Theoria: An International Journal for Theory, History and Foundations of Science

In this paper we present an overview of Professor Newton C. A. da Costa’s work in logic, emphasizing the main results obtained by him in the several areas of his research activity. The text furnish a detailed bibliographic reference of his works, which are listed in the last section.

Castañeda’s most significant insights iie in his awareness of serious ontologicaI problems, which beset usual treatments. Among this outstandlng proposals are these about the structure of relational facts, guise theory and the bundle view of individuals, the I, and practitions. Castañeda’s metaphysics Is one of the most remarkable achievements in anaytical philosophy.

We state and comment our recent results on the incompleteness of elementary real analysis and their relevance for the axiomatized sciences.

En 1931 Gödel publica su Teorema de incompletud. Resultado clave en Lógica matemática, se interpretó como una limitación de los formalismos, un fracaso del Programa de Hilbert. Sin embargo, el método de aritmetización, la recursividad han propiciado una visión positiva y la creación de nuevas teorias - Teorías de la Complejidad, de Información algorítmica ... -. Con ellas, nuevas demostraciones del teorema y, consecuentes, nuevas discusiones en Filosofía de la Matemática. En especial, desde la Teoría de la Complejidad. Además de incompleta, la Matemática se ve por algunos como aleatoria y experimental. En este ensayo se pretende explicitar los supuestos de estas interpretaciones. Discusión que conduce, en principio, a su rechazo, dejando abierta la problemática sobre el aspecto epistemológico dei razonamiento matemático.

This paper deals with the ontological genesis of the series point-line-plane-solid in Plato’s philosophy. The texts of the Dialogues concerning this subject are presented, and passages of the Unwritten Doctrines that we know from Aristotle and other sources are specially considered. Certain problems within this context, such as the postulation of indivisible Iines, or the relation between each of the dimensions and the figures that can be placed in them, are considered in detail.

Dedekind used to refer to Riemann as his main model concerning mathematical methodology, particularly regarding the use of abstract notions as a basis for mathematical theories. So, in passages written in 1876 and 1895 he compared his approach to ideal theory with Riemann’s theory of complex functions. In this paper, I try to make sense of those declarations, showing the role of abstract notions in Riemann’s function theory, its influence on Dedekind, and the importance of the methodological principle of avoiding ‘forms of representation’ in shaping ideal theory. In order to emphasize the abstract viewpoint of Riemann and Dedekind, I compare their work with that of their great german contemporaries, Weierstrass and Kronecker; so, an influential ‘Göttingen group’ is confronted with the ‘Berlin school’ of mathematics. Some light is also thrown on the relation between Riemann and Dedekind, particularly with respect to Dedekind’s interest on Riemann’s ideas --in function theory, geometry and topology--, beginning in the 1850s and through later works. The abstract approach of the ‘Göttingen group’ was determinant for the turn to abstract mathematics in our century, and the direct influence of Riemann on Dedekind shows how this approach developed.

The “Centro de Estudios Cientificos” (CEC) of San Sebastian was established to palliate the absence of universitary studies in the Basque Country, in base of an idea suggested by J. Rey Pastor. It began itsactivities in 1932 and ended in 1936, because of Spanish Civil War. It was supported by the “Sociedad de Estudios Vascos” and sponsored by the “Diputación Foral de Gipuzkoa” and the San Sebastian Council. In this paper, we describe and comment the different aspects relatives to the CEC such as constitution, statutes, members, sections, courses, teachers, students, ... , and scientific journals supported by the CEC.

This paper analyses the ancient atomism from a modern point of view. It focuses its attention on the essential properties of clasical atoms. First, it considers the principles of the atomist ontology and the concept of void. Then, it studies the concept of atom and its fundamental properties: shape, size, weight, and motion. It discusses three significant problems of the clasical atomism: i) the theoretical character to the atoms and their resemblances and differences with respect to the macroscopical bodies; ii) the atomist explanation and its limits; and iii) the visual representation of the atoms. Finally, it outlines some reasons which indicate a radical change and the very dissolution of the classical concept of particle in the field of the contemporary physical science.

Roger Boscovich, belonging to XVIII century, halfway from Newton to Faraday, is traditionally considered as a newtonian philosopher. Nevertheless, following Berkson’s suggestion, he could be a Field Theory forerunner. In this work, we will try to go on with the idea of this suggestion in order to show this possible Boscovich’s contribution.