje, a saber un sustantivo y un verbo. Por esto es importante, para mi lectura, el pasaje 432d11 – 433a1, en el cual se muestra el carácter indispensable del lógos para mostrar lo que una cosa verdaderamente es. (shrink)

El eikôs-lógos es un concepto central del Timeo. Este ensayo se propone mostrar que con este concepto Platón recoge y resume sus reflexiones anteriores sobre el lógos, que se pueden sintetizar diciendo que al lógos en general le es inherente un momento de imagen y que la dialéctica platónica se realiza en la tensión entre imagen y concepto. Por esto, el eikôs-lógos no es algo propio únicamente de la cosmología, sino principalmente de la filosofía. Eikoôs señala a la dimensión finita, (...) terrena, del lógos filosófico. Por último, se quiere mostrar que el eikôs-lógos tiene sus raíces en el lenguaje y el número y que en las imágenes que nos hacemos del cosmos participan lenguaje y número. (shrink)

La teología desarrollada en el Timeo de Platón constituye la base del lógos cosmológico. En ese sentido, el discurso mítico que recorre el diálogo intenta esbozar una explicación plausible del predominio de la razón, la belleza y el orden en el mundo (kosmos) por encima del azar y la necesidad. Esta tesis se sustenta en el análisis filosófico de la función que cumple el demiurgo divino, en contraste con el Dios cristiano y con la deidades de la poesía griega. En (...) el primer punto, no se habla de una creación exnihilo, sino del ordenamiento de un material predado; en el segundo punto, no se trata de un dios antropomórfico, sino de un "dios filantrópico", responsable de la existencia de un kosmos cognoscible, que además constituye un modelo de la praxis humana. (shrink)

In this article, I examine the conceptual and methodological points of convergence and divergence of two intellectual currents frequently referred to as the decolonial and ontological turns in social and anthropological theory. Salient points considered are the ways both theoretical projects unsettle modernity’s dominant ontological and epistemological foundations by seriously engaging the conceptual potential of thinking with alterity (ethical dimension) and from exteriority (geopolitical dimension). I compare their subversive methodological contributions, examining, in particular, Enrique Dussel’s analectical hermeneutic approach and Eduardo (...) Viveiros de Castro’s ethnographic method of controlled equivocation. Lastly, I discuss how both theories and approaches complement each other’s efforts to destabilize Western modernity’s philosophical and anthropological foundations. (shrink)

Mi propósito en este ensayo es discutir y entender las relaciones entre mito e Ilustración en la Dialéctica de la Ilustración, que Horkheimer y Adorno expresan con la enigmática frase de que “el mito es ya Ilustración y la Ilustración recae en la mitología”. Para estos autores, la barbarie del siglo XX, que ha tenido una expresión inhumana en los campos fascistas y estalinistas de trabajo y de muerte, los lleva a repensar las relaciones entre los conceptos ya mencionados. Esta (...) inhumanidad no es meramente producto de “fuerzas oscuras”, sino que en ellos se manifiesta una forma limitada de la racionalidad, que no ha podido superar el elemento mítico de la venganza y el sacrificio. Esta limitación es producto de una razón que se ha puesto a sí misma como algo libre de toda naturaleza y corporeidad. Para aclarar esto, discuto los conceptos de razón identificadora, progreso, humanidad, sí mismo, sacrificio de sí y del otro. En la parte final, se analiza el viaje al Hades de Odiseo, que para estos autores encarna el momento de reconciliación entre ser humano y naturaleza, entre mito y razón, que no señala a un más allá de la historia, sino a la capacidad de la razón de darle voz a lo reprimido y de no olvidar sus determinaciones naturales. (shrink)

El artículo pretende mostrar que son cuatro los momentos esenciales que constituyen el Timeo de Platón y que sólo la consideración conjunta de ellos nos da una combinación adecuada para abrir el enigma del diálogo. Estos cuatro momentos son: l. El problema de lo divino y su relación con el mundo; 2. La relación entre Eikos- Logos, Cosmos y Filosofía; 3. La teoría de los elementos y su relación con el espacio; 4. El problema del Tiempo y la necesidad.

En el ensayo logos y oúcría en el Sofista de Platón se discute la relación entre lenguaje y ser, y el modo como Platón concibe dicha relación. Primero se reconstituye las críticas que platón le hace a las teorías ontológicas anteriores así como la famosa gigantomaquia alrededor del ser. Todas tienen algo en común: desconocen el papel esencial del Lógos en el conocimiento del ser. Luego se discute la función de la imagen en el conocimiento, y qué relación hay entre (...) lenguaje e imagen. Por último se busca dar una interpretación del término lógos, utilizado al final del diálogo, y que se traduce normalmente como frase u oración. (shrink)

Este ensayo se propone explorar algunas de las relaciones de Adorno con la literatura, teniendo como hilo conductor los conceptos de memoria, catástrofe y utopía. La memoria de la literatura es rebelión contra el manto de olvido con que los medios de comunicación y las esferas del poder quieren cubrir las catástrofes, crueldades y mentiras vividas por una sociedad, y de ésta contribuye a mantener la utopía: el deseo de que la sociedad puede ser organizada más justamente. En el centro (...) de la reflexión de Adorno sobre la literatura está la relación entre obra literaria y sociedad, y las relaciones de la primera, mediante la forma, con la segunda. ¿En qué consiste la verdad de una obra de arte? ¿Cómo pensar la relación entre obra de arte y sociedad sin caer en el sociologismo? ¿Cumple la obra de arle una función cognoscitiva? Responder a estas preguntas es un objetivo de este ensayo. (shrink)

En este ensayo quiero ocuparme de la manera como Marcuse piensa la muerte y el morir, una manera de pensar que considera a la muerte no solamente como un acontecimiento biológico, sino ante todo como un acontecimiento marcado profundamente por la política y social prevaleciente, que es algo que comparte con otros pensadores de la llevada Teoría Crítica. El caso es que muchos temen la muerte como el peor y más terrible mal. ¿Es posible pensarla de otra manera, sin dejar (...) de verla como el hecho biológico que es, pero sin caer en la tentación de pensarla como algo religiosa o metafísicamente sancionado, algo que pueda ser asumido sin temor ni pavor, sin angustia y sin autoengaños políticos o sociales? (shrink)

Este ensayo tiene como objetivo presentar algunos de los temas centrales de la Dialéctica negativa de Adorno. Especial atención se le dedica al concepto de mediación, sus relaciones con el concepto de no-identidad y a los de sujeto y objeto. Importante en Adorno es su intento de superar inmanentemente la dualidad férrea sujeto-objeto de la filosofía moderna, y el concepto de conciencia que le subyace, y por ello es difícil considerar a Adorno, como lo hace Habermas, a un filósofo preso (...) en las aporías de la filosofía de la conciencia. (shrink)

RESUMEN El diálogo propuesto entre Heidegger y Adorno, especificamente en lo que concierne a las ideas de crítica y de resistencia, no desconoce la importancia que adquiere para el último la crítica extensa que él mismo dirige a la ontología de Heidegger, ni para este su cuestionamiento de la filosofia dialéctica. El artículo busca mostrar que, en ambos autores, la idea de crítica y de resistencia se resuelve como indagación del sentido negativo de la experiencia. Se concluye que, destacando semejanzas (...) y diferencias en sus planteamientos, la reivindicación del carácter negativo de la experiencia equivale reivindicar el poder crítico de la filosofia y del arte. ABSTRACT The dialogue between these two authors, concerning specifically the ideas of critique and resistance, does not ignore the importance attributed by Adorno to the extensive critique that himself addresses to Heidegger's ontology, nor the one attributed by the latter to his questioning of the dialectical philosophy. The paper shows that in both authors the ideas of critique and resistance can be settled as an inquiry into the negative sense of experience. It is concluded that, highlighting similarities and differences in their approaches, the recognition of the negative trait of experience equals to claiming the critical power of philosophy and art. (shrink)

Este ensayo se propone explorar algunas de las relaciones de Adorno con la literatura, teniendo como hilo conductor los conceptos de memoria, catástrofe y utopía. La memoria de la literatura es rebelión contra el manto de olvido con que los medios de comunicación y las esferas del poder quieren cubrir las catástrofes, crueldades y mentiras vividas por una sociedad, y de ésta contribuye a mantener la utopía: el deseo de que la sociedad puede ser organizada más justamente. En el centro (...) de la reflexión de Adorno sobre la literatura está la relación entre obra literaria y sociedad, y las relaciones de la primera, mediante la forma, con la segunda. ¿En qué consiste la verdad de una obra de arte? ¿Cómo pensar la relación entre obra de arte y sociedad sin caer en el sociologismo? ¿Cumple la obra de arle una función cognoscitiva? Responder a estas preguntas es un objetivo de este ensayo. (shrink)

Este ensayo tiene como objetivo presentar algunos de los temas centrales de la Dialéctica negativa de Adorno. Especial atención se le dedica al concepto de mediación, sus relaciones con el concepto de no-identidad y a los de sujeto y objeto. Importante en Adorno es su intento de superar inmanentemente la dualidad férrea sujeto-objeto de la filosofía moderna, y el concepto de conciencia que le subyace, y por ello es difícil considerar a Adorno, como lo hace Habermas, a un filósofo preso (...) en las aporías de la filosofía de la conciencia. (shrink)

This work explores the later Wittgenstein’s philosophy of mathematics in relation to Lakatos’ philosophy of mathematics and the philosophy of mathematical practice. I argue that, while the philosophy of mathematical practice typically identifies Lakatos as its earliest of predecessors, the later Wittgenstein already developed key ideas for this community a few decades before. However, for a variety of reasons, most of this work on philosophy of mathematics has gone relatively unnoticed. Some of these ideas and their significance as precursors for (...) the philosophy of mathematical practice will be presented here, including a brief reconstruction of Lakatos’ considerations on Euler’s conjecture for polyhedra from the lens of late Wittgensteinian philosophy. Overall, this article aims to challenge the received view of the history of the philosophy of mathematical practice and inspire further work in this community drawing from Wittgenstein’s late philosophy. (shrink)

In this paper, I carry out a comparative study of the philosophical views of Edmund Husserl and Hermann Weyl on issues such as mathematical existence and mathematical intuition, the validity of classical logic, the concept of logical definiteness, the nature of symbolic mathematics, the role of mathematics in empirical science, the relation of scientific theories with perception, space representation and the philosophy of geometry, and intentional constitution in general. My main goal is not simply to assess the extent of Husserl’s (...) influence on Weyl, although this is an ever present concern, but to clarify the views of one by contrasting them with those of the other. (shrink)

In the first year of the twentieth century, in Gottingen, Husserl delivered two talks dealing with a problem that proved central in his philosophical development, that of imaginary elements in mathematics. In order to solve this problem Husserl introduced a logical notion, called “definiteness”, and variants of it, that are somehow related, he claimed, to Hilbert’s notions of completeness. Many different interpretations of what precisely Husserl meant by this notion, and its relations with Hilbert’s ones, have been proposed, but no (...) consensus has been reached. In this paper I approach this question afresh and thoroughly, taking into consideration not only the relevant texts and context, as others have also done before, but, more importantly, Husserl’s philosophy, his intuition-based epistemology in particular. Based on a system of clearly defined concepts that I here present, I reinforce an interpretation—definiteness as a form of syntactic completeness—that has, I believe, some advantages vis-à-vis alternative interpretations. It is in conformity with the available texts; it makes clear that Husserl’s notion of definiteness is indeed close to Hilbert’s notions of completeness; it solves the important problem of imaginaries for which it was created; and last, but not least, it fits naturally into Husserl’s system of concepts and ideas. (shrink)

The conditions involved in the applicability of mathematics in science are the subject of ongoing debates. One of the best‐received approaches is the inferential account, which involves structural mappings and pragmatic considerations in a three‐step model. According to the inferential account, these pragmatic considerations happen in the immersion and interpretation stages, but not during derivation (symbol‐pushing in a mathematical formalism). In this work, I draw inspiration from the later Wittgenstein and make the case that the applicability of mathematics also rests (...) on pragmatic considerations at the mathematical derivation level. I make the further case that pragmatic considerations at the mathematical derivation level may substitute pragmatic considerations in the other two stages, thus showing how they are holistically interrelated. I illustrate these two points with the solution of a simple geometrical problem. (shrink)

In this paper I discuss the version of predicative analysis put forward by Hermann Weyl in "Das Kontinuum". I try to establish how much of the underlying motivation for Weyl's position may be due to his acceptance of a phenomenological philosophical perspective. More specifically, I analyze Weyl's philosophical ideas in connexion with the work of Husserl, in particular "Logische Untersuchungen" and "Ideen I". I believe that this interpretation of Weyl can clarify the views on mathematical existence and mathematical intuition which (...) are implicit in "Das Kontinuum". (shrink)

Husserl left many unpublished drafts explaining (or trying to) his views on spatial representation and geometry, such as, particularly, those collected in the second part of Studien zur Arithmetik und Geometrie (Hua XXI), but no completely articulate work on the subject. In this paper, I put forward an interpretation of what those views might have been. Husserl, I claim, distinguished among different conceptions of space, the space of perception (constituted from sensorial data by intentionally motivated psychic functions), that of physical (...) geometry (or idealized perceptual space), the space of the mathematical science of physical nature (in which science, not only raw perception has a word) and the abstract spaces of mathematics (free creations of the mathematical mind), each of them with its peculiar geometrical structure. Perceptual space is proto-Euclidean and the space of physical geometry Euclidean, but mathematical physics, Husserl allowed, may find it convenient to represent physical space with a non-Euclidean structure. Mathematical spaces, on their turn, can be endowed, he thinks, with any geometry mathematicians may find interesting. Many other related questions are addressed here, in particular those concerning the a priori or a posteriori character of the many geometric features of perceptual space (bearing in mind that there are at least two different notions of a priori in Husserl, which we may call the conceptual and the transcendental a priori). I conclude with an overview of Weyl’s ideas on the matter, since his philosophical conceptions are often traceable back to his former master, Husserl. (shrink)

In this paper I discuss Husserl's solution of the problem of imaginary elements in mathematics as presented in the drafts for two lectures hegave in Göttingen in 1901 and other related texts of the same period,a problem that had occupied Husserl since the beginning of 1890, whenhe was planning a never published sequel to Philosophie der Arithmetik(1891). In order to solve the problem of imaginary entities Husserl introduced,independently of Hilbert, two notions of completeness (definiteness in Husserl'sterminology) for a formal axiomatic (...) system. I present and discuss these notionshere, establishing also parallels between Husserl's and Hilbert's notions ofcompleteness. (shrink)

I carry out in this paper a philosophical analysis of the principle of excluded middle (or, as it is often called in the version I favor here, principle of bivalence: any meaningful assertion is either true or false). This principle has been criticized, and sometimes rejected, on the charge that its validity depends on presuppositions that are not, some believe, universally obtainable; in particular, that any well-posed problem is solvable. My goal here is to show that, although excluded middle does (...) indeed rest on certain presuppositions, they do not have the character of hypotheses that may or may not be true, or matters of fact that may or may not be the case. These presuppositions have, I claim, a transcendental character. Hence, the acceptance of excluded middle does not necessarily require, as some have claimed, an allegiance to ontological realism or some sort of cognitive optimism, construed as factual theses concerning the ontological status of domains of objects and our capability of accessing them cognitively. DOI:10.5007/1808-1711.2011v15n2p333. (shrink)

I carry out in this paper a philosophical analysis of the principle of excluded middle. This principle has been criticized, and sometimes rejected, on the charge that its validity depends on presuppositions that are not, some believe, universally obtainable; in particular, that any well-posed problem is solvable. My goal here is to show that, although excluded middle does indeed rest on certain presuppositions, they do not have the character of hypotheses that may or may not be true, or matters of (...) fact that may or may not be the case. These presuppositions have, I claim, a transcendental character. Hence, the acceptance of excluded middle does not necessarily require, as some have claimed, an allegiance to ontological realism or some sort of cognitive optimism, construed as factual theses concerning the ontological status of domains of objects and our capability of accessing them cognitively. (shrink)

In this paper I discuss the version of predicative analysis put forward by Hermann Weyl in Das Kontinuum. I try to establish how much of the underlying motivation for Weyl''s position may be due to his acceptance of a phenomenological philosophical perspective. More specifically, I analyze Weyl''s philosophical ideas in connexion with the work of Husserl, in particular Logische Untersuchungen} and Ideen .I believe that this interpretation of Weyl can clarify the views on mathematical existence and mathematical intuition which are (...) implicit in Das Kontinuum. (shrink)

In this paper I present and discuss Husserl’s concept of material a priori truth, particularly with respect to color-concepts, and show that Schlick’s criticism of Husserl’s notion is based on misunderstandings and misconceptions.

In this paper I argue for the view that structuralism offers the best perspective for an acceptable account of the applicability of mathematics in the empirical sciences. Structuralism, as I understand it, is the view that mathematics is not the science of a particular type of objects, but of structural properties of arbitrary domains of entities, regardless of whether they are actually existing, merely presupposed or only intentionally intended.

In this paper I argue for the view that structuralism offers the best perspective for an acceptable account of the applicability of mathematics in the empirical sciences. Structuralism, as I understand it, is the view that mathematics is not the science of a particular type of objects, but of structural properties of arbitrary domains of entities, regardless of whether they are actually existing, merely presupposed or only intentionally intended.

In the first year of the twentieth century, in Gottingen, Husserl delivered two talks dealing with a problem that proved central in his philosophical development, that of imaginary elements in mathematics. In order to solve this problem Husserl introduced a logical notion, called “definiteness”, and variants of it, that are somehow related, he claimed, to Hilbert’s notions of completeness. Many different interpretations of what precisely Husserl meant by this notion, and its relations with Hilbert’s ones, have been proposed, but no (...) consensus has been reached. In this paper I approach this question afresh and thoroughly, taking into consideration not only the relevant texts and context, as others have also done before, but, more importantly, Husserl’s philosophy, his intuition-based epistemology in particular. Based on a system of clearly defined concepts that I here present, I reinforce an interpretation—definiteness as a form of syntactic completeness—that has, I believe, some advantages vis-à-vis alternative interpretations. It is in conformity with the available texts; it makes clear that Husserl’s notion of definiteness is indeed close to Hilbert’s notions of completeness; it solves the important problem of imaginaries for which it was created; and last, but not least, it fits naturally into Husserl’s system of concepts and ideas. (shrink)

In the first year of the twentieth century, in Gottingen, Husserl delivered two talks dealing with a problem that proved central in his philosophical development, that of imaginary elements in mathematics. In order to solve this problem Husserl introduced a logical notion, called “definiteness”, and variants of it, that are somehow related, he claimed, to Hilbert’s notions of completeness. Many different interpretations of what precisely Husserl meant by this notion, and its relations with Hilbert’s ones, have been proposed, but no (...) consensus has been reached. In this paper I approach this question afresh and thoroughly, taking into consideration not only the relevant texts and context, as others have also done before, but, more importantly, Husserl’s philosophy, his intuition-based epistemology in particular. Based on a system of clearly defined concepts that I here present, I reinforce an interpretation—definiteness as a form of syntactic completeness—that has, I believe, some advantages vis-à-vis alternative interpretations. It is in conformity with the available texts; it makes clear that Husserl’s notion of definiteness is indeed close to Hilbert’s notions of completeness; it solves the important problem of imaginaries for which it was created; and last, but not least, it fits naturally into Husserl’s system of concepts and ideas. (shrink)

In this paper I discuss the version of predicative analysis put forward by Hermann Weyl in Das Kontinuum. I try to establish how much of the underlying motivation for Weyl's position may be due to his acceptance of a phenomenological philosophical perspective. More specifically, I analyze Weyl's philosophical ideas in connexion with the work of Husserl, in particular Logische Untersuchungen} and Ideen.I believe that this interpretation of Weyl can clarify the views on mathematical existence and mathematical intuition which are implicit (...) in Das Kontinuum. (shrink)

Husserl left many unpublished drafts explaining (or trying to) his views on spatial representation and geometry, such as, particularly, those collected in the second part of Studien zur Arithmetik und Geometrie (Hua XXI), but no completely articulate work on the subject. In this paper, I put forward an interpretation of what those views might have been. Husserl, I claim, distinguished among different conceptions of space, the space of perception (constituted from sensorial data by intentionally motivated psychic functions), that of physical (...) geometry (or idealized perceptual space), the space of the mathematical science of physical nature (in which science, not only raw perception has a word) and the abstract spaces of mathematics (free creations of the mathematical mind), each of them with its peculiar geometrical structure. Perceptual space is proto-Euclidean and the space of physical geometry Euclidean, but mathematical physics, Husserl allowed, may find it convenient to represent physical space with a non-Euclidean structure. Mathematical spaces, on their turn, can be endowed, he thinks, with any geometry mathematicians may find interesting. Many other related questions are addressed here, in particular those concerning the a priori or a posteriori character of the many geometric features of perceptual space (bearing in mind that there are at least two different notions of a priori in Husserl, which we may call the conceptual and the transcendental a priori). I conclude with an overview of Weyl’s ideas on the matter, since his philosophical conceptions are often traceable back to his former master, Husserl. (shrink)

In this paper I show the possibility of an ontology of mathematics that keeps some points in common with platonism and constructivism while diverging from them in other essencial ones. I understand that mathematical objects are simply the referential focus of mathematical discourse, I also understand that their existence is merely intentional but none the less objective, in the sense of being shared by all those who are engaged in the mathematical activity. However, the objective existence of mathematical entities is (...) not secured once and for all but only in so far as the mathematical discourse is consistent. This is the core of the criterium of objective existence put forward and that I believe should sustain a mathematical ontology without the presupposition of the independent existence of a domain of mathematical objects, and without the restrictions imposed on it by constructivism and formalism in their various versions.Neste artigo quero apontar para a possibilidade de uma ontologia da matemática que, mesmo mantendo alguns pontos em comum com o platonismo e com o construtivismo, desliga-se destes em outros pontos essenciais. Por objeto matemático entendo o foco referencial do discurso matemático, ou seja, aquilo sobre o qual a matemática fala. Entendo que a existência destes objetos é meramente intencional, presuntiva, mas, simultaneamente, objetiva, no sentido de ser uma existência comunalizada, compartilhada por todos aqueles engajados no fazer matemático. A existência objetiva das entidades matemáticas não está, entretanto, garantida de uma vez por todas, mas apenas enquanto o discurso matemático for consistente. Este é o espírito do critério de existência objetiva enunciado que, acredito, deve sustentar uma ontologia matemática sem o pressuposto da existência independente de um domínio de objetos matemáticos, sem o empobrecimento que lhe impõem as diferentes versões construtivistas e sem a aniquilação que lhe infringe o formalismo sem objetos. (shrink)