Roughly, a proof of a theorem, is “pure” if it draws only on what is “close” or “intrinsic” to that theorem. Mathematicians employ a variety of terms to identify pure proofs, saying that a pure proof is one that avoids what is “extrinsic,” “extraneous,” “distant,” “remote,” “alien,” or “foreign” to the problem or theorem under investigation. In the background of these attributions is the view that there is a distance measure (or a variety of such measures) between mathematical statements and (...) proofs. Mathematicians have paid little attention to specifying such distance measures precisely because in practice certain methods of proof have seemed self- evidently impure by design: think for instance of analytic geometry and analytic number theory. By contrast, mathematicians have paid considerable attention to whether such impurities are a good thing or to be avoided, and some have claimed that they are valuable because generally impure proofs are simpler than pure proofs. This article is an investigation of this claim, formulated more precisely by proof- theoretic means. After assembling evidence from proof theory that may be thought to support this claim, we will argue that on the contrary this evidence does not support the claim. (shrink)

El artículo expone la evolución de las expectativas acerca del futuro y destino final del hombre tanto a nivel individual como colectivo. Hasta el siglo XVIII predominaban las preocupaciones referidas al destino personal de cada hombre y el significado de la muerte. Tras las revoluciones que acabaron con el antiguo régimen, empezó a dominar el interés por el futuro de la humanidad en su conjunto. Desde mediados del siglo XX hay una preocupación creciente ante el peligro de que la especie (...) humana desaparezca y sea suplantada por otro tipo de entidades, tal vez desarrolladas por el propio hombre. El transhumanismo se ha presentado como una respuesta, basada en la ciencia, a las ansias de supervivencia y mejora del hombre. No obstante, sus propuestas adolecen de varias debilidades, algunas de ellas posiblemente irreparables. (shrink)

Traditional geometry concerns itself with planimetric and stereometric considerations, which are at the root of the division between plane and solid geometry. To raise the issue of the relation between these two areas brings with it a host of different problems that pertain to mathematical practice, epistemology, semantics, ontology, methodology, and logic. In addition, issues of psychology and pedagogy are also important here. To our knowledge there is no single contribution that studies in detail even one of the aforementioned areas.

Gaisi Takeuti (1926–2017) is one of the most distinguished logicians in proof theory after Hilbert and Gentzen. He extensively extended Hilbert's program in the sense that he formulated Gentzen's sequent calculus, conjectured that cut-elimination holds for it (Takeuti's conjecture), and obtained several stunning results in the 1950–60s towards the solution of his conjecture. Though he has been known chiefly as a great mathematician, he wrote many papers in English and Japanese where he expressed his philosophical thoughts. In particular, he used (...) several keywords such as "active intuition" and "self-reflection" from Nishida's philosophy. In this paper, we aim to describe a general outline of our project to investigate Takeuti's philosophy of mathematics. In particular, after reviewing Takeuti's proof-theoretic results briefly, we describe some key elements in Takeuti's texts. By explaining these texts, we point out the connection between Takeuti's proof theory and Nishida's philosophy and explain the future goals of our project. (shrink)

English summary: This book attests that a study on the history of philosophy has its place in contemporary debates. This book attempts to reconstruct William of Ockham's position on the nature and function of real definitions in order to show that nominalism does not oblige a particular position on essences. French description: Guillaume d'Ockham, defenseur bien connu du nominalisme, a manifeste un interet soutenu pour les definitions reelles. Cet ouvrage s'efforce de reconstruire sa position sur la nature et la fonction (...) des definitions reelles. Un examen de ce qu'est une definition reelle appelle une etude de ce qu'est une essence et de la nature du lien a poser entre l'essentiel et le necessaire. Cet ouvrage a pour objectif de montrer que le nominalisme ne contraint pas a une position determinee sur les essences. Saul Kripke nous dit que l'essentiel se reduit au necessaire. Kit Fine, par contre, affirme qu'il faut revenir a Aristote et soutenir que l'essentiel est irreductible au necessaire. Notre examen atteste qu'une etude en histoire de la philosophie a toute sa place dans les debats contemporains. L'essentialisme de Guillaume d'Ockham incite a rendre a Aristote ce qui lui revient et a faire preuve de circonspection. En definitive, les deux positions en presence ne sont peut-etre pas incompatibles. (shrink)

Vattimo rethinks ontology at a time when modernity's concept of Being has been uprooted along with any faith in history as a unitary process characterized by progressive reappropriations of its own origins. Having dissolved the ground of the new, the end of modernity sees Being reduced to exchange value, the new for the sake of the new, which in turn science and technology make routine. The impasse is a radical one, for modernity cannot be left behind by offering a truer (...) foundation, since "overcoming" and "foundation" themselves are concepts central to modernity. The first section of the book consists in an account of this condition. (shrink)

During the last few months, several pioneer genome‐wide transcriptomic, proteomic and metabolomic studies have revolutionised the understanding of bacterial biological processes, leading to a picture that resembles eukaryotic complexity. Technological advances such as next‐generation high‐throughput sequencing and high‐density oligonucleotide microarrays have allowed the determination, in several bacteria, of the entire boundaries of all expressed transcripts. Consequently, novel RNA‐mediated regulatory mechanisms have been discovered including multifunctional RNAs. Moreover, resolution of bacterial proteome organisation (interactome) and global protein localisation (localizome) have unveiled an (...) unanticipated complexity that highlights the significance of protein multifunctionality and localisation in the cell. Also, analysis of a complete bacterial metabolic network has again revealed a high fraction of multifunctional enzymes and an unexpectedly high level of metabolic responses and adaptation. Altogether, these novel approaches have permitted the deciphering of the entire physiological landscape of one of the smallest bacteria, Mycoplasma pneumoniae. Here, we summarise and discuss recent findings aimed at defining the blueprint of any prokaryote. (shrink)

In contrast to the robust mutual interpretability phenomenon in set theory, Ali Enayat proved that bi-interpretation is absent: distinct theories extending ZF are never bi-interpretable and models of ZF are bi-interpretable only when they are isomorphic. Nevertheless, for natural weaker set theories, we prove, including Zermelo–Fraenkel set theory $\mathrm {ZFC}^{-}$ without power set and Zermelo set theory Z, there are nontrivial instances of bi-interpretation. Specifically, there are well-founded models of $\mathrm {ZFC}^{-}$ that are bi-interpretable, but not isomorphic—even $\langle H_{\omega _1},\in (...) \rangle $ and $ \langle H_{\omega _2},\in \rangle $ can be bi-interpretable—and there are distinct bi-interpretable theories extending $\mathrm {ZFC}^{-}$. Similarly, using a construction of Mathias, we prove that every model of ZF is bi-interpretable with a model of Zermelo set theory in which the replacement axiom fails. (shrink)

To what extent does perceptual language reflect universals of experience and cognition, and to what extent is it shaped by particular cultural preoccupations? This paper investigates the universality~relativity of perceptual language by examining the use of basic perception terms in spontaneous conversation across 13 diverse languages and cultures. We analyze the frequency of perception words to test two universalist hypotheses: that sight is always a dominant sense, and that the relative ranking of the senses will be the same across different (...) cultures. We find that references to sight outstrip references to the other senses, suggesting a pan-human preoccupation with visual phenomena. However, the relative frequency of the other senses was found to vary cross-linguistically. Cultural relativity was conspicuous as exemplified by the high ranking of smell in Semai, an Aslian language. Together these results suggest a place for both universal constraints and cultural shaping of the language of perception. (shrink)

In his 1978 paper “Mathematical Explanation”, Mark Steiner attempts to modernize the Aristotelian idea that to explain a mathematical statement is to deduce it from the essence of entities figuring in the statement, by replacing talk of essences with talk of “characterizing properties”. The language Steiner uses is reminiscent of language used for proofs deemed “pure”, such as Selberg and Erdős’ elementary proofs of the prime number theorem avoiding the complex analysis of earlier proofs. Hilbert characterized pure proofs as those (...) that use only “means that are suggested by the content of the theorem”, a characterization we have elsewhere called “topical purity”. In this paper we will examine the connection between Steiner’s account of mathematical explanation and topical purity. Are Steiner-explanatory proofs necessarily topically pure? Are topically pure proofs necessarily Steiner-explanatory? Answers to these questions will shed light on the general question of the relation between purity and explanatory power. (shrink)

This paper critically examines the debate between William of Ockham and his contemporary Peter Auriol on how to account for the intension and remission of forms. Peter Auriol denies that an added degree of a quality such as the theological virtue of charity could be anything other than something which is neither a universal nor an individual and which cannot be grasped by intuition, but must be posited in order to account for the possibility that an accidental form can vary (...) in intensity. Ockham aims at proving that Auriol’s account is inconsistent. In my opinion, Ockham does not succeed, nor does he succeed in defending a consistent account of the metaphysics underlying the intension and remission of forms. Indeed, he conceives of the degree of a quality as a part of the intensified quality which, while being an individual that is really distinct from the quality and can be picked out, forms a unity with it in such a way that the union of the degree and the quality is itself an individual. Ockham fails to distinguish the notion of maximal resemblance holding between a quality’s degree and the quality from the notion of maximal resemblance which holds between individuals of the same species. In the end, he is led to give up the idea that a degree is an individual that can be picked out in order to protect the core thesis of his nominalism about universals. (shrink)

AS IS WELL KNOWN, one of Kant's major concerns was the reconciliation of Newtonian science and metaphysics, a preoccupation made particularly acute by the need to provide a satisfactory explanation of organisms. It is in light of his claim that only the mechanistic principles of Newton's physics can provide scientific knowledge that the role to be played by purposiveness becomes problematic. Purpose appears to resist mechanistic explanation and is therefore a major impediment to unifying science under one set of principles. (...) Kant concludes that although organisms cannot be explained mechanistically, the impossibility is due to a limitation of reason. By appealing to the critical turn Kant thus avoids an antinomy between mechanism and finality while allowing that it is possible for mechanism and finality to be reconciled in the supersensible. This reconciliation, unfortunately, we will never know. (shrink)

Normal 0 21 false false false PT-BR X-NONE X-NONE MicrosoftInternetExplorer4 O estudo tem caráter qualitativo e busca em referenciais teóricos . C ontempla uma reflexão aberta a respeito do alvorecer da razão na modernidade, que lança o ser humano ao centro do universo . Ao fazê-lo, fragmenta a formação do ethos, vórtice que alcança a contemporaneidade. Na sequência traz suportes teóricos, com base, principalmente nas concepções de Agamben, Bauman,Vaz, entre outros, acerca da ainda possibilidade formativas de construção da comunidade humana. (...) Apesar dos numerosos e diversos conhecimentos produzidos, na atualidade, crescem as dificuldades de compreensão da condição humana e do significado dos fazeres humanos. A aposta na subjetividade como uma essência, marca da metafísica, também pos em questão a identidade pessoal . A contemporaneidade ampla e complexa clama por contextos educacionais capazes de dinamizar bases ontológicas com diferentes bases conceptuais e vivenciais. Dessa forma a problemática orientadora da reflexão é: como constituiu-se a formação do ethos contemporâneo e, quais são os desafios à educação? Como resultados destacamos que a racionalidade técnico-científica gerou despreocupação com a vivência de valores éticos voltados ao bem comum; a individualização fomentou a descrença em políticas democráticas e ceifou a esperança na formação de um ethos contemporâneo solidário. Concluimos que existe uma necessidade urgente de repensar a constituição ontológica do ser humano, como pressuposto imprescindível na reconstituição da relação entre o EU/TU ou EU/HUMANIDADE; a compreensão do Outro requer aceitar a pluralidade de perspectivas de novas vivências e formas relacionais; educar é vivenciar laços de sensibilidade social para construir comunidades nas quais a amizade seja o esteio político. Palavras-chaves: Educação. Ética. Comunidade. Amizade THE FORMATION OF CONTEMPORARY ETHOS: CHALLENGES TO EDUCATION: The study is qualitative and search for theoretical frameworks. Contemplates an open reflexion about the dawn of reason in modernity, which throws the human to the center of the universe . fragments forming the ethos vortex that reaches the contemporary. Following brings theoretical supports, based mainly on the concepts of Agamben, Bauman, Vaz, among others, about the possibility of building further training of the human community. Despite numerous and diverse knowledge produced, in actuality, there are growing difficulties in understanding the human condition and the meaning of human doings. The focus on subjectivity as an essence, brand of metaphysics, also calls into question the personal identity. The broad and complex contemporary calls for educational contexts capable of stimulating ontological bases with different conceptual and experiential bases. Thus the problem of guiding reflection is: how was the formation of the contemporary ethos and what are the challenges for education? The results highlight that the scientific-technical rationality generated unconcern with the experience of ethical values ​​directed to the common good; individualization fostered disbelief in democratic politics and claimed the hope of forming a contemporary ethos supportive. We conclude that there is an urgent need to rethink the ontological constitution of the human being as an indispensable prerequisite to rebuild the relationship between I / YOU or I / HUMANITY; understanding of the Other requires accepting the plurality of perspectives and experiences of new relational forms; educate social ties is experiencing sensitivity to build communities in which friendship is political mainstay . Keywords: Education. Ethics. Community. Friendship. (shrink)

Many mathematicians have cited depth as an important value in their research. However, there is no single widely accepted account of mathematical depth. This article is an attempt to bridge this gap. The strategy is to begin with a discussion of Szemerédi's theorem, which says that each subset of the natural numbers that is sufficiently dense contains an arithmetical progression of arbitrary length. This theorem has been judged deep by many mathematicians, and so makes for a good case on which (...) to focus in analyzing mathematical depth. After introducing the theorem, four accounts of mathematical depth will be considered. (shrink)

Words that describe sensory perception give insight into how language mediates human experience, and the acquisition of these words is one way to examine how we learn to categorize and communicate sensation. We examine the differential predictions of the typological prevalence hypothesis and embodiment hypothesis regarding the acquisition of perception verbs. Studies 1 and 2 examine the acquisition trajectories of perception verbs across 12 languages using parent questionnaire responses, while Study 3 examines their relative frequencies in English corpus data. We (...) find the vision verbs see and look are acquired first, consistent with the typological prevalence hypothesis. However, for children at 12–23 months, touch—not audition—verbs take precedence in terms of their age of acquisition, frequency in child‐produced speech, and frequency in child‐directed speech, consistent with the embodiment hypothesis. Later at 24–35 months old, frequency rates are observably different and audition begins to align with what has previously been reported in adult English data. It seems the initial orientation to verbalizing touch over audition in child–caregiver interaction is especially related to the control of physically and socially appropriate behaviors. Taken together, the results indicate children's acquisition of perception verbs arises from the complex interplay of embodiment, language‐specific input, and child‐directed socialization routines. (shrink)

The project of this Précis de philosophie de la logique et des mathématiques (vol. 1 under the direction of F. Poggiolesi and P. Wagner, vol. 2 under the direction of A. Arana and M. Panza) aims to offer a rich, systematic and clear introduction to the main contemporary debates in the philosophy of mathematics and logic. The two volumes bring together the contributions of thirty researchers (twelve for the philosophy of logic and eighteen for the philosophy of mathematics), specialists (...) in the history or philosophy of logic or mathematics. Each volume consists of ten chapters; each chapter is about forty pages, is independent from the others and deals with a particular philosophical question about logic or mathematics. The objective is to offer to the French-speaking reader a reference book. On many of the issues addressed, this Précis offers the first clear and thorough synthesis that is available in French. -/- This book is intended for third year / master's students, but also for teachers of philosophy in secondary or higher education, for logicians and mathematicians willing to take a philosophical look at the fundamental concepts of their discipline, and for any reader interested in the basic issues discussed in the philosophy of logic and mathematics. Each chapter is self-contained, although some basic training in logic (or in mathematics, as the case may be) is required. The book should be looked at as a comprehensive framework for contemporary discussions in the philosophy of logic and mathematics, while also giving some guidance on the disciplinary content required by these discussions. -/- The first three chapters of the present volume (on the philosophy of mathematics) are devoted to the history of the philosophy of mathematics: from antiquity to the modern era, and from this period to the foundational crisis of the nineteenth century, to the twentieth century. Next, four chapters deal with the crucial questions for the philosophy of mathematics of the twentieth century: the opposition and/or comparison of set theory and category theory as foundational frameworks for mathematics; mathematical constructivism; the analysis of calculability; and Benaceraff's problem. The two following chapters are focused on the philosophy of mathematical practice, by treating the notion of ideals of proof, in particular explanation and purity, and the notion of informal proof and the use of visual artifacts in mathematical argumentation. Finally the last chapter deals with the applicability of mathematics, including the role of probability. -/- In Chapter 1, Sébastien Maronne and David Rabouin present the history of the philosophy of mathematics from antiquity to the modern era. The goal is to show that the philosophy of mathematics has a history as ancient as mathematics and philosophy itself, a history largely continuous with the latter, often by staying out of sync with respect to the former. In Chapter 2, Sébastien Gandon discusses the question of the foundations of mathematics from Kant to the end of the nineteenth century. By tying this question to Kant, he is able to show that the discussion of the foundations of mathematics that occupied a large part of these two centuries has much older roots. Accent is placed on the differences between the framework proposed by Kant for making sense of classical mathematics and the use made by Frege and then Russell of logical tools for giving a different reading of arithmetic and real analysis. The link between between these different conceptions and the evolution of mathematics itself is also underlined. -/- Chapter 3, written by Hourya Benis-Sinaceur and Mirna Džamonja, deals with the evolution of mathematics in the twentieth century, notably the structuralist turn of Dedekind, E. Noether, and Bourbaki, and certain more recent developments. They devote themselves, among other things, to explaining how these developments can be understood as responses to and extensions of questions posed during the foundational debate of the immediately preceding era. -/- In Chapter 4, Jean-Pierre Marquis and Jean-Jacques Szczeciniarz discuss the different foundational options coming from set theory and category theory. They discuss, among other things, Lawvere's program to replace ZF with an axiomatization of the category of sets, the latter judged better adapted to the needs of contemporary mathematics. Different reactions, as much philosophical as technical, raised by this program are also taken into account. -/- Constructive mathematics are the object of Chapter 5, in which Gerhard Heinzmann and Mark van Atten examine in a critical and comparative manner a variety of constructivisms, including intuitionism, constructive type theory, predicativism, and finitism. -/- In Chapter 6, Guido Gherardi and Maël Pegny present computability theory, stressing its philosophical consequences. They discuss the basic concepts of this theory, for the computability of both the natural numbers and the real numbers. They then tackle the theory of computational complexity. -/- In Chapter 7, Andrea Sereni and Fabrice Pataut present Benacerraf's problem, opposing every possible account of mathematical knowledge to the availability of a theory of truth founded on an abstract ontology of mathematical objects. They give an overview of the original problem and its reformulation by Hartry Field, and reconstruct the debate that this problem has raised, by meeting at once the major opposing positions today of the analytic affiliation of the philosophy of mathematics, as much those on the platonist side as on that of nominalism. -/- In Chapter 8, Valeria Giardino and Yacin Hamami treat certain crucial aspects of the philosophy of mathematical practice. They deal in particular with the notion of an informal proof and on the role of artifacts in mathematical reasoning. They consider how such proofs, making use of these means, can contribute to the understanding of theorems and mathematical theories, thanks also to the role of visualisation and the aid of computer tools. -/- Why do mathematics often give several proofs of the same theorem? This is the question that Andrew Arana raises in Chapter 9, introducing the notion of an epistemic ideal and discuss two such ideals, the explanatoriness and purity of proof. -/- Chapter 10 is devoted to the applicability of mathematics in the study of empirical phenomena. After summarizing the history of this question, going back to Plato and Aristotle and passing by Mill and Kant, Daniele Molinini and Marco Panza reconstruct the contemporary debate, in relation, among other things, to questions raised in the philosophy of science. -/- Two appendices complete the volume. In the first Frédéric Patras treats the French tradition in philosophy of mathematics of the 20th century. In the second, Maria Carla Galavotti discusses the notion of probability and its use in science. (shrink)

Over the years many mathematicians have voiced a preference for proofs that stay “close” to the statements being proved, avoiding “foreign”, “extraneous”, or “remote” considerations. Such proofs have come to be known as “pure”. Purity issues have arisen repeatedly in the practice of arithmetic; a famous instance is the question of complex-analytic considerations in the proof of the prime number theorem. This article surveys several such issues, and discusses ways in which logical considerations shed light on these issues.

El texto presenta la perspectiva de la filósofa y científica Vinciane Despret a partir de cuatro tópicos que se intersecan y atraviesan diferencialmente todas sus investigaciones: heredar, interesar, confiar, sostener. Esta delimitación permite acentuar distintas modulaciones del ethos investigativo que nos propone y amplificar la originalidad de su apuesta al hacerla dialogar con otras perspectivas filosóficas que resultan afines a ella y no son explicitadas por la autora (Spinoza, Foucault). Considero que la perspectiva de Despret condensa de manera ejemplar las (...) principales líneas de un ethos de investigación que despierta el deseo de sostenerlo saliendo de las encrucijadas actuales donde normalmente se lo aplasta o desprecia. En la introducción aclaro por qué me centro en la perspectiva de esta autora en particular, y en las consideraciones finales sugiero un enlace con mi propia propuesta de investigación que vengo desarrollando hace tiempo. (shrink)

La filosofía y el filosofar adquieren en la vida de Juan Bautista Ferro lasdimensiones propias de un auténtico pensar que se reflejó en cada acción,en el discurrir diario de la vida académica universitaria o en las "tertuliasrnonologantes" que el propio Ferro concertaba con sus amigos en su casao en plena calle. Ferro fue el ejemplo vivo de una tradición filosófica quecentra su acción sobre el propio filósofo, y el cultivo de la "oralidad" corno"método" para enseñar a aprender. El texto muestra (...) todo esto a través deuna reconstrucción testimonial. (shrink)

C’est grâce à son essai Politische Romantik, publié en 1919, que Carl Schmitt fait une entrée remarquée sur la scène intellectuelle allemande. L’ouvrage se présente comme une charge systématique et radicale contre la tradition allemande du « romantisme politique », et fut vivement discuté dans les années qui suivirent sa publication. Mais aujourd’hui il se trouve relégué parmi les œuvres de jeunesse de Schmitt et reste rarement étudié par la recherche, qui le lit au mieux comme une belle contribution à (...) l’histoire culturelle du xixe siècle.Pourtant, une étude approfondie de l’essai, ainsi que du contexte historique dans lequel il s’insère, fait apparaître que ce n’est pas dans une telle perspective que Politische Romantik manifeste toute sa portée : bien plus qu’une étude historique, cet essai constitue un élément de réponse à la crise du conservatisme allemand tel qu’il apparaît dans l’après-guerre. Le rejet du « romantisme politique » marque la rupture de Schmitt avec l’idéal « apolitique » des intellectuels allemands de l’époque, et par là constitue le premier moment du projet schmittien, tel qu’il se verra développé de manière croisée dans son étude sur le catholicisme et sa Théologie politique. (shrink)

En este escrito realizo en nombre propio un punto de pasaje y conjunción entre Foucault y Spino-za. Tomo en función de ello a Lacan como “mediador evanescente” y produzco un desplazamiento en el plano teórico habilitado por la urgencia de la coyuntura política, signada esta por el neolibe-ralismo y su posible dislocación, para engarzar el planteo crítico a la ética y a una necesaria refor-ma del entendimiento que active los afectos. Motiva y avala esta composición conceptual un trabajo teórico y (...) práctico sostenido que ha venido desplegándose sistemáticamente en torno al pensamien-to llamado posfundacional. (shrink)

Sound and listening are intrinsically linked to how we experience and engage with places and communities. This guide invites landscape architects and urban designers to become soundscape architects and offers practical advice on sound and listening applicable to each stage of a design project: from reading the environment to intervening on it. This book foregrounds listening as an affective mediator between subjects and multispecies environments, and a vehicle to think and conceptualise environmental design beyond prevailing visual and human-centred modes. The (...) guide expands landscape architects' and urban designers' tools and skills to assess existing soundscapes, predict how those soundscapes will be altered through their designs, consider sound as a creative and active part of the design process, and envisage how users might perceive and be affected by those soundscapes as they evolve in time. The volume sits in the interface of research and practice, and incorporates theoretical, methodological and creative contributions from acoustic ecology, ecoacoustics, bioacoustics and sound art. Each of the design stages is illustrated through project examples that demonstrate the many advantages of incorporating attentive listening and sound into Landscape Architecture and Urban Design Practice. The book shows how incorporating listening and sounding as part of the design process promotes slow and subtle ways of practice, adds social and ecological value through the reduction of noise pollution and by monitoring the health of habitats, and enables the design of soundscapes that complement the character and design intent of a scheme and elicit joy and wonder. The book will be of interest to landscape architects, together with other design professionals such as urban designers, architects, geographers and engineers, who play a primary role in the composition of the soundscape. (shrink)

Why do mathematics often give several proofs of the same theorem? This is the question raised in this article, introducing the notion of an epistemic ideal and discussing two such ideals, the explanatoriness and purity of proof.

Many mathematicians have sought ‘pure’ proofs of theorems. There are different takes on what a ‘pure’ proof is, though, and it’s important to be clear on their differences, because they can easily be conflated. In this paper I want to distinguish between two of them.

Throughout history, mathematicians have expressed preference for solutions to problems that avoid introducing concepts that are in one sense or another “foreign” or “alien” to the problem under investigation. This preference for “purity” (which German writers commonly referred to as “methoden Reinheit”) has taken various forms. It has also been persistent. This notwithstanding, it has not been analyzed at even a basic philosophical level. In this paper we give a basic analysis of one conception of purity—what we call topical purity—and (...) discuss its epistemological significance. (shrink)

This article will consider imagination in mathematics from a historical point of view, noting the key moments in its conception during the ancient, modern and contemporary eras.

Este artículo es parte de un trabajo de investigación bastante más amplio, que llevó casi cinco años de duración, en el cual nos propusimos una lectura rigurosa como singular del concepto de sujeto en Alain Badiou y Jacques Lacan. En vistas a delimitar claramente el tema de estudio, la pregunta que nos formulamos como punto de partida fue la siguiente: ¿qué lugar ocupa el concepto de sujeto en la articulación lógica del pensamento de ambos autores, particularmente en sus dimensiones ontológicas (...) y políticas? Como se podrá apreciar a continuación, dicho trabajo despliega una topología conceptual que a su vez presenta ciertos hitos históricos. Pues es sabido que uno de los temas centrales en torno al cual ha girado la fi losofía política contemporánea ha sido la cuestión del sujeto, y así también, correlativamente, de los saberes y verdades que lo fundamentan. Se ha interrogado por ejemplo ¿cuál puede ser el lugar de este sujeto? ¿Es posible especifi car a priori su existencia? ¿De dónde deriva su estatuto? ¿Se trata de una función trascendental o sustancial, vacía o refl exiva? Estas preguntas responden, en cierta forma, a una concepción de sujeto más bien propia de la metafísica moderna que intentaremos desplazar. (shrink)

RESUMEN En este texto argumento que la filosofía de Alain Badiou, al contrario de lo que se sostiene habitualmente, no es una ontologia matemática, ni un platonismo recidivo, ni un comunismo de la Idea, ni siquiera una teoría poscartesiana o poslacaniana del sujeto. En verdad, es todo eso junto y aún más. O lo que es lo mismo, es un montaje; un gran montaje o una gran ficción compuesta de heteróclitos fragmentos discursivos que se suplementan entre sí. La operación filosófica (...) pasa desapercibida entre ellos, aunque guarda su especificidad. ABSTRACT In this text, I will argue that the philosophy of Alain Badiou, contrary to what many believe, is not a mathematical ontology, nor a recurrence Platonism, neither communism of the Idea, and not even a post-Cartesian or post-lacanian theory of the subject. Is all together and more. Or what is the same thing: a montage. A great montage or a great fiction composed of diverse discursive fragments that supplement each other. (shrink)

An important issue for visual argumentation is its relationship to propositions, since it has been argued that, in order to be arguments, images should be propositional. The first part of the paper will approach this debate from a theoretical perspective. After quickly surveying the field on the issue, I will address the relationship between images and propositions. Three specific questions will be examined: can propositions accurately account for the way images express arguments?; are verbal propositions necessary to reconstruct arguments that (...) images alone cannot convey, due to their lacking linguistic tools?; are images essentially non-propositional because they don’t have truth-value? The second part of the paper will include a detailed analysis of two posters. From these analyses, I will ultimately conclude that some images can display a visual argument without necessarily being propositional. (shrink)