Logicism

The aim of this paper is twofold: first to explicate how Riemann's philosophy of geometry is organized around the concept of manifold. Second, to argue that Riemann's philosophy of geometry does not dismiss Kant's spatial intuition. To this end, first I analyse Riemann's Habilitationsvortrag with respect to interaction between philosophical, mathematical and physical perspectives. Then I will argue that although Riemann had no particular commitment to the truth of Euclidean geometry his alternative geometry does not necessarily dismiss Kant's spatial intuition.

Galileo found the idea of larger or smaller infinities impossible to comprehend, then unintentionally made them equal. Over 200 years later, Cantor insisted that his formulation of transfinite numbers was not arbitrary, but then unintentionally inherited Galileo's methodological error in the way one-to-one correspondence was used. Genuine one-to-one correspondence produces exact agreement between set size, cardinality, natural density, and probability measures. The error is subtle, and literally not visible, enabling it to evade detection for more than 380 years. It takes more than a few pages to unpick the detail, since every living mathematician has had to prove that some extremely low density (and some extremely high density) sets are countably infinite as a minor part of their undergraduate studies.

The article surveys the early work of Zhang Dongsun on topics like the logicism of Bertrand Russell and scientific philosophy, which aimed to criticize its foundations and replace them with a Neo-Kantian alternative. It tries to show how a series of Zhang's articles from the early 1920s, in which he sought to create a new "neutral" variety of logicism, can be used to better understand the intellectual foundations of the neovitalist "philosophy of life" of Zhang Junmai. By delving deeper into the underlying ideas and possible motivations behind Zhang's philosophical endeavours from the early 1920s, the article argues for a different kind of understanding of the historical basis of humanism in modern Chinese philosophy. Moreover, it strives to show how the "Science and the View on Life" controversy, as initiated by Zhang Junmai in 1923, might be rooted in or at least directly related to a syncretistic ideal, to conjoin science and the view of life in a new kind of harmonistic outlook. Most importantly, the article will try to show how Zhang Dongsun's critical engagement with Russell's philosophy, modern logic and physical science could be understood as the theoretical nucleus of the so-called "view on life" philosophy, not only in the context of the 1923 controversy, but possibly the entire Republican Period. Due to limited space, the article does not offer a concise introduction to Zhang's life and philosophy, but instead provides a focused discussion of particular fragments of his work from the early 1920s.

In §1 I discuss Dedekind and Frege on the logical and structural analysis of natural numbers and present my view that the logical analysis of the notion of number involves a combination of their analyses. In §2 I answer some of the specific questions that Abel raises in connection with Chapter 9 of Logical Forms.

Este volumen de Metatheoria incluye traducciones al castellano de los tres famosos trabajos sobre las escuelas de fundamentos de la matemática, el logicismo, el intuicionismo y el formalismo, expuestos en el Simposio de Königsberg sobre Fundamentos de la Matemática en septiembre de 1930, y que fueron finalmente publicados en la revista Erkenntnis en 1931. Los tres constituyeron un hito en la filosofía de la matemática del siglo pasado. En esta introducción a las traducciones, los compiladores del volumen esbozan el contexto histórico en el cual se concibieron los trabajos originales e incluyen detalles de su publicación original. También ponen de relieve el papel decisivo que tuvieron en la filosofía de la matemática del siglo pasado. Se hace un resumen de sus contenidos, y se discuten las ideas subyacentes del Círculo de Viena acerca de la naturaleza de la matemática y el impacto de los teoremas de Gödel en los programas de fundamentos. Además, se hace una breve introducción a las ...

In 1882, Richard Dedekind and HeinrichWeber offer an arithmetico-algebraic re-definition of the Riemann surface, using concepts and methods introduced by Dedekind in algebraic number theory. In an attempt to investigate Dedekind’s works beyond the mere idea of a “conceptual approach”, this works proposes to identify the elements of practice specific to Dedekind, starting from the paper co-written with Weber. I put forward the idea thatin Dedekind’s works, arithmetic can play an essential and active role in the elaboration of mathematical knowledge. For this, I propose to study, inDedekind’s mathematical practice, the conception of arithmetic, the place and role of arithmetical notions and the possible evolutions in Dedekind’sideas about arithmetic. This study is based on a careful analysis of a selection of Dedekind’s texts. For this, I study Dedekind’s early works, his 1854Habilitationsvortrag and his first works in number theory. Then, I propose a comparison between the first two ...

SUMMARY. — At the end of the 19th century, what is referred to as « modern » mathematics made its first appearance when Georg Cantor and Richard Dedekind introduced their theory of infinite sets. At the beginning of the 20th century, this mathematics became universalized, when the axiomatic method and the definitions of algebraic and topological structures, which were already household words at David Hilbert's school at Göttingen, became systematically used. In this article, I examine the epistemological options of mathematical modernity - especially those that were widely shared beyond the Göttingen circle. I attempt to locate modernity transversally, without concern for methodical, philosophical, ideological, or circumstantial borders. What I aim at are epistemological invariants of modern mathematical practices and argumentation. I arrive at the themes of the autonomy of mathematics — which is a particular aspect of the general theme of the autonomy of science -, the promotio...

The division of modern science into disciplines is a fundamental fact that, as far as we know, no one disputes. At most, individual cases have differences of opinion about the relative degree of independence of the field in question (e.g., about whether it is a particular intradisciplinary field, a unique discipline, or a discipline "proper"). It is no less evident that systematic processes of forming new disciplines characterize recent science, that the associated tendency to increase the number of disciplines continues, and that there is no end to this process, even in the foreseeable future (Stichweh, 1984).

The paper explores the alleged connection between indefinite extensibility and the classic paradoxes of Russell, Burali-Forti, and Cantor. It is argued that while indefinite extensibility is not per se a source of paradox, there is a degenerate subspecies—reflexive indefinite extensibility—which is. The result is a threefold distinction in the roles played by indefinite extensibility in generating paradoxes for the notions of ordinal number, cardinal number, and set respectively. Ordinal number, intuitively understood, is a reflexively indefinitely extensible concept. Cardinal number is not. And Set becomes so only in the setting of impredicative higher-order logic—so that Frege’s Basic Law V is guilty at worst of partnership in crime, rather than the sole offender.

The naturalization of phenomenology is a project of integrating philosophical analyses of conscious experiences into the explanatory framework of natural sciences. Prominent members of the phenomenological movement had divided attitudes on the possibility of implementing such a project. Martin Heidegger strictly separated »essential thinking«, which lets beings show themselves in their modes of being, from methodologically guided scientific research, while Edmund Husserl remained ambivalent on the question of naturalization by criticizing psychologism and later pointing to the foundation of sciences in the life-world, but also requiring rigorous scientificity, adherence to the methodological steps of reductions, and refrainment from metaphysical prejudices. Only in the works of Maurice Merleau-Ponty did the naturalization of phenomenology become accepted as a positive possibility which should improve both science and philosophy by connecting empirical research with transcendental analyses. Contemporary currents of formalized, experimental, and neurophenomenology, although different from each other, follow Merleau-Ponty in this sense, pointing to the indispensability of phenomenological descriptions when studying complex aspects of the conscious mind. We believe that the fruitful cooperation between phenomenological philosophy and integrative cognitive science is conditioned by the necessity of a multi-perspective and interdisciplinary approach in order to fully understand and explain our world-conscious life implemented in a dynamic system of interactive processes occurring in the brain, body and environment.

In Chapter III, Łukasiewicz turns his attention to the psychological formulation of the principle, which he discusses over the span of three chapters. He presents an analysis and elucidation of Aristotle’s account of the psychological version of the principle. Łukasiewicz argues that Aristotle considers the psychological PC as a consequence of the logical and ontological principles and that Aristotle presents an argument which aims to show that this is so.

O objetivo deste artigo é retomar e requalificar a distinção entre conceitos e objetos, tal como proposta por Frege, no célebre Sobre o Conceito e o Objeto (1892), reforçando sua importância para a compreensão da distinção entre dizer e mostrar, proposta pelo Wittgenstein do Tractatus. O texto também estabelece a influência da mencionada distinção para o tratamento pelo primeiro Wittgenstein dos paradoxos por impredicatividade. Retomamos então as análises de Charles Travis sobre o assunto, em: Where Words Fail (2020), levando-se em conta o ponto de vista da filosofia do segundo Wittgenstein. Para Travis, o obstáculo envolvido na distinção entre conceitos e objetos não passaria de uma confusão filosófica, devendo ser ela mesma dissolvida. Os argumentos propostos por Travis são também aqui brevemente passados em revista, mas nosso foco consiste principalmente em concluir que a importância da distinção deve ser rastreada nos limites do que pode ser descrito ou definido adicionalmente, ...

Curiosity, passion, erudition, high standards, perseverance, uprightness, humility, listening, bonhomie: this is what characterised Alain Gallay and made him so endearing. The breadth of his work makes him one of the major figures in anthropology in its broadest sense. To go further: https://journals.openedition.org/africanistes/12491

Curiosité, passion, érudition, exigence, persévérance, droiture, humilité, écoute, bonhomie : voila ce qui caractérisait Alain Gallay et le rendait si attachant. Son œuvre, de par son ampleur, en fait l’une des figures majeures de l’anthropologie dans son acception la plus large. Pour aller plus loin : https://journals.openedition.org/africanistes/12491

Résumé. — Le texte ci-après propose une interprétation de l'idée fameuse de Jean Cavaillès exhortant la philosophie de la science à devenir une « philosophie du concept ». Cette interprétation conjugue deux sources : 1) Les écrits de Cavaillès, y compris les fragments de lettres publiés par Gabrielle Ferrières dans son livre Jean Cavaillès, philosophe dans la guerre, et les lettres inédites à Albert Lautman, dont nous publions les plus significatives, selon nous, dans ce numéro de la Revue d'Histoire des Sciences. 2) Dans la mesure où nous avons pu le reconstituer, le contexte mathématique de ces écrits. Cette interprétation donne leur portée maximale à deux notions clés de l'épistémologie de Cavaillès : celles de « concept » et de « structure », en observant qu'elles furent également au centre du développement de la mathématique dite « moderne ». Ce parallélisme mis à nu induit un rapport, naturel mais peu remarqué, entre la mathématique des structures, florissante ...

Frege está considerado como uno de los puntos de partida de la lógica contemporánea (incluso de los orígenes de la filosofía analítica), al cambiar los fundamentos de esta ciencia, entre otras cosas al introducir en lógica el uso de funciones y argumentos (de variables), y ensayar con una simbología específica basada en la matemática, y en intentar reducir ésta (fundamentalmente la aritmética) a las reglas de la lógica, en un primer 'programa logicista' que luego seguiría Bertrand Russell, intentando solventar las paradojas (contradicciones) del sistema de Frege.