O desenvolvimento das conceitualizações dos domínios de números como base para o novo rigor da matemática no século XIX

Abstract

The aim of this thesis is to analyze how different types of numbers were conceptualized throughout the 19th century in different mathematical communities in Europe. In the midst of the dominant epistemology of the time, the conception of the arithmetic of natural numbers was treated without controversy, while the foundation of other objects was more problematic, since they came into conflict with the basic conception of quantity. At first, we will address the methodology for extending the concept of number that was developed in the 19th century in line with epistemological changes that began to emerge in the discipline, opening space for new ways of understanding and substantiating arithmetic that go beyond the natural numbers. Notably, the analysis based on different mathematical communities reveals that, at first, the process takes place only in regions of Germany while, in France, until then the epicenter of mathematics, new ideas were barred for a long time, in view of the dominance of the physical-mathematical conception in the region. Furthermore, our analysis reveals in particular the different positions on the ontological role of "natural numbers" - showing how each mathematician's more individual convictions affected the way in which the emergence of number was justified. Separately, we analyze the development of new approaches to conceiving the irrational numbers that, until then, remained based on the traditional conception of continuous magnitudes. We were able to verify that the emergence of new conceptions that arise in the second half of the 19th century, often named "modern" by historiography to separate them from the previous ones, arise in in the midst of a broader movement towards the search for the foundation of analysis on a new concept of number. As we intend to demonstrate, this movement occurs, to a large extent, in the context of teaching analysis, especially in Germany. When analyzing different attempts, including by figures often ignored in historiography, we could see that many of these were not successful, from the point of view of their own authors, thus demonstrating the challenge of such an enterprise. Notably, in the midst of our research on the subject, we identified an unnoticed conception of “complex number”, developed by Weierstraß, which becomes key to understanding the peculiarities of his process of extending the concept of number that are obscured by traditional approaches to the history of mathematics. Finally, we analyze the new approaches to the fundamentals of the concept of natural number that, until now, consisted of the basis of arithmetic, without there being new elaborations on its foundation in view of its ontological “superiority”, even with the popularization of the extension process of the concept and the dissemination of the new epistemology. In this sense, the isolated approach of Hermann Graßmann in 1861 as well as the axiomatization made by Peano, at the end of the century, strongly influenced by Graßmann's approach, will be the focus of our analysis.