HERMANN WEYL AND HIS PHENOMENOLOGICAL RESEARCHES WITHIN INFINITESIMAL GEOMETRY

The present work focuses on the mathematical and philosophical works of Hermann Weyl (1885-1955). Weyl was a leading mathematician at the beginning of the twentieth century and his major contributions have concerned several fields of research, both within pure mathematics and theoretical physics. Many of them were pioneering works at that time and, most of all, they were carried out in the light of his peculiar philosophical view. As few mathematicians of his time, Weyl was able to manage both scientific and philosophical issues with an impressive competence. For this reason he represented a very peculiar figure among scientists and mathematicians of his time. This dissertation aims to clarify these works both from a philosophical and a mathematical perspective. Specifically, I will focus on those works developed through the years 1917-1927. The first chapter aims to shed some light on the philosophical reasons that underlie Weyl's foundational studies during this period. I will explore these works especially with respect his attempt to establish a connection between a descriptive analysis of phenomena and their exact determination. I will focus both on his mathematical formulation of Euclidean space and on his analysis of phenomenal continuum pointing out the main features of these studies. Weyl's investigations on the relations between what is intuitively given and the mathematical concepts through which we seek to construct the given in geometry and physics do not seem to be carried out by chance. These investigations indeed could be better understood within the phenomenological framework of Husserl's philosophy. Husserl's distinction between descriptive and exact concepts delineates the difference between a descriptive analysis of a field of inquiry and its exact determination. Clarifying how they are related is not an easy task. Nevertheless, Husserl points out that a connection might be possible if we were able to establish a connection by means of some idealizing procedure intuitively ascertained. Within this phenomenological framework we should interpret Weyl's investigations on the relation between phenomenal knowledge and theoretical construction. In the second chapter I will focus on Weyl's mathematical account of the continuum within the framework of his pure infinitesimal geometry developed mainly in \emph{Raum-Zeit-Materie}. It deserves a special attention. Weyl indeed seems to make use of infinitesimal quantities and this fact appears to be rather odd at that time. The literature on this issue is rather poor. For this reason I've tried to clarify Weyl's use of infinitesimal quantities considering also Weyl's historical context. I will show that Weyl's approach has not to be understood in the light of modern differential geometry. It has instead to be understood as a sort of algebraic reasoning with infinitesimal quantities. This approach was not so unusual at that time. Many mathematicians, well-known to Weyl, were dealing with kind of mathematics although many of these studies were works in progress. In agreement with that, Weyl's analysis of the continuum has to be understood as a work in progress as well. In the following Weyl's studies in combinatorial topology are proposed. I will then suggest that both these approaches should be understood within the phenomenological framework outlined in the first chapter. The latter, however, attempts to establish a more faithful connection between a descriptive analysis of the continuum and its exact determination and for this reason it can be regarded as an improvement with respect to the former from a phenomenological point of view. Finally, in the third chapter I will attempt a phenomenological clarification of Weyl's view. In the first and second chapter Weyl's studies are clarified showing how they are related with the phenomenological framework of Husserl's philosophy. Despite this, the theoretical proposal revealed by them is not so easy to understand. That issue seems to be shared by many other contemporary studies. The relevant literature on this author dealing with a phenomenological interpretation seems often to be hardly understandable. I'm going to outline the main problems involved in this field of research and how they are related with the peculiarity of Husserl's framework. I will then suggest a way to improve these studies. Specifically, I will attempt a phenomenological clarification of Weyl's writings. To this aim, I will argue for an approach that makes use of Husserl's writings as a sort of ``analytic tools'' so that a sort of phenomenologically-informed reconstruction of Weyl's thought can be achieved. I will finally consider Weyl's notion of surface as a case study to show a concrete example of this kind of reconstruction.