Hemeroteca de la sección ‘Educacion’

Whitehead’s Mereotopology and the Project of Formal Ontology

Por • 17 ago, 2017 • Category: Educacion

Mereology is the theory of wholes and parts. The first formal mereology was developed by Husserl in his third Logical Investigation at the beginning of the twentieth century. In 1916 Stanisław Lesnie ´ wski gave the first axiomatization of a classical extensional formal mereology. That same year, Alfred North Whitehead also gave a sketch of a mereology in “La théorie relationniste de l’espace”. It was developed in the perspective of a theory of space in which the concept of point is no longer considered as primitive, but is built in terms of the relations between objects. This project was then taken up and amplified in the wider perspective of the method of extensive abstraction presented in An Enquiry Concerning the Principles of Natural Knowledge and The Concept of Nature. Afterwards, Whitehead added to what was first a theory of the part-whole relation some definitions of topological notions such as junction.

Kant y la “revolución del modo de pensar” (KrV B XI): su significado metodológico general y su significado metafísico específico, analizados desde una perspectiva sistemática e histórico-evolutiva

Por • 10 ago, 2017 • Category: Educacion

En el prólogo a la segunda edición de la Crítica de la razón pura, ¿qué ha querido decir Kant con una “revolución del modo de pensar” que se debe aplicar a la meta-física para que tome el camino de la ciencia? Cabe distinguir dos significados: uno de carácter metodológico, que asume que el conocimiento rige a los objetos y ten-dría alcance. Aplicada a la metafísica, supone una transformación más profunda que permite hablar, en segundo lugar, de un significado metafísico específico, que implica que el conocimiento constituye aquello mismo a lo que se llama el objeto de las representaciones.

Paradoxes, Self-Referentiality, and Hybrid Systems: A Constructive Approach

Por • 28 jul, 2017 • Category: Educacion

Since the discovery of the paradoxes of self-referentiality or self-reference respectively logicians and mathematicians tried to avoid self-reference when constructing formal systems. Yet “real” complex systems like the mind are characterized by self-reference and can accordingly only be modeled by formal systems that are also basically self-referential. In this article I show that and how self-referential computer programs, understood as algorithmic formal systems, are not only possible but also since some time quite common in special branches of computer science. Examples for this argument are neural networks and so-called hybrid systems, i.e. combination of different sub systems. The hybrid system SOCAIN, a combination of a cellular automaton, a neural network and a genetic algorithm is an example for the fruitfulness of using self-reference in a systematic way. In particular, such systems consist of mutually dependent sub systems, i.e. form no static hierarchy.

Betting on Quantum Objects

Por • 26 jul, 2017 • Category: Educacion

Dutch book arguments have been applied to beliefs about the outcomes of measurements of quantum systems, but not to beliefs about quantum objects prior to measurement. In this paper, we prove a quantum version of the probabilists’ Dutch book theorem that applies to both sorts of beliefs: roughly, if ideal beliefs are given by vector states, all and only Born-rule probabilities avoid Dutch books. This theorem and associated results have implications for operational and realist interpretations of the logic of a Hilbert lattice. In the latter case, we show that the defenders of the eigenstate-value orthodoxy face a trilemma. Those who favor vague properties avoid the trilemma, admitting all and only those beliefs about quantum objects that avoid Dutch books.

Segal-type models of higher categories

Por • 16 jul, 2017 • Category: Educacion

Higher category theory is an exceedingly active area of research, whose rapid growth has been driven by its penetration into a diverse range of scientific fields. Its influence extends through key mathematical disciplines, notably homotopy theory, algebraic geometry and algebra, mathematical physics, to encompass important applications in logic, computer science and beyond. Higher categories provide a unifying language whose greatest strength lies in its ability to bridge between diverse areas and uncover novel applications. In this foundational work we introduce a new approach to higher categories. It builds upon the theory of iterated internal categories, one of the simplest possible higher categorical structures available, by adopting a novel and remarkably simple “weak globularity” postulate and demonstrating that the resulting model provides a fully general theory of weak n-categories. The latter are among the most complex of the higher structures, and are crucial for applications. We show that this new model of “weakly globular n-fold categories” is suitably equivalent to the well studied model of weak n-categories due to Tamsamani and Simpson.

Ontología, epistemología y semántica: sobre la teoría kantiana acerca de la estructura objetual del mundo

Por • 30 jun, 2017 • Category: Educacion

El objetivo de este trabajo es reivindicar una interpretación de la Analítica Trascendental como una “ontología”. Pero no en el sentido heideggeriano de una “ontología fundamental” del Dasein (cf. Heidegger 1927, 1929), sino en el sentido más general de una teoría a priori de los objetos: una teoría categorial que opera una reconstrucción meta-teórica de la estructura formal objetiva de nuestro mundo a partir de la estructura de nuestro pensamiento y nuestras facultades cognitivas. No obstante, reconozco que la teoría presenta consecuencias epistemológicas y semánticas.

On the arithmetic of graphs

Por • 24 jun, 2017 • Category: Educacion

The Zykov ring of signed finite simple graphs with topological join as addition and compatible multiplication is an integral domain but not a unique factorization domain. We know that because by taking graph complements, it becomes isomorphic to the strong Sabidussi ring with disjoint union as addition. We prove that the Euler characteristic is a ring homomorphism from the strong ring to the integers by demonstrating that the strong ring is homotopic to a Stanley-Reisner Cartesian ring. More generally, the Kuenneth formula holds on the strong ring so that the Poincare polynomial is compatible with the ring structure. The Zykov ring has the clique number as a ring homomorphism. Furthermore, the Cartesian ring has the property that the functor which attaches to a graph the spectrum of its connection Laplacian is multiplicative. The reason is that the connection Laplacians do tensor under multiplication, similarly to what the adjacency matrix does for the weak ring. The strong ring product of two graphs contains both the weak and direct product graphs as subgraphs.

Quantitative ergodic theorems and their number-theoretic applications

Por • 8 jun, 2017 • Category: Educacion

We present an account of some recent applications of ergodic theorems for actions of algebraic and arithmetic groups to the solution of natural problems in Diophantine approximation and number theory. Our approach is based on spectral methods utilizing the unitary representation theory of the groups involved. This allows the derivation of ergodic theorems with a rate of convergence, an important phenomenon which does not arise in classical ergodic theory. Combining spectral and dynamical methods, quantitative ergodic theorems give rise to new and previously inaccessible applications. We demonstrate the remarkable diversity of such applications by deriving general uniform error estimates in non-Euclidean lattice points counting problems, explicit estimates in the sifting problem for almost-prime points on symmetric varieties, best-possible bounds for exponents of intrinsic Diophantine approximation on homogeneous algebraic varieties, and quantitative results on fast distribution of dense orbits on compact and non-compact homogeneous spaces.

Cosmology and the Origin of the Universe: Historical and Conceptual Perspectives

Por • 5 jun, 2017 • Category: Educacion

From a modern perspective cosmology is a historical science in so far that it deals with the development of the universe since its origin some 14 billion years ago. The origin itself may not be subject to scientific analysis and explanation. Nonetheless, there are theories that claim to explain the ultimate origin or “creation” of the universe. As shown by the history of cosmological thought, the very concept of “origin” is problematic and can be understood in different ways. While it is normally understood as a temporal concept, cosmic origin is not temporal by necessity.

Constructive mathematics

Por • 29 may, 2017 • Category: Educacion

This text was published in the book “Penser les mathematiques: seminaire de philosophie et mathematiques de l’Ecole normale superieure (J. Dieudonne, M. Loi, R. Thom)” edited by F. Guenard and G. Lelievre, Paris, editions du Seuil, 1982, pp. 58-72. It is reproduced with the kind authorisation of Francois Apery.