Artículos con la etiqueta ‘infinito matemático’

The Infinite as Method in Set Theory and Mathematics

Por • 23 ene, 2014 • Category: Opinion

El infinito como método en la teoría de conjuntos y la matemática. Este artículo da cuenta de la aparición histórica de lo infinito en la teoría de conjuntos, y de cómo lo tratamos dentro y fuera de las matemáticas. La primera sección analiza el surgimiento de lo infinito como una cuestión de método en la teoría de conjuntos. La segunda sección analiza el infinito dentro y fuera de las matemáticas, y cómo deben adoptarse.

Husserl, Cantor & Hilbert: La Grande Crise des Fondements Mathematiques du XIXeme Siecle

Por • 13 nov, 2013 • Category: Educacion

Three thinkers of the 19th century revolutionized the science of logic, mathematics, and philosophy. Edmund Husserl (1859-1938), mathematician and a disciple of Karl Weierstrass, made an immense contribution to the theory of human thought. The paper offers a complex analysis of Husserl’s mathematical writings covering calculus of variations, differential geometry, and theory of numbers which laid the ground for his later phenomenological breakthrough. Georg Cantor (1845-1818), the creator of set theory, was a mathematician who changed the mathematical thinking per se. By analyzing the philosophy of set theory this paper shows how was it possible (by introducing into mathematics what philosophers call ‘the subject’). Set theory happened to be the most radical answer to the crisis of foundations. David Hilbert (1862-1943), facing the same foundational crisis, came up with his axiomatic method, indeed a minimalist program whose roots can be traced back to Descartes and Cauchy. Bringing together these three key authors, the paper is the first attempt to analyze how the united efforts of philosophy and mathematics helped to dissolve the epistemological crisis of the 19th century.

Infinitesimals, Imaginaries, Ideals, and Fictions

Por • 12 abr, 2013 • Category: Filosofía

Leibniz entertained various conceptions of infinitesimals, considering them sometimes as ideal things and other times as fictions. But in both cases, he compares infinitesimals favorably to imaginary roots. We agree with the majority of commentators that Leibniz’s infinitesimals are fictions rather than ideal things. However, we dispute their opinion that Leibniz’s infinitesimals are best understood as logical fictions, eliminable by paraphrase. This so-called syncategorematic conception of infinitesimals is present in Leibniz’s texts, but there is an alternative, formalist account of infinitesimals there too. We argue that the formalist account makes better sense of the analogy with imaginary roots and fits better with Leibniz’s deepest philosophical convictions. The formalist conception supports the claim of Robinson and others that the philosophical foundations of nonstandard analysis and Leibniz’s calculus are cut from the same cloth.

Character and object

Por • 21 sep, 2012 • Category: Filosofía

In 1837, Dirichlet proved that there are infinitely many primes in any arithmetic progression in which the terms do not all share a common factor. We survey implicit and explicit uses of Dirichlet characters in presentations of Dirichlet’s proof in the nineteenth and early twentieth centuries, with an eye towards understanding some of the pragmatic pressures that shaped the evolution of modern mathematical method. We also discuss similar pressures evident in Frege’s treatment of functions, and the nature of mathematical objects.

The Shape of Infinity

Por • 21 sep, 2012 • Category: Opinion

In these expository notes, intended for students without background in point-set topology, we develop the basic theory of the Stone-Cech compactification without reference to open sets, closed sets, filters, or nets. In particular, this means we cannot use any of the usual definitions of topological space. This may seem like proposing to run a marathon while hopping on one foot, but it is easier than it may appear, and not devoid of interest. We use gauge spaces (uniform spaces presented by a family of pseudometrics); we define compactness as total boundedness plus completeness; and we define completeness using a variation on Lawvere’s categorical characterization of completeness for metric spaces.

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.