MULTITUDES, COLECCIONES E INFINITO: LA EMERGENCIA DEL ENFOQUE CONJUNTISTA EN LA OBRA DE BERNHARD BOLZANO

El artículo tiene por objetivo analizar ciertos pasajes fundamentales de la Wissenschaftslehre y de las Paradoxien des Unendlichen de Bernard Bolzano en cuanto al análisis conjuntista se refiere. En dichos pasajes, Bolzano desarrolla conceptos fundamentales tales como multitud, colección e infinito que anticipan el carácter conjuntista y del análisis matemático moderno. Asimismo, se presentará un breve estudio de las Contribuciones a una más fundada exposición de la matemática y el apéndice, Sobre la teoría kantiana de la construcción de conceptos a través de intuiciones, textos donde Bolzano muestra su rechazo por Kant.The article addresses the treatment of certain fundamental passages from Bolzano’s Wissenschaftslehre and Paradoxien des Unendlichen. In these passages, Bolzano describes some of his key concepts such “Multitude”, “Collections” and “Infinite” in the context of mathematical analysis and Set theory. Therefore, I discuss at length the (almost) unknown Contributions to a Better-Grounded Presentation of Mathematics and the appendix On the Kantian Theory of the Construction of Concepts through Intuitions, texts where Bolzano shows his rejection of Kant.

The problem of intentionality in the school of Brentano

From the moment Franz Brentano formulated his definition of intentionality, it imme­diately began to undergo modifications in the works of his students. Brentano’s original definition included reference to the scholastic tradition, but it differs from the one that was formulated by the scholastics. In his work “Psychology from an Empirical Point of View”, Brentano defines intentionality both as an orientation towards an object and as a relation to some content, but at no later time, neither in this work, nor in other published works, does he clarify the meaning of the concept of «content». In this regard, the stu­dents and interpreters of Brentano’s works had a question: does the scheme of inten­tionality consist exclusively of an intentional act and an object, or does it also include the content of a representation? Brentano’s disciples did not view this definition as clear and unambiguous. In order to clarify this concept, they often studied other similar philo­sophical conceptions in search of a more precise definition. In particular, they looked for a similar concept in the theory of Bernard Bolzano. The first version of the schema of in­tentionality, including the content of representation, appeared in the works of Hoeffler and Twardowski. For this reason, for a long time they were considered by historians to be the discoverers of the distinction between object and content. However, after the notes of Brentano’s lectures, which he also read to his students, were recovered, it became clear that Brentano himself made this distinction. In this regard, it seems extremely important to interpret the history of the relationships in the Brentano school through the prism of the discussions devoted to the definition of intentionality and the structure of an inten­tional act, as well as to understand the origins of each individual interpretation of this concept proposed by Brentano’s students

Bolzano’s Badiou

This article raises a series of points of confluence between Badiou’s philosophy and that of Bernard Bolzano, whom Badiou has identified as a historical predecessor but never directly engaged. These points include their respective critiques of Kant and Hegel, as well as their various concepts of sets, platonist realism, axiomatisation, the infinite, adequate demonstration, structure, and mathematics as the adequate language of being.

Bolzano on Continuity

Bernard Bolzano (1781-1848) was a philosophical mathematician, especially interested in foundations and the analysis of important mathematical concepts. The notion of continuity was a subject of sustained reflection throughout his life. He deals with the notion in many settings: the theory of space (geometry), the theory of time (chronometry), the theory of functions (analysis), physics (continuous processes, matter), and numerical continuity (the theory of measurable numbers). One can also distinguish earlier and later versions of most of his work in these areas. This chapter provides an overview of his thoughts on these matters, along with some indications of the historical and philosophical context of his work.