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.

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)

An algebraic interpretation of multigraph networks is introduced in relation to conscious experience, brain and body. These multigraphs have the ability to merge by an associative binary operator \(\odot \), accounting for biological composition. We also study a mathematical formulation of splitting layers, resulting in a formal analysis of the transition from conscious to non-conscious activity. From this construction, we recover core structures for conscious experience, dynamical content and causal constraints that conscious interactions may impose. An important result is the (...) prediction of structural topological changes after conscious interactions. These results may inspire further use of formal mathematics to describe and predict new features of conscious experience while aligning well with formal tries to mathematize phenomenology, phenomenological tradition and applications to artificial consciousness. (shrink)

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)

Facultés, concepts, théories et modules innés... Le débat philosophique contemporain sur la structure naturelle de l'esprit multiplie les références aux idées innées. Noam Chomsky lui-même, défenseur d'une faculté innée de langage, s'inscrit dans la lignée de la philosophie moderne. Mais les idées innées contemporaines ont-elles le même sens que celles posées par Descartes et Leibniz? Les progrès des sciences biologiques et cognitives permettent-ils d'attester ou d'infirmer l'existence des idées innées? L'étude analyse les théories modernes et contemporaines des idées innées puis (...) propose une définition de l'innéisme qui démontre l'actualité du débat amorcé à l'âge classique entre innéisme et empirisme."--Page 4 of cover. (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)

We compare two different elicitation methods for measuring risk attitudes on a sample of French farmers. We consider the lottery tasks initially proposed by Holt and Laury (Econ Rev 92:1644–1655, 2002) and by Eckel and Grossman (Evol Hum Behav 23:281–295, 2002; J Econ Behav Org 68:1–7, 2008). The main empirical result from this within-subject study is that risk preference measures are affected by the type of mechanism used. We first show that this risk preference instability can be related to non-expected (...) utility preferences of farmers. Using a risk-taking psychometric questionnaire, we then demonstrate that risk preferences of farmers are context-dependent. This may be another explanation of the observed risk preference instability. (shrink)

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)

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)

La pregunta acerca de qué es un humano tiene una respuesta trivial: cualquier individuo perteneciente a la especie Homo sapiens. Pero si nos interrogamos por el sentido de lo que es un ser humano, cuáles son los rasgos distintivos respecto de otras especies, cómo evolucionaron y por qué, en qué medida los antecesores nuestros disponían de ellos y dónde y cuándo vivieron esos ancestros, entonces entramos en un terreno de difícil recorrido. En los últimos siete millones de años existieron varias (...) especies a las que cabe llamar «humanas» o, por lo menos, «homínidas». Es éste un repaso breve y muy sintético de lo que supuso tal panorama diverso y dinámico que conduciría, a la postre, a los humanos actuales. Nuestras propias características adquieren una nueva luz cuando se contemplan a lo largo del proceso de su filogénesis. (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.

In this paper we consider three potential simplifications to a result of Solovay’s concerning the Turing degrees of nonstandard models of arbitrary completions of first-order Peano Arithmetic (PA). Solovay characterized the degrees of nonstandard models of completions T of PA, showing that they are the degrees of sets X such that there is an enumeration R ≤T X of an “appropriate” Scott set and there is a family of functions (tn)n∈ω, ∆0 n(X) uniformly in n, such that lim tn(s) s→∞.

This study focuses on the differences in the perception of business ethics across two groups of management students from France and Romania (n = 220). Data was collected via the ATBEQ to measure preferences for three business philosophies: Machiavellianism, Social Darwinism, and Moral Objectivism. The results show that Romanian students present more favorable attitudes toward Machiavellianism than French students; whereas, French students valued Social Darwinism and Moral Objectivism more highly. For Machiavellianism and Moral Objectivism the results are consistent with the (...) literature and our hypotheses. However, contrary to our expectations, we find that Social Darwinism is more important in France than Romania. The results indicate that religious practice does not influence preferences for the three business philosophies. In terms of gender differences, women have less favorable attitudes toward Machiavellianism and more favorable attitudes toward Moral Objectivism than men. (shrink)

La psychologie des peuples a connu un grand succès dans les années 1930 dans les milieux les plus divers: sur les bancs de l'Université, tant des facultés de lettres que de celles de médecine, comme dans le monde colonial. Elle demeure imprégnée des postulats raciologiques du siècle précédent: hérédité raciale, influence du milieu géographique et du climat dans la formation des caractères nationaux, représentation inégalitaire de la différence et ses usages politiques ont été manifestes dans le domaine de la politique (...) coloniale. Les enseignements de la psychologie ethnique ont, en effet, eu des applications directes dans la politique scolaire à travers la mise en place d'un enseignement rudimentaire et à tendance ségrégationniste qui a réduit considérablement les ambitions de la mission civilisatrice. (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.

Why do we have social norms—of fairness, cooperation, trust, property, or gender? Modern-day Humeans, as I call them, believe these norms are best accounted for in cultural evolutionary terms, as adaptive solutions to recurrent problems of social interaction. In this paper, I discuss a challenge to this “Humean Program.” Social norms involve widespread behaviors, but also distinctive psychological attitudes and dispositions. According to the challenge, Humean accounts of norms leave their psychological side unexplained. They explain, say, why we share equally, (...) but not why we disapprove of those who don’t. I defend the Humean Program against this challenge. In particular, I suggest an idea for how to extend the Program to account for the psychological side of norms. Socially adaptive behaviors aren’t just likely to emerge in a group; They are also likely to be widely taught within it. The transmission of these behaviors through instruction explains why they are associated with distinctive normative attitudes and dispositions. These attitudes play a pedagogical role in helping transmit these behaviors to children and newcomers. (shrink)

Resumen En la siguiente discusión presento algunas objeciones al artículo de Federico Raffo Quintana “La noción de ‘espacio’ en los escritos juveniles de Leibniz”.1 Contra la interpretación de Raffo, quien considera que la concepción del espacio como lugar universal de las cosas es una idea que Leibniz abandona de manera muy temprana -según el autor en 1671-, intento mostrar que esta concepción trasciende sin duda el periodo de los escritos juveniles de Leibniz y que ciertas confusiones conceptuales en la lectura (...) de Raffo hacen que, en vez de ver una armónica continuidad en la teoría del espacio de Leibniz, vea en ella una ruptura en su evolución histórica, a la que, además, asocia algunas dicotomías discutibles o espurias.In this paper I introduce objections to Federico Raffo Quintana’s paper “The Notion of ‘Space’ in the Young Leibniz Writings”. Against Raffo, who considers that the Leibnizian conception of space as the universal place of things is an idea that Leibniz early abandons -Raffo says in 1671-, I will try to show that such a conception goes beyond the Leibniz’s young writings and that it is because of some conceptual confusions in Raffo’s survey that he, instead of seeing a harmonic continuity in Leibniz’s theory of space, sees a rupture in its historical evolution, associating to it some disputable or spurious dichotomies. (shrink)

In this paper I begin by extending two results of Solovay; the first characterizes the possible Turing degrees of models of True Arithmetic (TA), the complete first-order theory of the standard model of PA, while the second characterizes the possible Turing degrees of arbitrary completions of P. I extend these two results to characterize the possible Turing degrees of m-diagrams of models of TA and of arbitrary complete extensions of PA. I next give a construction showing that the conditions Solovay (...) identified for his characterization of degrees of models of arbitrary completions of PA cannot be dropped (I showed that these conditions cannot be simplified in the paper. (shrink)

En 1933-1934, Simone Weil était professeur à Roanne. Dans un petit pavillon isolé au fond d'un parc, sur l'herbe ou sous la neige, elle enseigna alors la philosophie à quelques élèves. On découvrira ici, à travers les notes recueillies par une de ces jeunes filles, ce que furent ces leçons, véritables entretiens socratiques. Le lecteur y reconnaîtra l'unité profonde de Simone Weil dont la vie brève fut intensément consacrée à la recherche de la vérité.

Une preuve est pure si, en gros, elle ne réfère dans son développement qu’à ce qui est « proche » de, ou « intrinsèque » à l’énoncé à prouver. L’infinité des nombres premiers, un théorème classique de l’arithmétique, est un cas d’étude particulièrement riche pour les recherches philosophiques sur la pureté. Deux preuves différentes de ce résultat sont ici considérées, à savoir la preuve euclidienne classique et une preuve « topologique » plus récente proposée par Furstenberg. D’un point de vue (...) naïf, il semblerait que la première soit pure et la seconde impure. Des objections à cette vue naïve sont ici considérées et réfutées. Concernant la preuve euclidienne, la question relève de la logique, notamment de la définissabilité arithmétique de l’addition en termes de successeur et de divisibilité telle que démontrée par Julia Robinson, tandis qu’en ce qui concerne la preuve topologique, la question relève de la sémantique, notamment pour tout ce qui touche au problème de savoir ce qui est « inclus » dans le contenu d’énoncés particuliers.A proof is pure, roughly, if it draws only on what is « close » or « intrinsic » to the statement being proved. The infinitude of prime numbers, a classical theorem of arithmetic, is a rich case study for philosophical investigation of purity. Two different proofs of this result are considered, namely the classical Euclidean proof and a more recent « topological » proof by Furstenberg. Naively the former would seem to be pure and the latter to be impure. Objections to these naive views are considered and met. In the case of the former the issue rests on logical matters, specifically the arithmetic definability of addition in terms of successor and divisibility shown by Julia Robinson, while in the case of the latter the issue rests on semantic matters, specifically with respect to what is « contained » in the content of particular statements. (shrink)

Political sociologists have paid closer attention of late to the territoriality of political communities, and have even begun theorizing the theme of territoriality’s legitimation. To date, however, the field has mostly overlooked the topic of maps, the quintessential territorial tool. Thus, we know little regarding maps’ crucial role in shaping modern subjects’ relationship to territory. This article argues that “map-mindedness”—i.e., the effects of map imagery on how subjects experience territory—can be productively theorized by working through the social-scientific concept of “place.” (...) Using a range of modern and contemporary examples, I illustrate how maps can draw on and manipulate political subjects’ experience of place. Maps, I submit, allow political communities to render themselves more place-like, thus bridging the phenomenological distance between these abstract, territorially vast units and their “emplaced” subjects. More specifically, maps solve this “problem of distance” through three ideal-typical processes: 1) they render the political community as a proximate “object in the world”; 2) they present the political community as a body-like target for cathexis and identification; and 3) they mediate the traffic of meaning between the local and the national to produce a multi-scalar sense of place that can be harnessed in the service of the political community. Maps are a potent means of “re-personalizing” politics; their study suggests that territoriality is not only a form of “impersonal rule,” as recent works have observed, but always also implicated in the production of political subjects. (shrink)

En este artículo examinamos los aspectos más relevantes de la concepción del espacio de Leibniz. A través de un enfoque genético, nuestro propósito es mostrar que, en su desarrollo evolutivo, dicha concepción sufre dos transiciones destacables, las que separan y distinguen tres versiones teóricas. En su juventud, Leibniz defiende una concepción substancialista del espacio. Sin embargo, debido a la adopción del nominalismo, dicha concepción sufre un primer giro que decanta en un monismo substancialoide, el que se consolida en el período (...) medio de la filosofía de Leibniz. Ya en el período tardío, Leibniz reformula, en una segunda transición, su concepción del espacio, la que adopta la forma definitiva de -lo que llamamos- un relacionismo substancialoide. (shrink)

Blanco, Carlos. Historia de la neurociencia: el conocimiento del cerebro y la mente desde una perspectiva inter-disciplinar. Madrid: Biblioteca Nueva, 2014. 296 pp.

For more than a century, Harlem has been the epicenter of black America, the celebrated heart of African American life and culture—but it has also been a byword for the problems that have long plagued inner-city neighborhoods: poverty, crime, violence, disinvestment, and decay. Photographer Camilo José Vergara has been chronicling the neighborhood for forty-three years, and Harlem: The Unmaking of a Ghetto is an unprecedented record of urban change. Vergara began his documentation of Harlem in the tradition of such (...) masters as Helen Levitt and Aaron Siskind, and he later turned his focus on the neighborhood’s urban fabric, both the buildings that compose it and the life and culture embedded in them. By repeatedly returning to the same locations over the course of decades, Vergara is able to show us a community that is constantly changing—some areas declining, as longtime businesses give way to empty storefronts, graffiti, and garbage, while other areas gentrify, with corporate chain stores coming in to compete with the mom-and-pops. He also captures the ever-present street life of this densely populated neighborhood, from stoop gatherings to graffiti murals memorializing dead rappers to impersonators honoring Michael Jackson in front of the Apollo, as well as the growth of tourism and racial integration. Woven throughout the images is Vergara’s own account of his project and his experience of living and working in Harlem. Taken together, his unforgettable words and images tell the story of how Harlem and its residents navigated the segregation, dereliction and slow recovery of the closing years of the twentieth century and the boom and racial integration of the twenty-first century. A deeply personal investigation, Harlem will take its place with the best portrayals of urban life. (shrink)

Gaisi Takeuti extended Gentzen's work to higher-order case in 1950's–1960's and proved the consistency of impredicative subsystems of analysis. He has been chiefly known as a successor of Hilbert's school, but we pointed out in the previous paper that Takeuti's aimed to investigate the relationships between "minds" by carrying out his proof-theoretic project rather than proving the "reliability" of such impredicative subsystems of analysis. Moreover, as briefly explained there, his philosophical ideas can be traced back to Nishida's philosophy in Kyoto's (...) school. For the proving the consistency of such systems, it is crucial to prove the well-foundedness of ordinals called "ordinal diagrams" developed for it. Takeuti presented such arguments several times in order to show that they are admitted in his stand point. As a starting point of investigating his finitist stand point, we formulate the system of ordinal notations up to ε0 and reconstruct the well-foundedness arguments of them. (shrink)

O objetivo principal deste trabalho é apresentar a distinção entre ciência e cientificismo e, com base no trabalho de van Fraassen, intitulado A imagem científica (2006), discutir sobre as circunstâncias em que o cientificismo poderia ser repudiado. O cientificismo é uma corrente de pensamento que somente considera válido um conhecimento se ele for científico. Segundo essa corrente, os procedimentos da ciência natural seriam mais especiais, uma vez que, dentre outros motivos, eles são capazes de descrever regularidades e possibilitar predições. Dessa (...) forma, o cientificismo defende que os procedimentos não científicos não possuem valor algum, pois não seriam capazes de descrever regularidades e realizar predições como fazem os procedimentos da ciência natural. Trataremos de maneira crítica essa concepção, apresentando o argumento de van Fraassen de que a descrição de uma regularidade também pode ser construída por explicações não científicas. Em sua opinião, o poder explicativo de uma teoria vai além de sua importância empírica, ele é radicalmente dependente do contexto. Isso significa que as regularidades descritas por uma crença científica são também dependentes de explicações não científicas, como as explicações da psicologia, da história, da filosofia, da sociologia, etc. Assim, a forte predição que encontramos na ciência natural não é realizada somente com base na adequação empírica e científica, mas com base em um sistema de explicações não científicas. Isso torna incorreta a classificação do conhecimento científico como o mais verdadeiro ou especial por ser capaz de descrever regularidades e realizar a predições. Pretende-se, com isso, refletir se o conhecimento pode ser construído por outras áreas não científicas, sem que o conhecimento seja com isso prejudicado. Uma reflexão necessária que servirá para apresentar um dos problemas que tornam o cientificismo indesejável. (shrink)

Comunicado del Claustro de Profesores del Programa de Filosofía - Universidad del Quindío Llamado al diálogo, cuidado de la vida y solidaridad por las víctimas de las protestas sociales iniciadas el 28 de abril de 2021 en Colombia La protesta social expresa los deseos de un pueblo por salir de las situaciones políticas y económicas desfavorables, es una expresión de esperanza. Precisamente, las distintas acciones ciudadanas en la mayoría de los municipios quindianos, y en general, en el territorio colombiano, son (...) la muestra de un profundo deseo por aportar a la construcción de un país donde la vida en todas sus manifestaciones sea respetada y valorada. (shrink)

This paper aims to bring together the study of normative judgments in mathematics as studied by the philosophy of mathematics and verbal hygiene as studied by sociolinguistics. Verbal hygiene (Cameron 1995) refers to the set of normative ideas that language users have about which linguistic practices should be preferred, and the ways in which they go about encouraging or forcing others to adopt their preference. We introduce the notion of mathematical hygiene, which we define in a parallel way as the (...) set of normative discourses regulating mathematical practices and the ways in which mathematicians promote those practices. To clarify our proposal, we present two case studies from 17th century France. First, we exemplify a case of mathematical hygiene proper: Descartes' algebraic geometry and Newton's subsequent criticism of it, a case of (im)purity in mathematics. Then, we compare Descartes' and Newton's mathematical hygiene with verbal hygiene from this period, as exemplified by the work of the grammarian Claude Favre de Vaugelas (Ayres-Bennett 1987). We argue that these early modern normative discourses on mathematics and language respectively can be seen as emanating from a common socio-political program: the development of a new bourgeois intellectual class. We conclude that the study of mathematical hygiene has the potential to yield new understandings of the social aspects of mathematical practice, and that similarities between mathematical and verbal hygiene at certain time periods, such as 17th century France, open up a new area of inquiry at the borders of linguistics and the philosophy of mathematics. (shrink)

La flexibilidad y la mercantilización de lo espiritual ¿son aspectos característicos de la cultura contemporánea? Centrado en los casos de Alemania, Inglaterra y Francia, el artículo detalla cómo el romanticismo literario se articula como un movimiento pionero de la espiritualidad flexible, proceso que se entrelaza con su rol seminal en la formación del campo literario moderno. El crecimiento económico y demográfico sostenidos en Europa desde el s. XV, así como la alfabetización impulsada por el protestantismo, construyen un moderno público lector (...) que da pie —durante el s. XVIII— a la revolución y diversificación de la lectura, lo que promueve la difusión de incipientes tendencias modernas (ciencia, Ilustración, prensa). Oponiéndose parcialmente a estas tendencias y extendiendo la diversificación de lo impreso, surge la tendencia romántica. Sus autores flexibilizan la espiritualidad más allá de las convenciones religiosas de la época —y aportan a la formación del campo literario— (1) caracterizando el desencanto y otras formas de malestar moderno; (2) multiplicando el lenguaje imaginario-trascendente; (3) brindando connotaciones metafísicas al genio artístico; (4) promoviendo la autonomía y libertad expresiva del arte. La evidencia indica que el mercado fue una condición histórica básica para la EF. Se sugiere que esta flexibilidad se muestra como una adaptación de la espiritualidad a las circunstancias empíricas de la modernidad, la cual conlleva nuevas necesidades de sentido y una mayor capacidad para divergir de los relatos religiosos tradicionales. (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.