In a previous work, we have shown that the null result of the Michelson–Morley experiment in vacuum is deeply connected with the notion of time. The same is true for the postulate of constancy of the two-way speed of light in vacuum in all frames independently of the state of motion of the emitting body. The argumentation formerly given is very general and has to be true not only within Special Relativity and its “equivalence” of all inertial frames, but as (...) well as in Lorentz-Poincaré scenario of a preferred reference frame. This paper is the second of a trilogy intending to revisit the foundations of Special Relativity, and addresses the question of the constancy of the one-way speed of light and of the differences and similarities between both scenarios. Although they manifestly differ in philosophy, it is debated why and how the assumption of a “special system of reference experimentally inaccessible” is indeed compatible with Einstein’s Special Relativity, as beautifully outlined and discussed by Bell [Speakable and Unspeakable in Quantum Mechanics (Cambridge University Press, Cambridge, 1988)]. This rather trivial statement is still astonishing nowadays to a big majority of scientists. The purpose of this work is to bring such assertion into perspective, widening the somewhat narrow view of Special Relativity often presented in textbooks and scientific papers. (shrink)

El presente artículo ofrece una propuesta de intervención para la enseñanza de la Guerra Civil en el País Vasco a través del cómic para alumnos de segundo de Bachillerato en el aula de historia. Para ello, primero se examinan las contribuciones de varios autores que han valorado positivamente la utilidad didáctica de la novela gráfica en el aula de historia. Más concretamente, este artículo se hace eco de los últimos estudios publicados en los Estados Unidos sobre la utilidad (...) del cómic como recurso didáctico en las clases de historia. Posteriormente se diseña una propuesta didáctica de mejora para la enseñanza de la Guerra Civil en el País Vasco dirigida a alumnos de segundo de Bachillerato para la asignatura de Historia de España. Se propone utilizar el cómic Tristísima ceniza (2011) de Mikel Begoña e Iñaket como medio para ayudar a los alumnos a construir las capacidades de identificar, contextualizar y corroborar. (shrink)

En este artículo se ofrece una visión sucinta de los ochenta años (1918- 1998) de historia de la Real Academia de la Lengua Vasca, presididos siempre por el objetivo de la normalización de la lengua. En la primera parte del trabajo se presenta la historia de la Institución: fundación, los tiempos oscuros de la guerra y de la posguerra. En 1968 la normalización de la lengua recibirá un nuevo y definitivo impulso. En la segunda parte se hace un repaso (...) de la labor realizada hasta el presente en el campo de la Lexicografía, la Gramática, la Onomástica, la Dialectología y la Literatura. Finalmente, se explica el funcionamiento interno de la institución. (shrink)

En este artículo se analiza la labor docente de Jesús Galíndez Suárez en diferentes instituciones universitarias americanas. El objetivo es desmitificar a este político nacionalista vasco y delegado del Gobierno vasco en el exilio en Nueva York, cuya figura ha estado rodeada por la conspiración y el espionaje de la Guerra Fría debido a su misteriosa desaparición, y poner en valor una faceta habitualmente olvidada: su puesto de profesor en la Universidad de Columbia, una de las más (...) prestigiosas de Estados Unidos. En esta investigación se han utilizado fuentes inéditas de los archivos de la Universidad de Columbia y de la Universidad de Rutgers. (shrink)

El artículo analiza cualitativamente la relación entre infancia y violencia en el contexto de la guerra civil en El Salvador (1980-1992). La reflexión es vehiculada por el examen de la película Voces inocentes (2004), como una obra socialmente producida y como una representación de la realidad histórica de índole prospectiva. El trabajo destaca el mérito del filme al visibilizar y denunciar la responsabilidad de los sustentadores primarios de la violencia ‒aún hoy impune‒ hacia la infancia salvadoreña, cuyas secuelas psicosociales (...) constituyeron, a la postre, un entorno sociohistórico centroamericano signado por una debacle humanitaria que comenzó hace cuatro décadas, en un tránsito infausto “de la locura a la esperanza”. Para ello, he empleado el marco teórico conceptual sobre la violencia desarrollado por el jesuita vasco-salvadoreño, Ignacio Ellacuría, según el cual, las violencias resultantes solo pueden ser valoradas a partir de la crítica inicial a la violencia originaria de carácter estructural. (shrink)

En este articulo abordo la concepcion educativa del pensador espanol Miguel de Unamuno. haciendo especial hincapie en los principios e ideas que respecto a la eduaction aparecen en sus primeros articulos y en su promera novela, Paz en la guerra (la cual contiene muchas de las ideas que seran el tema central de Amor y pedagogia, su novela pedagogica, o mas bien antipedagogica, por exelencia. Hasta el momento, no se habian tenido presente estos escritos a la hora de analizar (...) su labor educativa y sus ideas pedagogicas, a pesar de que son de gran importancia a la hora de ver como se va fraguando su conception de la education u como esta va evolucionado. Considero que las ideas que el vasco mantuvo en el ambito educativo son todavia de actualidad y dignas de tener en cuenta. Su propuesta educativa encaja perfectamente y defiende los principios de educacion continua, integral y personalizada o indyvidualizada que ahora estan en boga. Por ello, es nescesario hacer una relectura de los escritos unamunianos desde la perspectiva educativa u ver que nos aportan para el momento presente. (shrink)

Ofrecemos dos cartas inéditas entre los filósofos Manuel García Morente y Xavier Zubiri. La primera carta, escrita por el pensador giennense, viene a iluminar el itinerario de su conversión religiosa que le llevó a ordenarse sacerdote en la Iglesia católica. Dicho proceso espiritual había empezado a fraguarse el año anterior en París, durante el exilio que ambos pensadores habían compartido en la capital francesa, tras el estallido de la guerra civil española. En la segunda carta, fechada casi dos años (...) y medio después de la anterior, Zubiri dirige al pensador andaluz unas palabras entrañables en respuesta a la invitación recibida de asistir a su primera misa; una carta en la que el filósofo vasco desvela la inquebrantable amistad entre ambos pensadores, intensificada en los últimos años por la fe cristiana. (shrink)

We discuss computability properties of the set of elements of best approximation of some point xX by elements of GX in computable Banach spaces X. It turns out that for a general closed set G, given by its distance function, we can only obtain negative information about as a closed set. In the case that G is finite-dimensional, one can compute negative information on as a compact set. This implies that one can compute the point in whenever it is uniquely (...) determined. This is also possible for a wider class of subsets G, given that one imposes additionally convexity properties on the space. If the Banach space X is computably uniformly convex and G is convex, then one can compute the uniquely determined point in . We also discuss representations of finite-dimensional subspaces of Banach spaces and we show that a basis representation contains the same information as the representation via distance functions enriched by the dimension. Finally, we study computability properties of the dimension and the codimension map and we show that for finite-dimensional spaces X the dimension is computable, given the distance function of the subspace. (shrink)

Normative theories of argumentation tend to assume that logical and dialectical rules suffice to ensure the rationality of argumentative discourse. Yet, in everyday debates people use arguments that seem valid in light of such rules but nonetheless biased and tendentious. This article seeks to show that the rationality of argumentation can only be fully promoted if we take into account its ethical dimension. To substantiate this claim, I review some of the empirical evidence indicating that people’s inferential reasoning is systematically (...) affected by a variety of biases and heuristics. Insofar as these cognitive illusions are typically unintentional, it appears that arguers may be biased despite their well-intended efforts to follow the rules of critical argumentation. Nevertheless, I argue that people remain responsible for the rationality of their arguments, given that there are a number of measures that they can (and ought to) take to avoid such distortions. I highlight the importance of argumentational virtues and critical thinking to rational debates, and describe a set of indirect strategies of “argumentative self-control”. (shrink)

In this article, we propose to question the accuracy of Jean Wahl’s definition of Vladimir Jankélévitch’s Philosophie Première (1953) as one of the “ultimate efflorescences of negative theology.” In fact, we maintain that—although conceiving the apophatic discourse as a necessary (but insufficient) stepping-stone to the absolute—the aforementioned work constitutes, if anything, an effort to overcome it by means of what we shall call a tesiphatic philosophy, i.e., one that identifies the first principle, not as the absolute negation of being, but (...) as the absolute position of being. To defend our thesis, we shall start by discussing the meaning of the five divine names that Jankélévitch bestows upon his absolute, namely, (1) the “totally-other-order,” (2) the “super-truth,” (3) “the itself,” (4) the “I-know-not-what,” and (5) the “almost-nothing.” In this section of our paper, we will try to show how Jankélévitch’s discourse is, in fact, shaped by the whole negative theology tradition (from Plotinus to Nicholas of Cusa). Finally, in the last part of our paper, we will strive to reflect upon Jankélévitch’s interpretation of the limits of a apophatic metaphysical discourse, questioning whether his Philosophie Première effectively overcomes negative theology’s conception of the absolute. (shrink)

We systematically study the completion of choice problems in the Weihrauch lattice. Choice problems play a pivotal rôle in Weihrauch complexity. For one, they can be used as landmarks that characterize important equivalences classes in the Weihrauch lattice. On the other hand, choice problems also characterize several natural classes of computable problems, such as finite mind change computable problems, non-deterministically computable problems, Las Vegas computable problems and effectively Borel measurable functions. The closure operator of completion generates the concept of total (...) Weihrauch reducibility, which is a variant of Weihrauch reducibility with total realizers. Logically speaking, the completion of a problem is a version of the problem that is independent of its premise. Hence, studying the completion of choice problems allows us to study simultaneously choice problems in the total Weihrauch lattice, as well as the question which choice problems can be made independent of their premises in the usual Weihrauch lattice. The outcome shows that many important choice problems that are related to compact spaces are complete, whereas choice problems for unbounded spaces or closed sets of positive measure are typically not complete. (shrink)

In this paper we study a new approach to classify mathematical theorems according to their computational content. Basically, we are asking the question which theorems can be continuously or computably transferred into each other? For this purpose theorems are considered via their realizers which are operations with certain input and output data. The technical tool to express continuous or computable relations between such operations is Weihrauch reducibility and the partially ordered degree structure induced by it. We have identified certain choice (...) principles such as co-finite choice, discrete choice, interval choice, compact choice and closed choice, which are cornerstones among Weihrauch degrees and it turns out that certain core theorems in analysis can be classified naturally in this structure. In particular, we study theorems such as the Intermediate Value Theorem, the Baire Category Theorem, the Banach Inverse Mapping Theorem, the Closed Graph Theorem and the Uniform Boundedness Theorem. We also explore how existing classifications of the Hahn—Banach Theorem and Weak Kőnig's Lemma fit into this picture. Well-known omniscience principles from constructive mathematics such as LPO and LLPO can also naturally be considered as Weihrauch degrees and they play an important role in our classification. Based on this we compare the results of our classification with existing classifications in constructive and reverse mathematics and we claim that in a certain sense our classification is finer and sheds some new light on the computational content of the respective theorems. Our classification scheme does not require any particular logical framework or axiomatic setting, but it can be carried out in the framework of classical mathematics using tools of topology, computability theory and computable analysis. We develop a number of separation techniques based on a new parallelization principle, on certain invariance properties of Weihrauch reducibility, on the Low Basis Theorem of Jockusch and Soare and based on the Baire Category Theorem. Finally, we present a number of metatheorems that allow to derive upper bounds for the classification of the Weihrauch degree of many theorems and we discuss the Brouwer Fixed Point Theorem as an example. (shrink)

In this paper we study a reducibility that has been introduced by Klaus Weihrauch or, more precisely, a natural extension for multi-valued functions on represented spaces. We call the corresponding equivalence classes Weihrauch degrees and we show that the corresponding partial order induces a lower semi-lattice. It turns out that parallelization is a closure operator for this semi-lattice and that the parallelized Weihrauch degrees even form a lattice into which the Medvedev lattice and the Turing degrees can be embedded. The (...) importance of Weihrauch degrees is based on the fact that multi-valued functions on represented spaces can be considered as realizers of mathematical theorems in a very natural way and studying the Weihrauch reductions between theorems in this sense means to ask which theorems can be transformed continuously or computably into each other. As crucial corner points of this classification scheme the limited principle of omniscience LPO, the lesser limited principle of omniscience LLPO and their parallelizations are studied. It is proved that parallelized LLPO is equivalent to Weak Kőnig's Lemma and hence to the Hahn—Banach Theorem in this new and very strong sense. We call a multi-valued function weakly computable if it is reducible to the Weihrauch degree of parallelized LLPO and we present a new proof, based on a computational version of Kleene's ternary logic, that the class of weakly computable operations is closed under composition. Moreover, weakly computable operations on computable metric spaces are characterized as operations that admit upper semi-computable compact-valued selectors and it is proved that any single-valued weakly computable operation is already computable in the ordinary sense. (shrink)

The investigation of computational properties of discontinuous functions is an important concern in computable analysis. One method to deal with this subject is to consider effective variants of Borel measurable functions. We introduce such a notion of Borel computability for single-valued as well as for multi-valued functions by a direct effectivization of the classical definition. On Baire space the finite levels of the resulting hierarchy of functions can be characterized using a notion of reducibility for functions and corresponding complete functions. (...) We use this classification and an effective version of a Selection Theorem of Bhattacharya-Srivastava in order to prove a generalization of the Representation Theorem of Kreitz-Weihrauch for Borel measurable functions on computable metric spaces: such functions are Borel measurable on a certain finite level, if and only if they admit a realizer on Baire space of the same quality. This Representation Theorem enables us to introduce a realizer reducibility for functions on metric spaces and we can extend the completeness result to this reducibility. Besides being very useful by itself, this reducibility leads to a new and effective proof of the Banach-Hausdorff-Lebesgue Theorem which connects Borel measurable functions with the Baire functions. Hence, for certain metric spaces the class of Borel computable functions on a certain level is exactly the class of functions which can be expressed as a limit of a pointwise convergent and computable sequence of functions of the next lower level. (shrink)

Matthias Schröder has asked the question whether there is a weakest discontinuous problem in the topological version of the Weihrauch lattice. Such a problem can be considered as the weakest unsolvable problem. We introduce the discontinuity problem, and we show that it is reducible exactly to the effectively discontinuous problems, defined in a suitable way. However, in which sense this answers Schröder’s question sensitively depends on the axiomatic framework that is chosen, and it is a positive answer if we work (...) in Zermelo–Fraenkel set theory with dependent choice and the axiom of determinacy $\mathsf {AD}$. On the other hand, using the full axiom of choice, one can construct problems which are discontinuous, but not effectively so. Hence, the exact situation at the bottom of the Weihrauch lattice sensitively depends on the axiomatic setting that we choose. We prove our result using a variant of Wadge games for mathematical problems. While the existence of a winning strategy for Player II characterizes continuity of the problem (as already shown by Nobrega and Pauly), the existence of a winning strategy for Player I characterizes effective discontinuity of the problem. By Weihrauch determinacy we understand the condition that every problem is either continuous or effectively discontinuous. This notion of determinacy is a fairly strong notion, as it is not only implied by the axiom of determinacy $\mathsf {AD}$, but it also implies Wadge determinacy. We close with a brief discussion of generalized notions of productivity. (shrink)

As palavras que se seguem constituem uma tentativa de explorar os diversos sentidos que revestem a figura do «com» [mit], tal como surge configurada ao longo dos §§ 25–27 de Sein und Zeit 2 . Escutaremos atentamente aquilo que Heidegger tem a dizer a respeito da necessária determinação do ser-no-mundo [in-der-Welt-sein] como ser-com [Mitsein] e veremos, depois, em que medida a sua analítica existencial faz (ou não) depender a constituição do ser-si-mesmo [Selbstsein] da figura do outro. Por último, e em (...) jeito de conclusão, tentaremos mostrar como Heidegger articula a interrogação pela ipseidade ou pelo «quem?» [Wer?] do ser-aí quotidiano [alltägliche Dasein], com uma breve reflexão acerca da inautenticidade [Uneigentlichkeit] fáctica em que o mesmo muitas vezes parece soçobrar. Será então, estamos em crer, o momento apropriado para debatermos o significado da controversa noção heideggeriana de «eles» [das Man], aproveitando também o ensejo para lançarmos a questão relativa à responsabilidade (ética?) do Dasein. Metodologicamente, contornaremos (sempre que possível) os tradicionais comentários a respeito das infinitas modalidades alternativas de tradução do jargão filosófico heideggeriano. (shrink)

Given a program of a linear bounded and bijective operator T, does there exist a program for the inverse operator T−1? And if this is the case, does there exist a general algorithm to transfer a program of T into a program of T−1? This is the inversion problem for computable linear operators on Banach spaces in its non-uniform and uniform formulation, respectively. We study this problem from the point of view of computable analysis which is the Turing machine based (...) theory of computability on Euclidean space and other topological spaces. Using a computable version of Banach’s Inverse Mapping Theorem we can answer the first question positively. Hence, the non-uniform version of the inversion problem is solvable, while a topological argument shows that the uniform version is not. Thus, we are in the striking situation that any computable linear operator has a computable inverse while there exists no general algorithmic procedure to transfer a program of the operator into a program of its inverse. As a consequence, the computable version of Banach’s Inverse Mapping Theorem is a powerful tool which can be used to produce highly non-constructive existence proofs of algorithms. We apply this method to prove that a certain initial value problem admits a computable solution. As a preparation of Banach’s Inverse Mapping Theorem we also study the Open Mapping Theorem and we show that the uniform versions of both theorems are limit computable, which means that they are effectively -measurable with respect to the effective Borel hierarchy. (shrink)

We develop some parts of the theory of compact operators from the point of view of computable analysis. While computable compact operators on Hilbert spaces are easy to understand, it turns out that these operators on Banach spaces are harder to handle. Classically, the theory of compact operators on Banach spaces is developed with the help of the non-constructive tool of sequential compactness. We demonstrate that a substantial amount of this theory can be developed computably on Banach spaces with computable (...) Schauder bases that are well-behaved. The conditions imposed on the bases are such that they generalize the Hilbert space case. In particular, we prove that the space of compact operators on Banach spaces with monotone, computably shrinking, and computable bases is a computable Banach space itself and operations such as composition with bounded linear operators from left are computable. Moreover, we provide a computable version of the Theorem of Schauder on adjoints in this framework and we discuss a non-uniform result on composition with bounded linear operators from right. (shrink)

This paper presents a synthesis of the methodology used when doing historical research from a positivist standpoint. It points out that the efforts of the historian to strictly imitate the scientific method is a fruitless effort which stems from the mechanistic worldview of the 17th century. In the ..

This article advances three claims. The first is that the standard instrumentalist case for minimal age and sanity requirements for voting is weak and inconclusive in such a way that the evaluation of such requirements should be made exclusively on the basis of procedural fairness considerations. The second claim is that fairness requires the inclusion of all and only those persons who have the franchise capacity: the minimum necessary cognitive and moral powers to experience the benefits of enfranchisement. The third (...) and final claim is that current age and sanity prerequisites for voting in most places fail to comply with the demands of fairness and ought to be revised. (shrink)

The denial of voting rights to certain types of persons continues to be a moral problem of practical significance. The disenfranchisement of persons with mental impairments, minors, noncitizen residents, nonresident citizens, and criminal offenders is a matter of controversy. This book makes a contribution to this largely neglected yet key topic.

In this article I argue that most biases in argumentation and decision-making can and should be counteracted. Although biases can prove beneficial in certain contexts, I contend that they are generally maladaptive and need correction. Yet critical thinking alone seems insufficient to mitigate biases in everyday contexts. I develop a contextualist approach, according to which cognitive debiasing strategies need to be supplemented by extra-psychic devices that rely on social and environmental constraints in order to promote rational reasoning. Finally, I examine (...) several examples of contextual debiasing strategies and show how they can contribute to enhance critical thinking at a cognitive level. (shrink)

We study the uniform computational content of different versions of the Baire category theorem in the Weihrauch lattice. The Baire category theorem can be seen as a pigeonhole principle that states that a complete metric space cannot be decomposed into countably many nowhere dense pieces. The Baire category theorem is an illuminating example of a theorem that can be used to demonstrate that one classical theorem can have several different computational interpretations. For one, we distinguish two different logical versions of (...) the theorem, where one can be seen as the contrapositive form of the other one. The first version aims to find an uncovered point in the space, given a sequence of nowhere dense closed sets. The second version aims to find the index of a closed set that is somewhere dense, given a sequence of closed sets that cover the space. Even though the two statements behind these versions are equivalent to each other in classical logic, they are not equivalent in intuitionistic logic, and likewise, they exhibit different computational behavior in the Weihrauch lattice. Besides this logical distinction, we also consider different ways in which the sequence of closed sets is “given.” Essentially, we can distinguish between positive and negative information on closed sets. We discuss all four resulting versions of the Baire category theorem. Somewhat surprisingly, it turns out that the difference in providing the input information can also be expressed with the jump operation. Finally, we also relate the Baire category theorem to notions of genericity and computably comeager sets. (shrink)

We prove that the Weihrauch lattice can be transformed into a Brouwer algebra by the consecutive application of two closure operators in the appropriate order: first completion and then parallelization. The closure operator of completion is a new closure operator that we introduce. It transforms any problem into a total problem on the completion of the respective types, where we allow any value outside of the original domain of the problem. This closure operator is of interest by itself, as it (...) generates a total version of Weihrauch reducibility that is defined like the usual version of Weihrauch reducibility, but in terms of total realizers. From a logical perspective completion can be seen as a way to make problems independent of their premises. Alongside with the completion operator and total Weihrauch reducibility we need to study precomplete representations that are required to describe these concepts. In order to show that the parallelized total Weihrauch lattice forms a Brouwer algebra, we introduce a new multiplicative version of an implication. While the parallelized total Weihrauch lattice forms a Brouwer algebra with this implication, the total Weihrauch lattice fails to be a model of intuitionistic linear logic in two different ways. In order to pinpoint the algebraic reasons for this failure, we introduce the concept of a Weihrauch algebra that allows us to formulate the failure in precise and neat terms. Finally, we show that the Medvedev Brouwer algebra can be embedded into our Brouwer algebra, which also implies that the theory of our Brouwer algebra is Jankov logic. (shrink)

The classical Hahn–Banach Theorem states that any linear bounded functional defined on a linear subspace of a normed space admits a norm-preserving linear bounded extension to the whole space. The constructive and computational content of this theorem has been studied by Bishop, Bridges, Metakides, Nerode, Shore, Kalantari Downey, Ishihara and others and it is known that the theorem does not admit a general computable version. We prove a new computable version of this theorem without unrolling the classical proof of the (...) theorem itself. More precisely, we study computability properties of the uniform extension operator which maps each functional and subspace to the set of corresponding extensions. It turns out that this operator is upper semi-computable in a well-defined sense. By applying a computable version of the Banach–Alaoglu Theorem we can show that computing a Hahn–Banach extension cannot be harder than finding a zero in a compact metric space. This allows us to conclude that the Hahn–Banach extension operator is ${\bf {\Sigma^{0}_{2}}}$ -computable while it is easy to see that it is not lower semi-computable in general. Moreover, we can derive computable versions of the Hahn–Banach Theorem for those functionals and subspaces which admit unique extensions. (shrink)