Hemeroteca de la sección ‘Leyes’

Necessity or Contingency. The Master Argument

Por • 30 nov, 2017 • Category: Leyes

The Master Argument, recorded by Epictetus, indicates that Diodorus had deduced a contradiction from the conjoint assertion of three propositions. Epictetus adds that three solutions of the aporia had been obtained by denying one or another of the propositions advanced while maintaining the other two. The argument, which has to do with necessity and contingency and therefore with freedom, has attracted the attention of logicians above all. In any case there have been many attempts at reconstructing it in logical terms, without excessive worry about historical plausibility and with the foregone conclusion that it was sophistic since it directly imperiled our common sense notion of freedom. On both of these counts I have taken exception to recent tradition. The success of the argument with the Ancients, and with Ancients who were no mean logicians, seemed reason for presuming that the Master Argument is not sophistic and that the contradiction it produces is a real one. On the other side, I looked for a classical text containing the propositions stated by Epictetus and which could have furnished Diodorus with the material for his argument. I believe to have found such a text in Aristotle’s De Caelo. In order to demonstrate the contradiction in the propositions thus restored, I had in my turn to translate them into logical terms. It is unlikely that Diodorus proceeded in such a way.



La proposición del Fundamento

Por • 9 nov, 2017 • Category: Leyes

La proposición del fundamento reza: Nihil est sine ratione. Se traduce: nada es sin fundamento. Cabe transcribir lo que la proposición enuncia de la forma siguiente: todo, es decir, toda suerte de cosas que, de algún modo, sean, tiene un fundamento. Omne ens habet rationem. Aquello que cada vez es efectivamente real tiene un fundamento de su realidad efectiva. Aquello que cada vez es posible tiene un fundamento de su posibilidad. Aquello que cada vez es necesario tiene un fundamento de su necesidad. Nada es sin fundamento.



Hausdorff Measure: Lost in Translation

Por • 4 nov, 2017 • Category: Leyes

In the present article we describe how one can define Hausdorff measure allowing empty elements in coverings, and using infinite countable coverings only. In addition, we discuss how the use of different nonequivalent interpretations of the notion “countable set”, that is typical for classical and modern mathematics, may lead to contradictions.



Properly ergodic structures

Por • 1 nov, 2017 • Category: Leyes

We consider ergodic Sym(N) -invariant probability measures on the space of L -structures with domain N (for L a countable relational language), and call such a measure a properly ergodic structure when no isomorphism class of structures is assigned measure 1 . We characterize those theories in countable fragments of L ω 1 ,ω for which there is a properly ergodic structure concentrated on the models of the theory. We show that for a countable fragment F of L ω 1 ,ω the almost-sure F -theory of a properly ergodic structure has continuum-many models (an analogue of Vaught’s Conjecture in this context), but its full almost-sure L ω 1 ,ω -theory has no models.



The system of integer functions, an efficient version of discrete mathematical analysis

Por • 23 oct, 2017 • Category: Leyes

Abstract: The proposed system of integer functions is logically fully independent from the traditional mathematical analysis of the real functions, but there is a well-defined mutual correspondence between the two disciplines. The system of integer functions intends to help to make the transition from the present approach to the problems of the calculus to a more computer-centric one as smooth and efficient as possible , and to find a way to some kind of synthesis of the discrete and continuous



Looking backward: From Euler to Riemann

Por • 13 oct, 2017 • Category: Leyes

We survey the main ideas in the early history of the subjects on which Riemann worked and that led to some of his most important discoveries. The subjects discussed include the theory of functions of a complex variable, elliptic and Abelian integrals, the hypergeometric series, the zeta function, topology, differential geometry, integration, and the notion of space. We shall see that among Riemann’s predecessors in all these fields, one name occupies a prominent place, this is Leonhard Euler.



Is Objectivity a Useful Construct?

Por • 30 sep, 2017 • Category: Leyes

Humanity’s efforts to transmute lead into gold have impelled civilizations. Our efforts to transmute human experience into objective laws have enjoyed similar success. Through thinkers such as Oliver Wendell Holmes, William James, Felix S. Cohen, Carol E. Cleland, Russell K. Standish and Christopher A. Fuchs we can see that a source of the difficulty in understanding phenomena via objective laws is that the law can best be understood as a quantum system, not a classical one. Law resembles a quantum system because maximal legal information is not complete and cannot be completed.



El concepto de Probabilidad en la obra de Lord Keynes

Por • 26 sep, 2017 • Category: Leyes

La interpretación logicista propuesta por Keynes condujo a un modelo en el que la probabilidad se traduce en un grado de creencia racional concebido como una relación entre un cuerpo de conocimiento y una proposición o conjunto de proposiciones. Un análisis detenido del “Treatise on probability” permite concluir: i) que el modelo Keynesiano no sólo es una consecuencia, sino que constituye una extensión de los “Principia mathematica” y los “Problems of philosophy” en la que la aproximación al concepto de probabilidad es perfectamente asimilable a la aproximación de Russell y Whitehead a la matemática y ii) que, más allá de la naturaleza innegablemente metafísica, la representación numérica de la probabilidad logicista comprende un número muy restringido de casos, debido a la calidad de heurístico del principio de indiferencia.



On the Decidability of the Ordered Structures of Numbers

Por • 18 sep, 2017 • Category: Leyes

The ordered structures of natural, integer, rational and real numbers are studied here. It is known that the theories of these numbers in the language of order are decidable and finitely axiomatizable. Also, their theories in the language of order and addition are decidable and infinitely axiomatizable. For the language of order and multiplication, it is known that the theories of N and Z are not decidable (and so not axiomatizable by a computably enumerable set of sentences). By Tarski’s theorem, the multiplicative ordered structure of R is decidable also; here we prove this result directly and present an axiomatization. The structure of Q in the language of order and multiplication seems to be missing in the literature; here we show the decidability of its theory by the technique of quantifier elimination and after presenting an infinite axiomatization for this structure we prove that it is not finitely axiomatizable.



Hilbert’s Program Then and Now

Por • 30 ago, 2017 • Category: Leyes

Hilbert’s program was an ambitious and wide-ranging project in the philosophy and foundations of mathematics. In order to “dispose of the foundational questions in mathematics once and for all, “Hilbert proposed a two-pronged approach in 1921: first, classical mathematics should be formalized in axiomatic systems; second, using only restricted, “finitary” means, one should give proofs of the consistency of these axiomatic systems. Although Godel’s incompleteness theorems show that the program as originally conceived cannot be carried out, it had many partial successes, and generated important advances in logical theory and meta-theory, both at the time and since. The article discusses the historical background and development of Hilbert’s program, its philosophical underpinnings and consequences, and its subsequent development and influences since the 1930s.