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)

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)

O foco deste trabalho repousa sobre um certo caminho histórico percorrido pela ciência, como um todo, no sentido desta se despir de qualquer revestimento metafísico e que teve como marco incontornável a criação, em 1929, do Círculo de Viena. Abordamos algumas tendências e limitações desse movimento e argumentamos que o metafísico terá sempre um espaço no seio mais íntimo da reflexão científica. Este trabalho foi realizado no âmbito da disciplina de Filosofia das Ciências I, parte do curso de Filosofia da (...) Faculdade de Letras da Universidade do Porto, Portugal. (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)

Neste trabalho abordamos a obra de Francis Bacon e as suas contribuições para a Filosofia da Ciência, incluindo a sua proposta de método científico e o seu posicionamento crítico face à anterior versão aristotélica e a influência que o pensamento baconiano exerceu na promoção de um novo quadro filosófico para o desenvolvimento científico na modernidade. Este trabalho foi realizado no âmbito da disciplina de Filosofia das Ciências I, parte do curso de Filosofia da Faculdade de Letras da Universidade do Porto, (...) Portugal. (shrink)

Neste breve ensaio, exploramos alguns caminhos de uma controvérsia milenar em torno do estatuto dos entes matemáticos e apresentamos alguns argumentos a favor de uma posição platonista, aproximadamente clássica, sobre o tema. Este trabalho foi realizado no âmbito da disciplina de Filosofia das Ciências II, parte do curso de Filosofia da Faculdade de Letras da Universidade do Porto, Portugal.

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)

Neste trabalho abordamos as contribuições de Larry Laudan para a questão do progresso científico. O modelo proposto, baseado na eficácia das teorias na resolução de problemas, serve de ponto de partida para uma breve reflexão sobre a influência contemporânea que as dificuldades de visão do mundo exercem sobre o desenvolvimento das ciências. Este trabalho foi realizado no âmbito da disciplina de Filosofia das Ciências II, parte do curso de Filosofia da Faculdade de Letras da Universidade do Porto, Portugal.

¿Qué cabida tienen, o pueden tener, las éticas y las políticas de la alteridad en el horizonte filosófico actual? En el contexto de la filosofía española reciente, la obra de Gabriel Bello Reguera (Dehesas, León, 1943) ha afrontado el reto ético-político que plantea el tomarse en serio la alteridad. Ha desarrollado su carrera académica en la Universidad de La Laguna como catedrático de Filosofía Moral y en su trayectoria cabría señalar tres giros fundamentales, si bien relacionados entre sí. El primero (...) podemos derivarlo del impacto de la lectura de la obra del neopragmatista Richard Rorty, que incidió en su crítica al fundamentalismo autoritario de la tradición filosófica despertó su interés por la narratividad, ligada a la conversación, como herramienta edificante de la ética. El segundo es debido a la influencia del pensamiento de Emmanuel Lévinas, a partir del cual problematiza las relaciones entre las categorías éticas de identidad, diferencia y alteridad. El tercero, y más reciente, le llevará al terreno del análisis de las migraciones internacionales en su contexto postcolonial como acuciante asunto necesitado, precisamente, de un nuevo marco filosófico, ético y político. En este libro colectivo, a través de la obra de Gabriel Bello, afrontamos con rigor la contextualización de las éticas y las políticas de la alteridad, proseguimos la discusión en torno a estas a la vez que elaboramos, conjuntamente, un balance crítico de una línea de investigación, de sus posibilidades y límites. Con todo ello, además, seguimos la tradición de prestar reconocimiento y gratitud a nuestros maestros y a nuestras maestras. En este caso, reconocemos la trayectoria docente e investigadora del Profesor Gabriel Bello Reguera y su dedicación rigurosa y concienzuda a la vida universitaria. Su obra, que, recorre cuatro décadas, es testimonio de cómo la filosofía en el contexto del Estado español, tras abandonar el erial instigado por el franquismo, participa de los debates transnacionales actuales en los que las éticas y las políticas de la alteridad, especialmente, plantean desafíos insoslayables. Hemos, pues, recreado el ritual académico del Homenaje, algo tan central en la filosofía alemana como la Festschrift, con el deseo de dotarlo del sentido más relevante que, a nuestro juicio, puede tener: el del debate intelectual crítico. (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)

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)

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)

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)

Ejes de la territorialidad patriarcal -- Fronteras y antifaces del signo mujer -- En el flujo heterogéneo de la liberación.

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.

The question of how we should engage with a philosopher’s racial thought is of particular importance when considering Kant, who can be viewed as particularly representative of Enlightenment philosophy. In this article I argue that we should take a stance of deep acknowledgment when considering Kant’s work both inside and outside the classroom. Taking a stance of deep acknowledgment should be understood as 1) taking Kant’s racial thought to be reflective of his moral character, 2) Kant being accountable for his (...) racial thought and 3) being willing to consider the possibility that Kant’s racial thought is consistent with and inextricable from his moral philosophy. Alternative forms of engaging with Kant’s racial work have either moral or pedagogical failings, which range from simply teaching the history of philosophy uncritically to outright deception. A stance of deep acknowledgement will allow philosophers to understand how Kant’s racial thought interacts with his moral philosophy and allow instructors to teach philosophy in a historically contextualized approach so as to not alienate students whose demographic was disparaged by Kant. (shrink)

We study the computational content of the Brouwer Fixed Point Theorem in the Weihrauch lattice. Connected choice is the operation that finds a point in a non-empty connected closed set given by negative information. One of our main results is that for any fixed dimension the Brouwer Fixed Point Theorem of that dimension is computably equivalent to connected choice of the Euclidean unit cube of the same dimension. Another main result is that connected choice is complete for dimension greater than (...) or equal to two in the sense that it is computably equivalent to Weak Kőnig’s Lemma. While we can present two independent proofs for dimension three and upward that are either based on a simple geometric construction or a combinatorial argument, the proof for dimension two is based on a more involved inverse limit construction. The connected choice operation in dimension one is known to be equivalent to the Intermediate Value Theorem; we prove that this problem is not idempotent in contrast to the case of dimension two and upward. We also prove that Lipschitz continuity with Lipschitz constants strictly larger than one does not simplify finding fixed points. Finally, we prove that finding a connectedness component of a closed subset of the Euclidean unit cube of any dimension greater than or equal to one is equivalent to Weak Kőnig’s Lemma. In order to describe these results, we introduce a representation of closed subsets of the unit cube by trees of rational complexes. (shrink)

‘Intentionalist’ approaches portray self-deceivers as “akratic believers”, subjects who deliberately choose to believe p despite knowing that p is false. In this paper I argue that the intentionalist model leads to a number of paradoxes that seem to undermine it. I claim that these paradoxes can nevertheless be overcome in light of the rival hypothesis that self-deception is a non-intentional process that stems from the influence of emotions upon cognitive processes. Furthermore, I propose a motivational interpretation of the phenomenon of (...) ‘hyperbolic discounting bias’ which highlights the role of emotional biases in akratic behavior. Finally, I argue that we are not the helpless victims of our irrational attitudes, insofar as we have the ability – and arguably the epistemic obligation – to counteract motivational biases. (shrink)