This PhD thesis analyzes a change in the definition of mathematics in the 19th century: from science of quantities to science of forms. A comparison of the concepts of quantity, magnitude, number, form, and structure, which have not always been sufficiently distinguished in the literature on the 19th century, shows interesting connections between philosophy, geometry and algebra.
The first part is devoted to a historical and philosophical analysis of the notion of magnitude, especially to its geometrical origin in Greek mathematics, to its algebraic development in the 14th century \u2013 which changes the notion of magnitude (quantity) into the notion of a general magnitude (quantity) \u2013, and to the comparison with Descartes's and Leibniz's remarks on operations and relations. The analysis of some works of Wolff, d'Alembert, Euler and Gauss \u2013 texts that appear for the first time in an Italian translation in the Appendix \u2013 shows that magnitudes, when mathematics is defined as Gr\uf6ssenlehre, are extensive quantities defined by an equality relation and by an additive operation that characterizes the relation of the parts to the whole.
The second part of the research (chapters 4-6) is entirely devoted to an analysis of the epistemology of Hermann Grassmann, and in particular to the notion of an extensive magnitude that he develops in the Ausdehnungslehre (some passages of the 1844 and 1862 editions are also translated into Italian in the Appendix). The aim of the work is to show the intimate connection between the mathematical research and the philosophical conception of Grassmann, which is considered as an original contribution to the foundation of a theory of extensive magnitudes independently from numbers and from a theory of measurement
