Hemeroteca de la sección ‘Leyes’

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.

Measuring technological complexity – Current approaches and a new measure of structural complexity

Por • 25 ago, 2017 • Category: Leyes

The paper reviews two prominent approaches for the measurement of technological complexity: the method of reflection and the assessment of technologies’ combinatorial difficulty. It reviews their central underlying assumptions and discusses potential problems related to these. A new measure of structural complexity is introduced as an alternative. The paper also puts forward four stylized facts of technological complexity that serve as benchmarks in an empirical evaluation of five complexity measures (increasing development over time, larger R&D efforts, more collaborative R&D, spatial concentration). The evaluation utilizes European patent data for the years 1980 to 2013 and finds the new measure of structural complexity to mirror the four stylized facts as good as or better than traditional measures. It is also shown to be less problematic in empirical applications.

A Final Solution to the Mind-Body Problem by Quantum Language

Por • 17 ago, 2017 • Category: Leyes

Recently we proposed “quantum language”, which was not only characterized as the metaphysical and linguistic turn of quantum mechanics but also the linguistic turn of Descartes = Kant epistemology. And further we believe that quantum language is the only scientifically successful theory in dualistic idealism. If this turn is regarded as progress in the history of western philosophy (i.e., if “philosophical progress” is defined by “approaching to quantum language”), we should study the linguistic mind-body problem more than the epistemological mind-body problem. In this paper, we show that to solve the mind-body problem and to propose “measurement axiom” in quantum language are equivalent. Since our approach is always within dualistic idealism, we believe that our linguistic answer is the only true solution to the mind-body problema.

A mathematically derived theory of truth and its properties

Por • 3 ago, 2017 • Category: Leyes

Hannes Leitgeb formulated in his paper ‘What Theories of Truth Should be Like (but Cannot be) eight norms for theories of truth, and stated in p.8: “In the best of all (epistemically) possible worlds, some theory of truth would satisfy all of those norms at the same time. Unfortunately, we do not inhabit such a world.” In the present paper a theory of truth is derived mathematically. That theory is shown to satisfy well all those eight norms. Thus the choice over truth theories which don’t satisfy all those norms, or logic different from the classical one (cf., e.g., Section 2 of Leitgeb’s paper) seems not to be necessary. This makes the present trend to accept the choice over ‘alternative’ truths or facts less plausible.

The unreasonable power of the lifting property in elementary mathematics

Por • 28 jul, 2017 • Category: Leyes

We illustrate the generative power of the lifting property (orthogonality of morphisms in a category) as means of defining natural elementary mathematical concepts by giving a number of examples in various categories, in particular showing that many standard elementary notions of abstract topology can be defined by applying the lifting property to simple morphisms of finite topological spaces. Examples in topology include the notions of: compact, discrete, connected, and totally disconnected spaces, dense image, induced topology, and separation axioms. Examples in algebra include: finite groups being nilpotent, solvable, torsion-free, p-groups, and prime-to-p groups; injective and projective modules; injective, surjective, and split homomorphisms. We include some speculations on the wider significance of this.

Logic of gauge

Por • 16 jul, 2017 • Category: Leyes

The logic of gauge theory is considered by tracing its development from general relativity to Yang-Mills theory, through Weyl’s two gauge theories. A handful of elements—which for want of better terms can be called emph geometrical justice, emph matter wave, emph second clock effect, emph twice too many energy levels—are enough to produce Weyl’s second theory; and from there, all that’s needed to reach the Yang-Mills formalism is a emph non-Abelian structure group (say SU(N)).