Artículos con la etiqueta ‘matemática’

Mathematics in the Age of the Turing Machine

Por • 13 feb, 2013 • Category: Opinion

The article gives a survey of mathematical proofs that rely on computer calculations and formal proofs. Computers have rapidly become so pervasive in mathematics that future generations may look back to this day as a golden dawn. A comprehensive survey is out of the question. It would almost be like asking for a summary of applications of symmetry to mathematics. Computability – like symmetry – is a wonderful structural property that some mathematical objects possess that makes answers flow more readily wherever it isfound.

Rational Lukasiewicz logic and DMV-algebras

Por • 4 dic, 2012 • Category: Filosofía

In this paper we present some results on the variety of divisible MV-algebras. Any free divisible MV-algebra is an algebra of continuous piecewise linear functions with rational coefficients. Correspondingly, Rational {\L}ukasiewicz logic is defined and its tautology problem is shown to be co-NP-complete.

Demostración leibniziana de las fórmulas numéricas

Por • 8 ago, 2012 • Category: Ambiente

El presente artículo examina puntos centrales de la teoría leibniziana de la prueba matemática en conexión con su concepción general de la ciencia. Se analizan, primeramente, las características generales del método leibniziano, oponiéndolo al método cartesiano (§§ 1–3). Puesto que para Leibniz las fórmulas numéricas no son verdades lógicas primitivas y por ello requiere una estricta prueba formal, a continuación se examina la demostración de ellas contenida en los Nuevos ensayos, mencionando las críticas que Frege y Poincaré le han dirigido, a fin de precisar y aclarar el significado del formalismo leibniziano (§§ 4–6). Se finaliza con un recuento evaluativo de lo realizado en este trabajo (§ 7).

Plato’s theory of knowledge of Forms by Division and Collection in the Sophistes is a philosophic analogue of periodic anthyphairesis (and modern continued fractions)

Por • 14 jul, 2012 • Category: Filosofía

The aim of this paper is to show that Plato’s theory of knowledge of Forms (intelligible Beings, Ideas) in the Sophistes, obtained by Division and Collection, is a close philosophic analogue of the geometric theory of periodic anthyphairesis, an ancient theory of incommensurability (developed by the Pythagoreans, Theodorus and Theaetetus) having its modern counterpart in the theory of continued fractions. Division corresponds to infinite anthyphairetic division, Collection to the Logos Criterion, resulting in periodicity and a self-similar One, precisely the One of a Platonic Form.

The Axiom of Multiple Choice and Models for Constructive Set Theory

Por • 4 may, 2012 • Category: Opinion

We propose an extension of Aczel’s constructive set theory CZF which is acceptable from a generalised-predicative point of view, because interpretable in Martin-Lof’s type theory; strong enough to prove the Set Compactness Theorem and capture a substantial part of formal topology; and preserved by the model constructions of exact completion, realizability and sheaves. More concretely, our proposal is to extend CZF with an axiom postulating the existence of W-types and a choice principle we call the Axiom of Multiple Choice (and which is a slightly simpler and weaker version of the axiom as originally introduced by Palmgren and the second author). We will also show that this extension has a certain robustness about it in that these additional axioms are also reflected by the model constructions we mentioned.

The notion of abstract Manifold: a pedagogical approach

Por • 20 abr, 2012 • Category: Ciencia y tecnología

A self-contained introduction is presented of the notion of the (abstract) differentiable manifold and its tangent vector fields. The way in which elementary topological ideas stimulated the passage from Euclidean (vector) spaces and linear maps to abstract spaces (manifolds) and diffeomorphisms is emphasized. Necessary topological ideas are introduced at the beginning in order to keep the text as self-contained as possible. Connectedness is presupposed in the definition of the manifold. Definitions and statements are laid rigorously, lots of examples and figures are scattered to develop the intuitive understanding and exercises of various degree of difficulty are given in order to stimulate the pedagogical character of the manuscript. The text can be used for self-study or as part of the lecture notes of an advanced undergraduate or beginning graduate course, for students of mathematics, physics or engineering.

Alternative Mathematics without Actual Infinity

Por • 13 abr, 2012 • Category: Opinion

An alternative mathematics based on qualitative plurality of finiteness is developed to make non-standard mathematics independent of infinite set theory. The vague concept “accessibility” is used coherently within finite set theory whose separation axiom is restricted to definite objective conditions. The weak equivalence relations are defined as binary relations with sorites phenomena. Continua are collection with weak equivalence relations called indistinguishability. The points of continua are the proper classes of mutually indistinguishable elements and have identities with sorites paradox. Four continua formed by huge binary words are examined as a new type of continua. Ascoli-Arzela type theorem is given as an example indicating the feasibility of treating function spaces.

Inequalities having Seven Means and Proportionality Relations

Por • 17 mar, 2012 • Category: Opinion

Eve (2003), studied seven means from geometrical point of view. These means are \textit{Harmonic, Geometric, Arithmetic, Heronian, Contra-harmonic, Root-mean square and Centroidal mean}. Some of these means are particular cases of Gini’s (1938) mean of order r and s. In this paper we have established some proportionality relations having these means. Some inequalities among some of differences arising due to seven means inequalities are also established.

The Emerging Art of Algorithmic Music

Por • 20 dic, 2011 • Category: Filosofía

The work of Ville-Matias Heikkila, a Finnish artist and computer programmer, might come as a shock. In the last year or so, he and others have been experimenting with the audio output of simple computer programs in an infinite loop. The output is a modulated stream of pulses that, when played through an audio speaker, sounds melodic. Today, he outlines this work and some of the techniques and tools that he uses to generate the code, listen to it and even visualise it. He’s posted some of these tunes along with their source code on Youtube. Heikkila says that these programs generate surprisingly interesting music, sometimes by repeating only two or three arithmetic operations. So he and others have been exploring the space of all possible simple algorithms, albeit in a rather disorganised way. Now Heikkila, who also goes by the online moniker viznut, is proposing a more methodical search of this space. He wants to set up a program that generates new formulas automatically and a website that allows people to rate the music it finds. In essence, he wants to crowdsource the task of music discovery. One half of this problem may have already been cracked for him. Ten years ago, Stephen Wolfram argued that the laws of physics are no more than a set of simple algorithms. In his book A New Kind of Science, he explores and characterises the entire space of simple algorithms for cellular automata and argues that the Universe is governed by rules like them. The difficult task is finding these rules.

Bayesian Causal Induction

Por • 16 nov, 2011 • Category: Educacion

Discovering causal relationships is a hard task, often hindered by the need for intervention, and often requiring large amounts of data to resolve statistical uncertainty. However, humans quickly arrive at useful causal relationships. One possible reason is that humans use strong prior knowledge; and rather than encoding hard causal relationships, they encode beliefs over causal structures, allowing for sound generalization from the observations they obtain from directly acting in the world. In this work we propose a Bayesian approach to causal induction which allows modeling beliefs over multiple causal hypotheses and predicting the behavior of the world under causal interventions. We then illustrate how this method extracts causal information from data containing interventions and observations.