A sociedade brasileira no que diz respeito a sua estrutura de relações sociais como conhecemos, tem início no Brasil colônia a qual a instituição de maior força era a escravagista. Nesse sentido, as formações e interações de classes da sociedade brasileira foram arregimentadas pelo racismo, sendo ele escancarado no passado, com leis e naturalização das práticas de disciminação racial, e mantendo grande influência atualmente com a utilização de “máscaras” que encontram para legitimar sua atuação na sociedade contemporânea. Nesse contexto, nos (...) últimos anos tem-se observado a atuação irregular de ações policiais no país, principalmente nas comunidades periféricas, sendo utilizada a brutalidade e a violência física e moral dos sujeitos, violando direitos humanos e fundamentais, e colocando em perigo a vida de crianças e adolescentes que residem nesses ambientes, sobretudo, as negras, reconhecendo assim a existência do racismo estrutural. Considerando que o país é democrático, e reconhecendo que os atores sociais possuem a capacidade de influenciar nas mudanças e construções de políticas capazes de discutir os fundamentos da vida pública e social, esta proposta de pesquisa tem por objetivo refletir como a brutalidade policial afeta a confiança de crianças negras no Estado democrático por meio de instituições de segurança e proteção. Trata-se de uma pesquisa de revisão bibliográfica, qualitativa e descritiva. (shrink)

“El Árbol de Gernika es símbolo del bienestar de nuestro pueblo: no de ningún otro. Ni dentro de nuestro solar puede coexistir con ningún otro árbol”, escribía Sabino Arana en Euzkadi en septiembre de 1901. Esta concepción aranista del “nosaltres sols!” es el suelo interpretativo de la cosmovisión vasca, también de su historia negra de violencia. “No hay lugar en el mismo espacio geográfico para dos violencias legítimas, para dos soberanías, y el espacio geográfico de Euskal Herria es vasco (...) por definición en el caso de ETA, y ETA es la cristalización definitiva del pueblo vasco”, enlaza Joseba Arregi en el primero de los libros reseñados. La dialéctica entre el uno y el dos es la clave de bóveda de la construcción discursiva del tribalismo identitario. (shrink)

This paper analyzes the narrative strategies of Radical Basque Nationalism, taking advantage of Thomas Lacoste's documentary, Pays Basque et liberté. It is made up of four parts. The first, as an introduction, focuses on general considerations about language to frame the narrative practices of the RBN. The second and third address the two central elements of the nationalist communication strategy: the use of international figures as an amplification mechanism, on the one hand, and the appropriation of a noble language that (...) functions as a legitimating and prestigious element, while it incorporates elements that can be decoded by the members of the collective as part of its canonical interpretation, on the other. The fourth summarizes the main argumentative axes conveyed by the participants in the documentary. The text is closed with a brief conclusion. The main methodological tool is the analysis of contents. (shrink)

A Gemaldegalerie de Dresden expõe uma pintura de Valdés Leal, executada entre os anos de 1656 e1658, uma das doze que executou para o convento dos Jerónimos de Sevilha mostrando os santos da Ordem. A figura histórica de Vasco Martins, leiriense, “ermjtam de prove vida”, fundador do convento da Penha Longa, é conhecida nos seus traços mais significativos, mas menos presente hoje está a sua iconografia e o culto que mereceu, aspectos sobre os quais nos debruçamos.

Online supplement, "Mulsow: Additions to Notes drawn from the 2002 edition of Moderne aus dem Untergrund" full versions of nearly 300 notes that were truncated in the print edition. Hosted on H. C. Erik Midelfort's website. Martin Mulsow's seismic reinterpretation of the origins of the Enlightenment in Germany won awards and renown in its original German edition, and now H. C. Erik Midelfort's translation makes this sensational book available to English-speaking readers. In Enlightenment Underground, Mulsow shows that even in the (...) late seventeenth century some thinkers in Germany ventured to express extremely dangerous ideas, but did so as part of a secret underground. Scouring manuscript collections across northern Europe, Mulsow studied the writings of countless hitherto unknown radical jurists, theologians, historians, and dissident students who pushed for the secularization of legal, political, social, and religious knowledge. Often their works circulated in manuscript, anonymously, or as clandestinely published books. Working as a philosophical microhistorian, Mulsow has discovered the identities of several covert radicals and linked them to circles of young German scholars, many of whom were connected with the vibrant radical cultures of the Netherlands, England, and Denmark. The author reveals how radical ideas and contributions to intellectual doubt came from Socinians and Jews, church historians and biblical scholars, political theorists, and unemployed university students. He shows that misreadings of humorous or ironic works sometimes gave rise to unintended skeptical thoughts or corrosively political interpretations of Christianity. This landmark book overturns stereotypical views of the early Enlightenment in Germany as cautious, conservative, and moderate, and replaces them with a new portrait that reveals a movement far more radical, unintended, and puzzling than previously suspected. (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)

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)

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)

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)

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 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)

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)

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)

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)

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)

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.

La obra que pretendo reseñar a continuación es de difícil clasificación. Por un lado, estamos ante el trabajo de un académico que utiliza el instrumental propio del historiador de las ideas. Por otro, se nos presenta un contenido heterogéneo sustentado por un enfoque sumamente original. No resulta extraño por tanto que haya sido merecedora del XVIII Premio Miguel de Unamuno 2017 del Ayuntamiento de Bilbao al mejor ensayo en lengua castellana de entre los presentados a concurso. En los dieciséis capítulos (...) que la componen, rematados por una "Postdata personal" en la que el autor explica cómo surgió su interés por la obra del escritor vasco que concluyó finalmente con la escritura de la obra, Mario Martín Gijón, pacense de Villanueva de la Serena y profesor de la Universidad de Extremadura, presenta la historia de la recepción del pensamiento unamuniano en una España que quedó preterida injusta e injustificadamente durante tantos años: la de los exiliados que desarrollaron el grueso de su producción intelectual en Europa y América Latina durante la dictadura franquista. La heterogeneidad del contenido viene dada, en consecuencia, por la gran cantidad de autores -pensadores, poetas, novelistas o dramaturgos- que desfilan ante los ojos del lector. (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.

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)

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 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)

In this paper, I argue that ordinary cases of self-deception and akrasia derive from the phenomenon of motivated irrationality. According to the ‘motivational’ account, self-deception is typically induced by the influence that desires and emotions exert upon our cognitive faculties, and thereby upon the process of belief formation. Crucially, I show that this hypothesis is consistent with the empirical research carried out by social psychologists, and that it avoids a number of paradoxes that undermine the ‘intentionalist’ account. But motivated irrationality (...) also seems to account for most cases of akrasia, insofar as desires are equally liable to affect the evaluative judgments through which we assess feasible options. This analysis thus set the premises for a unified account of irrationality. (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)

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)