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The paper is devoted to a comparison of the artistic and life paths of the famous British logician and mathematician Mary Everest Boole in terms of the development of her pedagogical, scientific, psychological, ethical and even mystical ideas related to the interpretation and understanding of George Boole’s algebra as the basis of knowledge in general., Статья посвящена сопоставлению твореческого и жизненного путей известной британской логика и математика Мэри Эверест Буль в свете развития ее педагогических, научных, психологических, этических и мистических идей, связанных с интерпретацией и осмыслением алгебры Джорджа Буля как основы знания вообще.

In the present paper, the small article by George Boole, published the next year after the publication of his famous work The Mathematical Analysis of Logic, is analyzed. We discuss to what extent it can be considered as an adequate and representative overview of his logical views. The mathematical, extralogical and philosophical foundations of Boole’s logic are explored. It is pointed out that the mathematical basis of his logical system was his research in the field of linear differential equations. Moreover, his pioneering discoveries were based on arbitrary philosophical and methodological assumptions that were not directly related to the field of logic and mathematics. Some of the original psychological theory of mind, its philosophy of mathematics, which presupposes the existence of a universal mathematical calculus, and some provisions of Immanuel Kant’s logico-philosophical doctrine should be recognized as such assumptions., Проводится разбор и анализ небольшой статьи Джорджа Буля, опубликованной на следующий год после выхода его знаменитой работы «Математический анализ логики». Исследуется вопрос о том, насколько ее можно рассматривать как адекватный и репрезентативный обзор его логической концепции. Также исследуются математические, экстралогические и философские основания логики Буля. Указывается, что математическим основанием его логической системы послужили его исследования в области линейных дифференциальных уравнений. Кроме того, в основании его пионерских открытий лежали произвольные философско-методологическим допущения, которые напрямую не относились к области логики и математики. Такими допущениями следует признать некоторую оригинальную психологическую теорию сознания, его философию математики, предполагающую существование универсального математического исчисления, и некоторые положения логико-философской концепции И. Канта.

George Boole is one of the first logicians who offered a systematic formalization of language. He developed a notation to encode ordinary language sentences to algebraic symbols and developed algebraic methods to manipulate those symbols and to deduce results that are interpretable in the ordinary language. His methods formed the basis of modern logic. Boole applied his formal methods to many of the contemporary questions of his time, one of them being scholastic logic. In this paper, I explain how Boole deals with Aristotelian logic. I will start with his notation and algebraic methods, then apply them to Aristotelian conversions and syllogisms. It must be noted Boole has two versions of notation and methods, one is developed in Mathematical Analysis of Logic and the other is in his seminal book The Laws of Thought. I focus on the later version. George Boole dili sistematik bir şekilde yapısallaştıran ilk mantıkçılardan biridir. Dildeki sıradan cümleleri cebirsel sembollerle ifade edebilmek için bir notasyon ve bu sembolleri sistematik bir şekilde değiştirerek gündelik dilde de yorumlananabilen sonuçlar bulmak için cebirsel yöntemler geliştirdi. Geliştiridiği bu yöntemleri modern mantığın temelini oluşturmuştur. Geliştirdiği metotları skolastik mantık da dahil zamanındaki bir çok probleme uygulamıştır. Bu makalede Boole’un metotlarının Aristoteles'in mantığına nasıl uygulandığını inceliyorum. Öncelikle notasyonuna ve cebirsel metotlarına değinip, daha sonra bunları Aristoteles’in evirmelerine ve kıyaslarına uygulanışını gösteriyorum. Boole’un notasyon ve metotlarının iki versiyonu bulunmaktadır, bir tanesi Mantığın Matematiksel Analizi’nde diğeri ise Düşüncenin Yasaları’nda geliştirilen. Bu makalede Düşüncenin Yasaları’nda geliştirdiği versiyon esas alınmıştır.

In this article, we are mainly presenting a tribute to the fathers of computer and information sciences, George Boole and Claude Elwood Shannon with their hardships and achievements. This piece of writing also elaborates the applications of George Boole's and Claude Shannon’s works in different disciplines.

El meu treball tracta sobre les contribucions de George Boole (1815-1864) a la branca de les equacions diferencials i equacions en diferències finites, posant especial èmfasi en l'ús de mètodes simbòlics per a la resolució d'aquestes. Els mètodes simbòlics poden considerar-se un precedent de la lògica de Boole, el que els hi dóna importància, donat que Boole és conegut principalment pels seus treballs sonre lògica. AIxí doncs, resulta interessant conéixer l'origen dels mètodes simbòlics, relacionar i comparar els seus antecedents. Això es farà comparant l'obra de Boole amb la d'altres autors contemporanis com Duncan Gregory (1813-1844) i John Hymers (1803-1887). EL treball cpnsta de dues parts, una centrada en equacions diferencials ordinàries i l'altre en equacions en diferències finites, i cadascuna està centrada en un tractat de Boole: un sobre equacions diferencials i l'altre sobre diferències finites. Tot i així, en tot moment es va comparant el treball de Boole amb el d'altres autors.

National audience; La logique se rapproche fortement des mathématiques au cours du XIXe siècle. Le logicien anglais George Boole (1815-1864) joue un rôle important dans ce développement. En s’appuyant sur certaines « analogies » entre les lois logiques et les opérations algébriques, il construit une algèbre nouvelle et particulière de la logique. Boole ne se contente pas de résoudre les problèmes anciens auxquels s’attaquait la syllogistique aristotélicienne, il élargit la portée de la logique en redéfinissant les problèmes qui sont l’objet même de la logique formelle. Jusque-là, le syllogisme était la forme classique d’un argument logique. Objet de nombreux travaux en deux millénaires, la théorie du syllogisme reste la doctrine dominante en logique jusqu’au XIXe siècle. Bien sûr, la syllogistique discute aussi des arguments plus complexes (sorites). Mais ils sont ramenés à une série de syllogismes. Boole, en revanche, traite des problèmes bien plus complexes, en éliminant à partir d’un nombre quelconque de propositions les termes que l’on ne souhaite pas retrouver dans la conclusion, ce qui permet de déterminer la relation logique entre les termes retenus. Dans cette présentation, nous montrerons comment ce problème de l’élimination rompt avec la tradition syllogistique, comment il a été adopté par les successeurs de Boole comme le problème fondamental de la logique et comment ils se sont employés à le résoudre. S’il n’a pas de disciple à proprement parler, on pourra néanmoins affirmer que Boole a inauguré un programme de recherche qui occupera les logiciens du XIXe siècle.

Historians have repeatedly asserted that invariant theory was born in two papers of George Boole (1841 and 1842). Although several themes and techniques of 19th-century invariant theory are enunciated in this work, in reacting to it (and thereby founding the British school of invariant theory), Arthur Cayley shifted Boole's research program.

This catalogue text discusses artists and artworks featured in the exhibition BOOLEAN EXPRESSIONS: Contemporary Art and Mathematical Data, presented at The Lewis Glucksman Gallery, University College Cork, Ireland, 23 July – 8 November 2015. For an overview of the exhibition see the link https://vimeo.com/137620854

Este libro pretende comunicar el espíritu de la matemática contemporánea a estudiantes y otras personas interesadas que no se han especializado en esta ciencia. Dentro del libro se abarca el desarrollo de la matemática y la lógica, desde los pitagóricos hasta la escuela estructuralista, con especial interés en aquellos temas que tienen particular inherencia en la epistemología. Es claro que los temas aquí tratados sobrepasan lo que se puede enseñar en un semestre universitario. Para ello, se ha dividido la obra en tres tomos de tal modo que uno cualquiera de ellos podría servir de texto, permitiendo que se traten los respectivos temas con algún detenimiento. Para el lector casual, se ha hecho una división de inspiración cronológica que refleja tres momentos de la epistemología del siglo XX, a saber, el lanzamiento y eventual fracaso de proyectos de fundamentación, la reelaboración de la matemática, ciencias naturales y humanas, y las artes desde una perspectiva estructuralista, y finalmente la inherencia del computador en un cambio de la cara de las matemáticas y el álgido tema de la inteligencia artificial, nuevos retos para la epistemología de fines del siglo XX y comienzos del siglo XXI.

In his pioneering papers of 1845 and 1846, Arthur Cayley (1821–1895) introduced several approaches to invariant theory, the most prominent being the method of hyperdeterminant derivation. This article discusses these papers in the light of Cayley's unpublished correspondence with George Boole, who exercised considerable influence on Cayley at this formative stage of invariant theory. In the 1850s Cayley put forward a new synthesis for invariant theory framed in terms of partial differential equations. In this period he published his memoirs on quantics, the first seven of which appeared in quick succession. This article examines the background of these memoirs and makes use of unpublished correspondence with Cayley's lifelong friend, J. J. Sylvester.

0º Las ecuaciones en diferencias finitas coristltuyen un material relativamente poco conocido, a pesar de que desde el siglo pasado George Boole escribir una obra ya clásica sobre ellas, especialmente sobre las del tipo ordinario, con una sola variable discreta independiente. Sin embargo, el caso de dichas ecuaciones ordinarias lineales, es bien conocido de los tratadistas [1,2,3,5,6] y sobre él se conocen los teoremas de existencia y unicidad de soluciones, de superposición de soluciones particulares y otros en los cuales se apoya el estudio de esas ecuaciones. Hay métodos conocidos y sencillos para resolver esas ecuaciones, sean homogéneas, cuando los coeficientes son constantes y en algunas formas especiales de coeficientes variables.