"Taurus"
Dice el refrán "el que a hierro mata, a hierro muere". Borges es un torero, es el toro y la espada literaria. Es también Emerson y "Moisés Literato", pero es también la pluma más excelsa del "masonismo católico" y un maravilloso constructor con mero "barro", es el correlato con la pluma de las estructuras arquitectónicas en Perú. Construcciones que buscan no perpetuarse sino coadyuvar al triunfo del Rey de Reyes, donde Borges que Sí perdurará en El Rey de acuerdo a la Promesa. Pero no pretende perdurar como Borges, saber esto es ser un auténtico guerrillero, nadie que persiga la "gloria" se atrevería a disparar esos comentarios que Borges espeta a diestra y siniestra con maestría serpentina. La obra de Borges no está hecha para ser citada selectivamente a conveniencia. La Misión de Borges es sus escritos y sus declaraciones. Que defienden de acuerdo a las necesidades históricas particulares de su tiempo. Por eso afirmo de nuevo, El Che Viejo es el verdadero "guerrillero de Cristo"!!! Y no obstante su gran faena taurina su espada está limpia de sangre inocente!!!
Por eso, Héctor y Aquiles, "Georgie" nos enseñó-con Tauromaquia excelsa- como ser el genio y la bestia para la Gloria de Cristo!!!
Letras Libres - "Borges y Bioy conversan sobre México" por Héctor Manjarrez
Ostensible (adjective)
Pronunciation: [ah-'sten-sê-bêl]
Definition: Apparent, evident, conspicuous.
Usage: Although today's word means "evident," it usually implies the concealment of something more important, more real. So one's ostensible reason for going to a restaurant would be to eat but the more important one might be to talk with a certain waitress.
Suggested Usage: In and of itself, today's word means something very close to "obvious," as in: "All of Malcolm's intentions were ostensible and amused everyone." It is used, however, more often to refer to a mask concealing another agenda: "Loretta ostensibly travels to Atlanta on business but I noticed some time ago that she only goes when the Braves are playing a home (baseball) game."
Etymology: Via French from Medieval Latin "ostensibilis" from "ostensus," the past participle of ostendere "to show." This verb is composed of ob- "to(ward)" + tendere "to stretch," probably from stretching the arm out to show things. "Tendere" is based on the same original root, *ten- "stretch," as English "thin" and German "dünn," since stretching tends to make things thinner. The same root also developed into Latin tenere "to hold," which is detectable in the English borrowings "tenet," "tenant," "tenacious," "tenable." "Tense" and "tension" are also relatives. Finally, "baritone" is based partially on Greek tonos "string" of the same origin. Guess where "tone" came from.
–Dr. Language, YourDictionary.com
Interesante artículo de Juan Morales en el diario Ecuatoriano El Tiempo que coincide en el tenor de su evaluación de Borges y su obra con mi apreciación, aunque al final se vuelve, bien intencionadamente, precavido al advertir que "el sentido del mundo" que vislumbró y vivió Borges no puede ser presentado como una verdad eterna y absoluta, lo cual es obvio-ninguna dirección o señalamiento puede ser eterno o absoluto, - ahora bien, se puede entonces afirmar que Borges es el "guerrillero de Cristo,"? en términos dialécticos sí, en términos filosóficos no, ya que formalmente sólo estará enunciando la Paradoja de Russell y sólo por sus frutos determinaremos la verdad humana de esta afirmación. Afirmar que Cristo es Eterno y Absoluto se vuelve una afirmación paradójica cuando traemos a la discusión la parte humana de Cristo, que para empezar se contradice con el postulado de la Omnipresencia de Dios que también es Jesucristo. Es obvio que a los ojos humanos escépticos la Divinidad de Cristo será manifiesta sólo a través de la observación indirecta sobre la Armonía, Paz y Justicia que su Reino traiga. No hay de otra aquí en la Tierra,! pero para los que tienen Fé, estos ya están en Cristo y a la Diestra de Dios Padre.
Borges y la contemporaneidad
Por: Juan Morales Correo:
La crítica literaria reconoce en el autor argentino a uno de los representantes más importantes de las letras de América Latina y del mundo. Además del manejo excelso del lenguaje, en Borges impacta su clarividencia que probablemente influenció en su forma de ser caracterizada por la prudencia, la sencillez y el asombro permanente frente a las afirmaciones de las diferentes manifestaciones científicas o del conocimiento humano en general.
Los latinoamericanos, especialmente, fuimos influenciados de manera decisiva por la obra y la vida de Jorge Luis Borges. Sus poemas, ensayos, cuentos y novelas, formaron parte de nuestra vida y del forjamiento de nuestra identidad y visión del mundo, tanto a nivel individual como colectivo. El Aleph, Historia Universal de la Eternidad, Ficciones y tantas otras obras, despertaron en sus lectores, el interés por el misterio de la complejidad del funcionamiento de la vida, así como por la validez de las distintas posibilidades de explicación de la misma.
Borges fue un sabio y comprendió la importancia de la búsqueda permanente y la relatividad de todas las afirmaciones y por eso vivió deslumbrado desde un nivel de conciencia de su propia existencia como sujeto integrante de un todo complejo. Formó parte de una gran corriente de seres humanos cuya voluntad estaba orientada a rebasar los límites preestablecidos para buscar continuamente renovadas y constantes evoluciones en armonía con el transcurrir del tiempo en una clarividente y permanente metamorfosis.
La contemporaneidad puede ser entendida como el momento histórico en el cual confluyen armoniosamente los conocimientos científicos y las otras formas de comprensión del universo. La época actual es un escenario que se nutre de diferentes puntos de vista sobre el funcionamiento de la vida y que valida esa diversidad de enfoques con la intención de construir un todo que incorpore y no excluya.
Pensadores de la talla de Borges se encuentran en los antecedentes inmediatos de este modelo de comprensión que es denominado por algunos como pensamiento sistémico y complejo. De alguna manera, Borges, al oponerse a la racionalización de la vida y reconocer la validez de las otras formas de explicación de la misma, se constituye en un referente importante para entender lo que hoy es generalmente aceptado.
Borges vislumbró y vivió un sentido del mundo que puede ser encontrado a través de la introspección individual y que no puede ni debe ser presentado como una verdad eterna y absoluta.
Russell's Paradox
Russell's paradox is the most famous of the logical or set-theoretical paradoxes. The paradox arises within naive set theory by considering the set of all sets that are not members of themselves. Such a set appears to be a member of itself if and only if it is not a member of itself, hence the paradox.
Some sets, such as the set of all teacups, are not members of themselves. Other sets, such as the set of all non-teacups, are members of themselves. Call the set of all sets that are not members of themselves "R." If R is a member of itself, then by definition it must not be a member of itself. Similarly, if R is not a member of itself, then by definition it must be a member of itself. Discovered by Bertrand Russell in 1901, the paradox has prompted much work in logic, set theory and the philosophy and foundations of mathematics.
- History of the paradox
- Significance of the paradox
- Bibliography
- Other Internet Resources
- Related Entries
History of the paradox
Russell appears to have discovered his paradox in the late spring of 1901,[1] while working on his Principles of Mathematics (1903). Cesare Burali-Forti, an assistant to Giuseppe Peano, had discovered a similar antinomy in 1897 when he noticed that since the set of ordinals is well-ordered, it too must have an ordinal. However, this ordinal must be both an element of the set of all ordinals and yet greater than every such element. Unlike Burali-Forti's paradox, Russell's paradox does not involve either ordinals or cardinals, relying instead only on the primitive notion of set.
Russell wrote to Gottlob Frege with news of his paradox on June 16, 1902. The paradox was of significance to Frege's logical work since, in effect, it showed that the axioms Frege was using to formalize his logic were inconsistent. Specifically, Frege's Rule V, which states that two sets are equal if and only if their corresponding functions coincide in values for all possible arguments, requires that an expression such as f(x) be considered both a function of the argument x and a function of the argument f. In effect, it was this ambiguity that allowed Russell to construct R in such a way that it could both be and not be a member of itself.
Russell's letter arrived just as the second volume of Frege's Grundgesetze der Arithmetik (The Basic Laws of Arithmetic, 1893, 1903) was in press. Immediately appreciating the difficulty the paradox posed, Frege added to the Grundgesetze a hastily composed appendix discussing Russell's discovery. In the appendix Frege observes that the consequences of Russell's paradox are not immediately clear. For example, "Is it always permissible to speak of the extension of a concept, of a class? And if not, how do we recognize the exceptional cases? Can we always infer from the extension of one concept's coinciding with that of a second, that every object which falls under the first concept also falls under the second? These are the questions," Frege notes, "raised by Mr Russell's communication."[2] Because of these worries, Frege eventually felt forced to abandon many of his views about logic and mathematics.
Of course, Russell also was concerned about the contradiction. Upon learning that Frege agreed with him about the significance of the result, he immediately began writing an appendix for his own soon-to-be-released Principles of Mathematics. Entitled "Appendix B: The Doctrine of Types," the appendix represents Russell's first detailed attempt at providing a principled method for avoiding what was soon to become known as "Russell's paradox."
Significance of the paradox
The significance of Russell's paradox can be seen once it is realized that, using classical logic, all sentences follow from a contradiction. For example, assuming both P and ~P, any arbitrary proposition, Q, can be proved as follows: from P we obtain P ∨ Q by the rule of Addition; then from P ∨ Q and ~P we obtain Q by the rule of Disjunctive Syllogism. Because of this, and because set theory underlies all branches of mathematics, many people began to worry that, if set theory was inconsistent, no mathematical proof could be trusted completely.
Russell's paradox ultimately stems from the idea that any coherent condition may be used to determine a set. As a result, most attempts at resolving the paradox have concentrated on various ways of restricting the principles governing set existence found within naive set theory, particularly the so-called Comprehension (or Abstraction) axiom. This axiom in effect states that any propositional function, P(x), containing x as a free variable can be used to determine a set. In other words, corresponding to every propositional function, P(x), there will exist a set whose members are exactly those things, x, that have property P.[3] It is now generally, although not universally, agreed that such an axiom must either be abandoned or modified.[4]
Russell's own response to the paradox was his aptly named theory of types. Recognizing that self-reference lies at the heart of the paradox, Russell's basic idea is that we can avoid commitment to R (the set of all sets that are not members of themselves) by arranging all sentences (or, equivalently, all propositional functions) into a hierarchy. The lowest level of this hierarchy will consist of sentences about individuals. The next lowest level will consist of sentences about sets of individuals. The next lowest level will consist of sentences about sets of sets of individuals, and so on. It is then possible to refer to all objects for which a given condition (or predicate) holds only if they are all at the same level or of the same "type."
This solution to Russell's paradox is motivated in large part by the so-called vicious circle principle, a principle which, in effect, states that no propositional function can be defined prior to specifying the function's range. In other words, before a function can be defined, one first has to specify exactly those objects to which the function will apply. (For example, before defining the predicate "is a prime number," one first needs to define the range of objects that this predicate might be said to satisfy, namely the set, N, of natural numbers.) From this it follows that no function's range will ever be able to include any object defined in terms of the function itself. As a result, propositional functions (along with their corresponding propositions) will end up being arranged in a hierarchy of exactly the kind Russell proposes.
Although Russell first introduced his theory of types in his 1903 Principles of Mathematics, type theory found its mature expression five years later in his 1908 article, "Mathematical Logic as Based on the Theory of Types," and in the monumental work he co-authored with Alfred North Whitehead, Principia Mathematica (1910, 1912, 1913). Russell's type theory thus appears in two versions: the "simple theory" of 1903 and the "ramified theory" of 1908. Both versions have been criticized for being too ad hoc to eliminate the paradox successfully. In addition, even if type theory is successful in eliminating Russell's paradox, it is likely to be ineffective at resolving other, unrelated paradoxes.
Other responses to Russell's paradox have included those of David Hilbert and the formalists (whose basic idea was to allow the use of only finite, well-defined and constructible objects, together with rules of inference deemed to be absolutely certain), and of Luitzen Brouwer and the intuitionists (whose basic idea was that one cannot assert the existence of a mathematical object unless one can also indicate how to go about constructing it).
Yet a fourth response was embodied in Ernst Zermelo's 1908 axiomatization of set theory. Zermelo's axioms were designed to resolve Russell's paradox by again restricting the Comprehension axiom in a manner not dissimilar to that proposed by Russell. ZF and ZFC (i.e., ZF supplemented by the Axiom of Choice), the two axiomatizations generally used today, are modifications of Zermelo's theory developed primarily by Abraham Fraenkel.
Together, these four responses to Russell's paradox have helped logicians develop an explicit awareness of the nature of formal systems and of the kinds of metalogical and metamathematical results commonly associated with them today.
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